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THE UNIVERSITY OF ALBERTA
RELEASE FORM
NAME OF AUTHOR: David William Scott
TITLE OF THESIS: Irrigation and Drainage as Influenced by
Weather: A Simulated Model*
DEGREE FOR WHICH THESIS WAS PRESENTED. Master of Science
YEAR THIS DEGREE GRANTED. 1975 (SPRING)
Permission is hereby granted to THE UNIVERSITY OF
ALBERTA LIBRARY to reproduce single copies for
private* scholarly or scientific research purposes
only*
The author reserves other publication rights* and
neither the thesis nor extensive extracts from it may
be printed or otherwise reproduced without the
author's written permission.
THE UNIVERSITY OF ALBERTA
IRRIGATION AND DRAINAGE
AS
INFLUENCED BY WEATHER:
A SIMULATED MODEL
by
DAVID WILLIAM SCOTT
A THESIS
SUBMITTED TO THE FACULTY OF GRADUATE STUDIES AND RESEARCH
IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE
OF MASTER OF SCIENCE
DEPARTMENT OF AGRICULTURAL ENGINEERING
EDMONTON, ALBERTA
SPRING, 1975
4
THE UNIVERSITY OF ALBERTA
FACULTY OF GRADUATE STUDIES AND RESEARCH
The undersigned certify that they have ready
and recommend to the Faculty of Graduate Studies
and Researchy for acceptance y a thesis entitled
"Irrigation and Drainage as Influenced by leather:
A Simulated Model, " submitted by David William
Scott in partial fulfilment of the requirements
for the degree of Master of Science*
Abstract
Due to the unpredictable nature of wea the r 7 crop
growth, crop water requirements and drainage are variables
of nature over which man has no control. It is therefore
desirable to know how these variables react to different
weather patterns over a period of time sufficient to include
most different combinations of weather. Average trends in
irrigation and drainage can then be studied.
The primary objective of this investigation was to
develop an accurate model of seasonal crop growth for the
Lethbridge area by including weather and crop growing
conditions. A digital computer was used to generate weather
via the Monte Carlo sampling technique and to simulate crop
growth and soil moisture during the growing season. The
distribution of drainage and irrigation was then evaluated.
The average rate of drainage occurrence per day and the
average yield per drainage period were the parameters upon
which this study was based.
The results indicated that the rate of increase in
daily consumptive use greatly affected the occurrence of
drainage while the daily rate of consumptive use did not
show any significant effect upon drainage occurrence.
Furthermore, the amount of drainage occurring on a
particular day is determined mostly by the consumptive use
rate. High water use results in low drainage while low
water use produces high drainage rates. A set of
probability tables is presented as a guide to the probable
iv
.
dates of irrigation.
.
•
ACKNOWLEDGMENTS
The author wishes to express his appreciation to all
those persons involved in the preparation of this thesis>
Special thanks go to Professor E* Rapp who gave his
advice and encouragment throughout this project*
Acknowledgment is made to Messer's E* H* Hobbs and
K* K* Krogman of the Research Station* Canada Department
of Agriculture, Lethbridge, for their assistance in
supplying the raw data for this thesis* Thanks also go to
R « T* Hardin for his advice concerning the statistical
evaluation of the data*
Finally, acknowledgment is due to the Department of
Energy, Mines and Resources for their financial support of
this project*
vi
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■ ' .
TABLE OF CONTENTS
CHAPTER Page
1 • Introduction • ••••««»*. •«*«****.*..**.******0**. 1
2* Review of Literature •••••••••••••••••••••••••*• 4
2*1 The Moisture Budget . . . 4
2*2 Evapot ranspira t ion «• •»••••••••«•••••••*•• * 5
2*3 Description of the Area ••••••••••••••••••• 12
2*3*1 Location *••••• ••••••••••••••••••••• 12
2*3*2 Cii mat e ••••••••••«••••••••••••••••• 12
2*3*3 Soils Description ••••••»*«•••••«••• 13
2*3*4 Drainage Studies *••»••••••••••••••• 15
3* The Consumptive Use Model 18
3*1 The New Versatile Soil Moisture Budget •••« 18
3*2 Potential Evapot ranspi rati on •••«•••••••••• 23
3*3 The Soil Moisture Zones •* •*••••••*••••••• • 24
3*4 Runoff •••••••«• e • • 25
4® Selection of the Proper K— C oef f i cl en ts ••••••••• 27
4*1 Experimental Soil Moisture Data ••••••••••• 27
4*2 Weather Data •••••••••••••*•••••••••••••••• 28
4.3 The Z - Table . . ••••••• 29
4*4 Method <»«•«•«••<•••••••••••••••••••••••••••• 29
5* The Weather Model 32
5*1 Monte Carlo Sampling •••••••••••••••••••••• 32
5*2 Weather Distributions ••••••••••••••••••••• 32
5*3 Wet and Dry Day Probabilities ••••••••••••• 34
vii
*
•n ■
.
TABLE OF CONTENTS
( cont inued )
CHAPTER P age
5*4 The Rainfall Model ••••••«•••»••••••••••••« 38
5 © 5 The Potential Evapo transpirat ion Model •••• 41
5*6 The Overwinter Precipitation Model •••••••• 50
6 • Programming • •••••»<••••••••••••••••••*•••«•«•••• 53
6*1 Random Number Generator . . 54
6*2 Monte Carlo Sampling « * •*••••••• • 55
6*2*1 Precipitation « ••••<*••«.»••••«•••••* • 55
6*2*2 Potential Evapo t ranspirat ion ••••••* 56
6*3 Decision to Irrigate •••••••••••••••••••••• 57
7* Results and Conclusions •••* «••••*•«••••••■•••• • 59
7»1 Actual vs Simulated Data •••«••*••••••••••• 59
7*2 Intermittent Processes ••••••*•••••*•««•••• 69
7*2*1 Drainage: Parameters •••••«•*••»• 71
7*2*2 Drainage: A2 Parameters •*«•*«•••••• 85
7*2*3 Irrigation Parameters ••••••«••••••• 87
7*2*4 Drainage on Nonirrigated Soil •••••• 87
7*3 Irrigation Lapse Dates • •••••«•*•••••••••• • 88
7*4 Summary of results «•«••••••••••••••••*•••• 99
8* Conclusions «•»••••••*••••••••« »*••••••••* *•••••• 102
9* Recooioiendat ions •••••••••••••*•••••*••••«•«••••• 106
10. References ••«••••■•••«••••••••••••••••••••••••• 107
Appendix A •••••• a>« ••«••••»••«••••••••• «•«•«••• • 113
viii
.
.
LIST OF TABLES
Table Description Page
1. A DESCRIPTION OF SORE SOUTHERN ALBERTA
SOILS. 16
2. Z- VECTORS OF SOIL DRYNESS CURVE A AND H. 22
3. CHI-SQUARED TEST - PRECIPITATION AND
POTENTIAL E VAPOTRANSPI RATION • 43
4. A LIST OF THE a AND /? PARAMETERS OF THE
INCOMPLETE GAMMA DISTRIBUTION FOR
PRECIPITATION. 44
5. BIMONTHLY PROBABILITIES OF POTENTIAL
EVAPOTRANSPIRATION ON WET AND DRY DAYS. 46
6. SUMMARY OF THE SM I2NOV- KOLMOGOROV STATISTIC
FOR DAILY PE VALUES OCCURRING ON DRY DAYS. 46
7. A LIST OF THE MEANS AND STANDARD DEVIATIONS
- POTENTIAL EVAPOTRANSPIRATION. 48
8. SUMMARY OF THE MINIMUM IRRIGATION LEVELS FOR
FOUR DIFFERENT CROPS. 58
9. SUMMARY OF SIMULATED AND ACTUAL WEATHER DATA
- 45 YEARS. 60
10. K - COEFFICIENTS FOR THE VARIOUS CROPS. 67
11. DESCRIPTION OF THE IRRIGATION PRO BAB I LITIY
CURVES. 93
12. SUMMARY OF THE SMI RNOV— KOLMOGOROV STATISTIC
FOR IRRIGATION DISTRIBUTIONS. 94
13. IRRIGATION DATES WITH PROBABILITY EQUAL OR
LESS THAN - WHEAT. 96
14. IRRIGATION DATES WITH PROBABILITY EQUAL OR
LESS THAN - POTATOES. 96
15. IRRIGATION DATES WITH PROBABILITY EQUAL OR
LESS THAN - SUGAR BEETS 97
16. IRRIGATION DATES WITH PROBABILITY EQUAL OR
LESS THAN - ALFALFA. 97
ix
.
LIST OF FIGURES
FIGURE
Description
Page
1.
Average total monthly precipitation for
Lethbridge.
14
2.
Various proposals for the relationship
between the AE:PE ratio and the current
available soil moisture.
21
3.
A sample output of the Versatile Budget
simulation for Sugar Beets during I960.
31
4.
Comparison of actual and predicted values of
daily rainfall probabilities for days
following a dry day and days following a wet
day •
37
5.
Comparison of actual and theoretical
cumulative distribution of precipitation
following a non— rainy day - May 15—30.
42
6 •
Comparison of actual and theoretical
cumulative distribution of daily PE
occurring on a non— rainy day: July 16—31.
47
7.
Relative frequencies of dry day runs for
actual and simulated data: April 1 to Oct
31 .
62
8 a •
Actual and simulated Aj_ values:— 45 years.
64
8 b*
Actual and simulated I/A2 values:— 45 years.
64
9.
Comparison of actual and simulated daily
consumptive use averages for Wheat.
75
10.
Comparison of actual and simulated daily
consumptive use averages for Potatoes.
76
11.
Comparison of actual and simulated daily
consumptive use averages for Sugar Beets.
77
12.
Comparison of actual and simulated daily
consumptive use averages for Alfalfa.
78
13a.
curves for Wheat.
79
13b.
Aj curves for Alfalfa.
79
x
•
LIST OF FIGURES
( continued )
FIGURE
PAGE
13c*
A* curves for Potatoes.
80
13d.
Ax curves for Sugar Beets
80
14a.
Standard deviation of the A* curves for
Wheat and Alfalfa.
81
14b.
Standard deviation of the Ai curves
Potatoes and Sugar Beets.
for
81
15a.
1/ A2 curves for Wheat.
82
15b.
1/ A2 curves for Alfalfa.
82
15c.
1/ A2 curves for Potatoes.
83
15d.
1/ A2 curves for Sugar Beets.
83
16a.
Standard deviation of the 1/ Ag curves for
Wheat and Alfalfa.
84
16b »
Standard deviation of the l/Ag curves
Potatoes and Sugar Beets.
for
84
17.
Cumulative distribution of irrigation
dates for Wheat.
lapse
89
18.
Cumulative distribution of irrigation
dates for Potatoes.
lapse
90
19.
Cumulative distribution of irrigation
dates for Sugar Beets
lapse
91
20.
Cumulative distribution of irrigation
dates for Alfalfa.
lapse
92
xi
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A. a. Introduc tlon
Irrigation has been practised primarily in arid and
semi— arid regions of the world where natural rainfall is
insufficient for good crop growth* In semi— arid regions^
such as southern Alberta, irrigation water has been used
mainly as a supplement to natural rainfall* Rainfall in
this region is sufficient to support crop growth throughout
the growing season* However, the summer months in which
crop consumptive use is maximum are relatively dry* The
main purpose, therefore, of irrigation is to provide a means
of controlling the moisture level of the soil in order that
optimum conditions for crop production are maintained* Both
the quality and the quantity of the crop will increase,
thereby decreasing the risk of crop damage or loss*
Drainage problems are sometimes a result of improper
irrigation practices® Water is often applied at the
irrigators convenience or according to a fixed schedule
which has little concern for the needs of the crop or the
interrelationship between the soil and the crop* Soils,
which have an impermeable layer close to the surface, often
experience a rise in the water table following an excessive
irrigation* Small temporary sloughs, either in the
Irrigated field itself or in neighbouring fields, and salt
accumulation on the surface are the end results*
Drainage problems, however, are not exclusively
attributable to improper irrigation practices- Often, as is
the case in southern Alberta, an irrigation during the early
1
.
.
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2
growth stages of the crop is followed by an untimely
rainfall and then by a prolonged period of drought* Excess
soil water during the early growth stages will damage the
crop making it more susceptible to drought later on* Proper
irrigation scheduling is therefore essential*
The two major factors* therefore* which limit crop
production in southern Alberta* are: 1) the lack of
sufficient rainfall during the months of peak consumptive
use and* 2) an excess of irrigation water during the early
crop growth stages when rainfall is maximum*
The purpose of this research is to evaluate which has
the greater influence on irrigation and drainage; crop
consumptive use or weather* Information regarding the
occurrence and the amount of irrigation was available from
the Irrigation Guide records* However* information
regarding drainage and flooding were non-existent* Hence*
it was decided to construct a model which would simulate the
weather distribution and daily soil moisture content from
April 1st to October 31st for a period of 200 years*
Lethbridge was chosen as the area for this study because it
represents the area of highest concentration of irrigation
in southern Alberta and because daily weather data were
readily available*
The objectives of this research are fourfold*
1* To obtain probability distributions of rainfall
and potential evapotranspi ra tion and to derive the
conditional probabilities for rainy and non-rainy days for
~
3
the Lethbridge area® Weather records dating from 1922 to
1966 are available for use®
2® To simulate the soi 1— crop— water system
throughout the entire growing season with the weather
probabilities as the inputs to the model® Four major
irrigated crops are used: Soft Wheat , Potatoes y Sugar Beets
and Alfalfa®
3® To obtain from the simulation model probability
distribution curves of irrigation lapse times for each
irrigation and each crop®
4® To Qualitatively analyse both irrigation and
drainage as intermittent stochastic processes in terms of
the average number of occurrences per day and the average
yield per occurrence®
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2-s. Sevle_w_of Li t era lure »
Many attempts to simulate the soil-plant-water system
have been made in order to aid in the farm decision process*
Some researchers ( 10*35*48*49*50) have developed models
which aid in the selection of machinery for harvesting
operations or for scheduling farm operations based on
weather probabilities* Other models have been developed to
aid
in
the
decision
of irrigation scheduling
( 1 * 9 * 1 4 * 20 * 30 ; 3S * 40 * 4 1 * 44 * 47? 59* 6 0 )* and to simulate the
plant response to environmental conditions (11*13 )• Still
other models have been built to simulate the movement of
water through the soil (6*34)* or the response of a
watershed to precipitation (45)*
2*1 The Moisture Budget.
The relationship between the essential components of
the plant-soil— water system can best be expressed
mathematically by the following differential equation*
^ = I - 0 = (Rn + IRR) - (CU + Dr + Ro)
dt
where: Q ~ amount of stored water in the soil at time t
I ~ inflow into the soil medium
O - outflow from the soil medium
Rn = precipitation
IRR - irrigation water
CU — crop consumptive use
Dr = drainage from the root zone
Ro — surface runoff
t = time
The above soil moisture budget represents a simple
accounting procedure which continually updates the soil
mo
isture content in discrete intervals of time ( dt might
4
-
V
.
5
represent minutes, hours, days, e tc • ) • The method can be
applied to the entire root zone or to distinct soil zones
within the root zone. Robertson et al (46) applied this
budgeting technique to predict the timing of irrigation on
two plots of land. A black Bellani plate was used to
determine the daily potential evapot r anspi ra t io n rates. The
amount of irrigation water required by the budgeting
technique and that specified by the electrical resistance
block was within one inch. The soil moisture budgeting
technique has since been used in the majority of
mathematical soil moisture models.
Various methods have been developed throughout the
years to estimate, either theoretically or empirically, each
of the individual terms of the moisture budget. Early
researchers realized that one of the most important and most
difficult variables to estimate was that of potential
evapotraaspiratlon. They realized that the evaporation of
wafer from both the soil and the plant required energy and
that this energy was a function of the immediate climatic
parameters such as temperature and radiation. The methods
of estimating e vapo t ransp ir at ion are classified as 1) mass
transfer methods, 2) energy balance methods, 3) combination
methods, and 4) empirical methods based on meteorological
data. The first three methods involve a complicated
theoretical approach to the energy balance between the heat
transfer to and from the plant and its environment. Many of
'
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■ . ’-1 .
6
the variables are extremely difficult to measure; however,
the results are fairly accurate* The last method estimates
evapo transpiration from readily available climatic data via
empirically or experimentally derived mathematical
expressions* Meteorological data such as radiation,
temperature, humidity and wind speed are usually available
for most areas and are the main parameters upon which the
expressions are based* However, satisfactory results under
all conditions necessarily may not be achieved* A few of
the empirical methods are described in the following text*
In 1950, Blaney and Griddle, as cited by Gray ( 19),
developed a simplified formula for estimating consumptive
use in the arid western regions of the United States* It
relates mean monthly temperature (T), monthly percent of
annual daytime hours Ip) and a monthly crop coefficient (k)
to consumptive use (CU)* Stated mathematically:
CU
kTmp
100
kf
This method gives reliable monthly and seasonal estimates*
Penman, as cited by Hardee (20), combined the energy
balance equation and experimentally derived aerodynamic
equations to obtain an expression which included such
weather variables as short wave and long wave radiation, wet
and dry bulb psychrometric constants, mean wind speed, and
saturation vapor pressure at both the mean air temperature
and at the dew point temperature* Jensen et al (30)
proposed a formula for estimating potential
-
■
7
cvapotranspiration by an approximate energy balance-
aerodynamic equation which employed mean daily temperature
and solar radiation* Actual cvapotranspiration was obtained
by multiplying potential evapo transpiration with a crop
coefficient which reflected the effects of sensible and
latent heat flux and net radiation* Linacre ( 36 )« in 1967,
related cvapotranspiration to radiation and temperature*
Such variables as latent heat of vaporization^ short and
long wave radiation, water vapor pressure, specific heat of
air, net flux of heat into the atmosphere, air density,
saturation deficit and two crop resistant parameters were
employed© The net flow of heat took into consideration the
percentage of bright sunshine, and the temperatures for both
cloudy and non-cloudy days* An attempt was made by Linacre
to incorporate two crop resistant parameters which measured
the ability of the plant to release water into the
atmosphere* These parameters had to be experimentally
determined and were unique to a specific crop and location.
Christiansen and Hargreaves, as sited by Hardee (20),
produced a formula which involves several dimensionless
coefficients, each of which expresses the effect of mean
temperature, mean wind velocity, mean relative humidity, and
elevation, respectively* Radiation and a crop coefficient
were also included* The result, when all the coefficients
we re
multiplied
toget her ,
yie Ided
potentia 1
evapotranspiratlon. If a weather variable was not
available, the respective coefficient could be set to unity*
'
s
■
u
8
Eaglemanj in 197 1 , (16) developed a third degree
regression model which related the soil moisture ratio to
the ratio of actual to potential evapot ransp i rati on • The
soil moisture ratio was defined as the ratio of the current
soil moisture content to the total water capacity of the
soil* Eagleman found the relationship to be curvalinear*
In 196S» Baier and Robertson (2) proposed a linear
regression model which would estimate daily latent
evaporation from simple meteorological observations and
astronomical data readily available from tables* The
versatility of this method was enhanced by the fact that any
combination of up to six variables could be used* Estimates
of potential evapot ranspir at ion were obtained directly from
the model by multiplying latent evaporation by a coefficient
of 0*0034* This model will be discussed in more detail in a
later section*
Holmes and Robertson (26,27) recognized that as the
plant roots expanded and the soil moisture decreased, the
rate of plant water use also decreased* Soil moisture
drying curves, which adjusted the evapotranspira tion rate in
relation to the season and the soil moisture content, were
derived experimentally from laboratory and field
observations for various soils and crops* Holmes also
recognized the fact that as the plant roots reached a
certain soil depth, the actual evapotranspiration rate
decreased sharply from the potential rate* From these two
important concepts, the Modulated Soil Moisture Budget was
-
\
9
developed* The soil was divided into five zones* each with
equal water holding capacities* The actual
evapotranspiration was determined by the above mentioned
soil moisture curves and the amount of water extracted from
each zone was determined by a set of arbitrary coefficients*
Kerr (32*33) had used the basic principles of the Modulated
Budget in the development of a moisture budget which
considered the effects of the crop height* soil and plant
rooting characteristics on the rate of moisture use by
crops ®
Baier and Robertson (3) later developed a model called
the Versatile Soil Moisture Budget which made use of the
basic
concepts of
the modulated
budget*
In
addi tion
* the
concept of
atmospheric demand
rates
as a
f u net ion
of the
AE/PE
rat i o
and
a matrix of
crop
coefficients
which
reflected the amount of water the root system could extract
from each soil zone were instituted® The coefficients were
varied for each soil zone and for each stage of growth of
the crop throughout the season in order to attempt to
simulate the probable water extraction pattern of the root
system®
Other soil moisture models have attempted to simulate
consumptive use in various ways* Weaver (56)* in 1967*
described the algorithm which Pierce had developed in 1966
to estimate soil moisture deficit under corn* meadow and
wheat* Consumptive use was calculated by multiplying
potential evapotranspiration together with several
-
'
10
correction factors which included day lengthy soil moisture
dryness, rainfall and crop stage. Each correction factor in
turn was determined by a nor*-* linear regression equation*
Windsor and Chow (59,60) incorporated the relationship
between crop potential evapot r anspir ation and turgor loss
point in order to determine moisture stress days* Crop
potential evapot ranspira t ion was estimated from a Weather
Bureau Class A evaporation pan and a dimensionless crop
coefficient which accounted for the type of crop and stage
of crop development* Soil dryness curves, similar to those
used by Holmes, were used to convert potential crop
e vapotranspira t ion to actual crop evapotranspir a tion*
David (14) and Sasheed et al (44) developed regression
models which related the day of the growing season to the
rate of actual evapotranspiration • Rochester and Busch
( 47 ), in 1972, developed a scheduling model to improve the
management of irrigation systems* Pan evaporation
measurements were multipled by a coefficient, which varied
according to the day of the growing season, to determine
daily actual evapotranspiration estimates* Richardson and
Ritchie (45) developed empirical relationships to predict
separately soil and plant evaporation from a watershed*
The problem with any soil moisture budgeting technique
is to properly estimate potential evapotranspiration and
thus crop consumptive use* To date, only the Versatile Soil
Moisture Budget contains crop, soil and water parameters to
estimate crop water use* For this reason, the Versatile
•:
t -
i*
11
Soil Moisture Budget was chosen as the model to simulate
soil moisture conditions under several irrigated crops for
this study.
Literature which deals with the relationship between
weather and irrigation is scarce. Many models have been
built to produce probability distributions of seasonal
irrigation water requirements. Colig&do et al C 12)
presented a risk analysis of irrigation requirements for
each week of the growing season for numerous stations across
Canada® The risks were computed for different combinations
of total available soil moisture capacities and consumptive
use factors* No analyses have been found by the author
which attempts to depict the behaviour of drainage water in
re lati on to irrigation and rainfall. Data concerning the
amount and the time of occurrence of deep percolation under
natural conditions over a period of several years is
virtually non-existent.
Soils which have a moisture content in excess of field
capacity have been reported by many researchers to take two
to three days to reach equilibrium. It is generally
accepted that deep percolation rates level off when field
capacity has been reached. However, Wilcox (57) reported
that drainage never ceases and that there is no leveling off
point. Wilcox concluded that e vapotranspi ration, measured
by common soil moisture depletion methods, contains some
unknown quantity of deep drainage. Willardson and Pope (58)
explained that unsaturated drainage is usually accounted for
•-
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*
,
'
12
in most moisture models by the ev apo t ranspirati on parameter*
Since very little is known about unsaturated drainage
and the tact that any unsaturated drainage is probably
accounted for by the consumptive use tera; the use of the
Versatile Soil Moisture Budget was further Justified® The
Budget assumes that no unsaturated drainage occurs between
soil layers and that deep percolation is that amount of
water in excess of field capacity*
2*3 Description of the Area. *
Daily weather data for 45 years for six Alberta
stations were available on magnetic tape at the Department
of Agricultural Engineering^ University of Alberta* Of
these six stations? only two? Lethbridge and Medicine Hat,
were located in the southern regions of the province* Since
Lethbridge has the largest concentration of irrigation, it
was chosen as the study area for this thesis* A general
description of the area follows* The climatic information
and soil description were taken from Hobbs (21 ) and Nielson
(40) respectively.
2«3.1 Location*
Lethbridge is located at north latitude 49° 42* and
west longitude 112°47** It is situated 2,961 feet above sea
level •
2 » 3^2 _c.l imaJLe g.
The climate of the Lethbridge area is extremely
variable from month to month* Short, warm summers followed
by long, cold winters are typical. Lethbridge lies within
-
.
13
the influence of the Chinook winds which tend to reduce the
severity of the cold winter months and to alleviate the
extreme summer heat* These windst being relatively warm and
dry* originate on the eastern slopes of the Rocky Mountains*.
During the winter months* the winds may displace cold air
masses while during the summer months* they may effect
cooler temperatures but cause high moisture stress and
drought injury to crops*
Lethbridge has an average annual precipitation of
16*18 inches ( 1902—1969)* Approximately 75 percent (12*43
inches) of the total occurs during the months of April to
October and 32 percent occurs during the critical growing
months of hay and June when the crops are young and shallow
rooted* June has the highest rainfall amount* averaging
about 3*21 inches as shown in figure 1 • These average
values were calculated from the 45 years of daily weather
data available on computer tape*
During the winter months* it is not unusual to have
one foot or less of snow cover or no snow cover at all*
Warm Chinook winds often raise the temperature sufficiently
to remove any snow cover within several days* A midwinter
rainfall is not uncommon*
2*3*3 Soils Description*
Most of the soils in the immediate Lethbridge area
fall into the order of Chernozemic soils* They are
characterized by a thick dark brown "A" horizon*
Chernozemic dark brown soils were formed under slightly more
c
'
14
(S3H3NI) NOIiVlldlD3dd
Figure 1. Average total monthly precipitation for Lethbridge.
15
humid semiarid conditions than the brown soils of the more
eastern parts of southern Alberta* The upper layer is of a
clay, silt and sand mixture called Glac i o— Lac us trine
deposits* The permeability of this layer varies
considerably, but is generally moderately to rapidly
permeable, affording good to very good irrigating
condi tions •
The lower layer is a glacial deposit called Till* It
is massive and largely structureless* The thickness varies
between 60 to 130 feet* Sand and gravel are present, but
relatively rare* In some areas, the till forms the present
land surface white in other areas it underlies the
Lacustrine deposits* The depth at which the till is
situated, where overburden is present, ranges between 2 feet
and 40 feet with the average depth being 5 feet* Since the
permeability of this layer is very low (0*2 iph or less),
drainage problems are often a result of the existance of the
till on irrigated lands* Table 1 presents a brief
description of some of the more common soil types of the
Letbridf e area*
3* 3 a 4 Prainflge Studies*.
Experiments by Rapp and van Schaik (43) in shallow
glacial till soils, indicated that the irrigation amount and
irrigation frequencies influenced the position of the water
table considerably more so than did natural rainfall. The
water table was observed to rise close to the surface after
an irrigation.
and the amount of rise was found to be
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'
; Vi
16
TABLE 1 : A DESCRIPTION OF SOME SOUTHERN ALBERTA SOILS.
(Bowser et al, 8)
Horizon
Depth
( i ns )
H. C.
( iph )
Descript ion
Chin Light Ah
0-4
1.5
brown loam
Loam Bj
4 —15
1.0
brown— dark brown loam
C ca
15-26
0.7
light brownish grey loam
Csk
26-48
0.7
yellowish brown loam to
silt loam
Till
48-
o
•
to
glac ia l till
Irrigabi l i ty—
good to
ve ry
good. Glacial till averages
4 feet from the surface.
Shal lo
w Chin
— horizon characteristics same as above
— glacial till averages 2 feet from surface
causing high water
root zone.
tables well within the
— irrigability fairly good to good.
Cavendi sh
A
0-7 2. 5
brown fine sandy loam
Loamy
Sand
B
7 -24 1.5
brown sandy loam
Cc
24-40 2.5
light yellowish brown sand
sand to sand
Ck
40-60 3.0
light yellowish brown loamy
sand to sand
Till
60-
g l ac i a l till
glacial
till
averages 5 feet
below the surface
i rri gabi l i ty
— good to very
good
Mai eb
Loam
Ah
0-4 1.0
brown loam — loose
Be
4 -12 0. 3
brown to dark brown heavy
loam to clay loam
Cea
12-18 0.5
Csk
18-24
clay loam till — blocky
C
at 36 0.2
granite, ironstone, coal
irritability good to very good if good topography
exists.
V
17
dependant upon the amount of irrigation* The subsequent
recession of the water table took three to four days and was
considered to be primarily due to crop consumptive use* A
duration of 3 to 4 days of high water table was found not to
be injurious to shallow rooted crops; however* a
considerable amount of dead roots were found on deep rooted
c srops «
Excessive irrigation was also observed to be a
problem* It was estimated by Rapp that some fields were
irrigated by as much as 2 to 3 inches of water in excess of
field capacity* Because of the low hydraulic conductivity
of the till* temporary potholes or sloughs could form
causing eventual crop root damage and salinity problems*
Sloughs reduce the productive acreage of the farm and
increase the cost of operation*
Drainage problems* although not entirely due to
irrigation mal-practice, can be alleviated by developing
efficient irrigation methods*
■
iLt The Consumptive Use Model*
Any soli moisture model which simulates soil moisture
on a daily basis must employ a fairly sophisticated means of
determining daily crop consumptive use*
As stated
previously, the method developed by Baier and Robertson (3)
is the most refined mathematical model of consumptive use
devised to date* A detailed description of the model
follows •
3*1 The New Versatile Soil Moisture Budget*
The Versatile Soil .Moisture Budget is a method by
which climatic, plant and plant— soil interrelationships are
implemented to estimate crop consumptive use*
The
expression is as follows:
AE
n
= Z
K.
J
s,.(i“1)
S .
J
Z . PE . e
j i
-w(PE.
- PE)
(1)
where: AE.
Kj
S' . ( i-1 )
J
s .
J
J
J
PE.
w
PE
actual evapo transpi ra t ion on day i
coefficient matrix accounting for the
amount of water in percent of PE extracted
by plant roots from different zones J
during the growing season
available soil moisture in the Jth zone at
the end of day i— 1
total available water capacity in the jth
zone
adjustment factor for different types of
soil dryness curves
soil zone number
potential evapo transpiration for day i
adjustment function accounting for the
effects of varying PE rates on the AE:PE
ra ti o
long term average daily PE value for the
month or season
The crop coefficients, K ^ , describe the percent of PE
hich is removed from each soil zone. In essence, K. is a
18
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19
matrix of consumptive use factors: the columns represent the
various stages of growth on a time scale and the rows
represent the individual soil moisture zones* He nc e , in
this manner, a particular Kj coefficient may only apply to
one soil moisture zone over a period of time defined by the
length of the current stage of growth* The Kj coefficients
must be determined by iterative comparisons between computed
and observed soil moistures* Alternatively, they may be
estimated so as to represent the most probable soil moisture
pattern under prevailing environmental conditions* A third
alternative, provided experimentally determined average
consumptive use curves are available for different crops, is
to compute on a short term basis ( i*e* 5 to 10 day
intervals), dai ly consumptive use values averaged over a
period of several years of simulated crop growth* Iterative
comparisons between the experimental and simulated curves
may then be performed* Although more expensive, the latter
method will provide accurate results on a long term basis*
The K coefficients for this study were determined using both
the first and the latter techniques*
The term S' . ( i — 1 )/S ^ describes the ratio of the
current available soil moisture to the total available soil
moisture capacity in zone J. This ratio is used in
conjunction with the Z term which is a vector of 100
coefficients corresponding to the value of the moisture
ratio* The product S'jCi— 1)/Sj * Zj represents the amount
of water, expressed as a percentage of PE, extracted from
-
»
20
zone j according to the current moisture content of that
zone* Various proposals for the relationship between the
AE/PE ratio and the soil moisture content are presented in
figure 2* Each curve (A through H) has associated with it a
Z-vector similar to the A and H vectors presented in table
2® Baler 14) concluded from a comparison of observed soil
moisture with estimates obtained from the Versatile Budget
using five types of re la tionships that the type G curve
would yield the best results for grass grown in Matilda loam
soil* He further recommended that this curve be used as a
"first approximation in most medium textured, non— i rriga ted
soil" ( 5 ,pp 10)® Baier also encouraged the use of the type
A curve for sandy soils as well as "for soils under
irrigation when a moisture content close to field capacity
is maintained throughout the growing season" (5, pp 9). The
type H curve, which is a compromise between the A and G
curves, was chosen for use in the model* The Z— vectors for
the A and the H curves are presented in table 2 •
The exponential term of the Versatile Budget accounts
for the varying daily atmospheric demand rates* The W terra
is a regression equation developed by Baier et al (3) and is
described below*
W = 7.91 - 0.11 S'^-1"'L-)- 100 <2>
This value is dependent on the soil moisture ratio of each
soil zone*
•i
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21
AVAILABLE SOIL MOISTURE {%)
Figure 2. Various proposals for the relationship between
the AE:PE ratio and the current available soil
moisture (Baier et al, 5)
22
TABLE 2. Z - TABLES SOIL DRYNESS CURVES A AND H.
99. 99
50.00
9.09
8.33
4. 76
4.55
3.23
3.13
2.44
2.38
1.96
1.92
1.96
1.92
1.64
1.61
1.41
1.39
1.23
1.22
1.10
1.09
33.00
25.00
7.69
7.14
4.35
4.17
3.30
2.94
2.33
2.27
1.89
1.82
1.89
1.85
1.59
1.56
1.37
1.35
1.20
1.19
1.08
1.06
A TABLE
20.00
16. 66
6. 67
6.25
4.00
3.85
2.86
2.78
2.22
2.17
1.85
1.82
1.82
1.79
1.54
1.52
1.33
1.32
1. 18
1. 16
1.05
1.04
14.28
12.50
5.88
5.56
3.70
3.57
2.70
2. 63
2.13
2.08
1.79
1.75
1 .75
1.72
1.49
1.47
1.30
1.28
1.15
1. 14
1.03
1.02
11.11
10.00
5.26
5.00
3.45
3.33
2.56
2.50
2.04
2.00
1.72
1.69
1.69
1.67
1.45
1.43
1.27
1.25
1.12
1.11
1.01
1.00
H TABLE
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
1.96
1.92
1. 64
1.61
1.40
1.38
1.23
1.21
1.10
1.08
2. 00
2.00
2. 00
2.00
2. 00
2.00
2.00
2.00
2.00
2.00
1.88
1.85
1.59
1 .56
1.35
1.34
1. 19
1 . 18
1.07
1.06
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
1.81
1. 78
1.53
1.52
1.33
1.31
1. 17
1.15
1.05
1. 04
2.00
2. 00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
1.75
1.72
1-49
1.47
1.29
1.28
1 .14
1 . 13
1.03
1.02
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
1.69
1.67
1.45
1.43
1.26
1.25
1.12
1.11
1.01
1 .00
•
.
■
23
3*2 BvftffotrftiiaaAgft.ULanjL
The value of PE in equation 1 may be determined by
either the Bellani Plate Atcometer? Penman's equation f or by
a regression equation developed by Baier and Robertson (2).
The latter method involves the estimation of daily latent
evaporation from a combination of simple meteorological
observations and astronomical data readily available from
tables® Three to six terms were employed in a series of
eight equations® As the number ©f terms included in the
equation increased from three to six the multiple
correlation coefficients increased from 0®68 to Ga84® The
expression using all six terms is described below®
EL = -53.39 + 0.337 MAX + 0.531 (MAX-MIN) + 0.017 Qo
+ 0.0512 Qs + 0.977 WIND + 1.77 (Ew-Es) (3)
where S EL — latent evaporation
MAX = maximum daily temperature
MIN = minimum daily temperature
G© = solar radiation received at the top of the
atmosphere
Qs = solar radiation received on a
surface
WIND - total daily wind mileage
Ew = saturation vapor pressure at
temperature
Es = saturation vapor pressure at mean dew point
horizontal
mean
air
The value of Qs may be determined from the expressions
Qs - Qo{0. 251 + 0.616 |}
(4)
where: n = daily hours of bright sunshine
N = total hours between sunrise and sunset
Qo and Qs are as above®
Because 33 of the 45 years of weather records
available for the Lethbridge area contained measurements of
c
' >
v
24
only daily temperatures and precipitation; it vas decided t o
use the equation containing only four terms as described
below •
EL = -108.8 + 1.13 MAX + 0.920 (MAX -MIN) + 0.359 Qo + 0.131 WIND (5)
Potential e vapotranspi ration is obtained by multiplying EL
by 0*0034*
Because the regression equations were developed from
daily weather data recorded across Canada over several
years; reasonable estimates of latent evaporation for most
parts of Canada can be expected with the use of this
equa tion «
3x:3_Xhg . Soil. Moisture Zones.
Baler et a l C 5) adopted six standard soil moisture
zones which contained respectively 5*0* 7* 5 » 12*5* 25*0*
25 *0} 2S®0 percent of the total available moisture in the
root zone* The adoption of the six zones made it possible
to describe the plant water extraction characteristics in
any soil type regardless of the depth at which the moisture
was located* Several assumptions were made wi th the use of
these soil moisture zones*
1* The soil zone receives water in successive order
from top to bottom in a step-wise fashion* If the amount of
water entering the first zone is greater than the capacity
of that zone* the remaining water enters the next zone* If
It is less than the capacity of the zone * the water will
remain in that zone and no drainage will occur into the next
r
■ .
25
zone*
2* Because of the above assumption! water is assumed
to infiltrate into the soil zones instantaneously*
3® Drainage is assumed to be that amount of water
above the total soil moisture deficit of all six zones*
This amount is assumed to leave the soil zone as deep
percolation on the same day that the water was applied*
3ul£..R}MLQ&£
In order to incorporate runoff into the Versatile
Budgets Baler and Robertson implemented a simple
relationship between soil moisture in the top zone and daily
precipitation*
RUNOFF = RRi - I (6)
S' (i-1)
I = 0.9177 + 1.811 In RR. - 0.00973 In RR. — — - 100 (7)
i i b ^
where: RR^ = the rainfall for a 24 hour period ending in the
morning of day ( i+1 )«
I = amount of infiltration into the soil
S 1 x (i~l)
— — — — = the available soil moisture in percent of
capacity of ( Sj ) in the top zone at the
end of day ( i—1 )•
Runoff is assumed to occur if the total daily rainfall
exceeds i®00 inch* The topography is assumed to be level*
In generals irrigation sprinkler nozzles used in
southern Alberta discharge water at a rate of 0*5 inches per
hour* The majority of soils in the Lethbridge area possess
hydraulic conduct i vl t i es above that of the nozzle discharge*
A list of the various types of soils and their respective
V
.
• .
26
hydraulic conductivities are presented in table !• It
therefore assumed that runoff from sprinkler irrigation
negligible and any runoff that did occur was due solely
precipitation exceeding 1.00 inch per day*
was
was
to
*
-
'
Selection of the Proper K-Coeff icien ts.
In order for the Versatile Budget to effectively
simulate the moisture withdrawal from each soil zone , the K—
coefficients had to be selected so as to represent the most
probable soil moisture extraction pattern for the four crops
under study. The K-coef ficients were obtained by iterative
comparisons between actual and estimated soil moisture. The
procedure followed is described below.
Before iterative comparisons could be made,
experimental field measurements of soil moiature had to be
obtained® Field data was necessary in order that
comparisons between the daily soil moisture contents of
different crops# as simulated by the Versatile Budget , could
be made against actual values as measured in the field.
Hobbs and Krogman ( 24 ) had carried out experiments at
Vaushall on the consumptive use rates of 12 irrigated crops,
each grown in 15 foor square plots of land. Vauxhall lies
approximately 30 miles east of Lethbridge. When the soil
moisture content of each plot had depleted to approximately
50 percent of the total soil moisture capacity, the plots
were irrigated. The soil moisture content was determined
prior to an irrigation and the amount of water applied was
Just sufficient to bring the soil moisture to field
capacity. It was assumed that deep percolation was
negligible. From the soil moisture content readings and the
total irrigation and rainfall water applied to each plot, a
27
-
r
■
28
reasonable estimate of the rate of consumptive use between
irrigations was obtained*
The soil moisture readings, the total available soil
moisture, and the irrigation dates and amounts for the years
1960 to 1963 were obtained from Hobbs (22) for Soft Wheat,
Potatoes, Sugar Beets and Alfalfa* This data was then used
to estimate the K-coef licients*
4*2 Weather Data.
The Versatile Budget requires that potential
evapo transpira t ion be estimated from daily maximum and
minimum temperatures , solar radiation and wind velocity*
The daily temperatures and precipitation for the Vauxhall
area were obtained from the "Monthly Records of
Meteorological Observations in Canada" (38)* Solar
radiation received at the top of the atmosphere was obtained
from Smithsonian Tables (37) and the monthly average wind
velocities were gathered from table 7 of Rutledge (48)* Ten
years of daily wind velocities (1956 — 1966) were taken from
the computer tape containing the daily weather data and
averaged on a monthly basis* Equation 5 was then used to
calculate dal ly potential evapot renspi ra ti on from April 1st
to October 31st for the years 1960 to 1963*
The long terra average PE value in the exponential term
of the Versatile Budget was taken from the monthly averages
for Lethbridge as determined by Rutledge in table 4 (48)*
Equation 3 was used by Rutledge to determine daily PE
values* According to the values , Medicine Hat and
4
-
\
29
Lethbridge showed very little difference in their monthly PE
values^ Hence, since Vauxhall lies approximately between
the two stations.
it was felt
that
the
condi tions
a t
Lethbridge would be
sufficiently
c lose
to
conditions
at
Vauxhall. This procedure of selecting long term averages of
PE values had to be done since daily weather data for the
Vauxhall station was not readily available on computer tape*
Furthermore® the purpose of performing the iterative
comparison between actual and simulated data was to obtain
only approximate K— coefficients for each crop* Later® the
K-coefficients would be readajusted, using accurate average
PE values for Lethbridge, to fit average consumptive use
curves for all of southern Alberta* Hence, the accuracy of
the PE term in the Versatile Budget is only minor at this
points
4*3 The Z-Table.
The data obtained from Hobbs indicated that the daily
rate of consumptive use was quite high* This suggested that
either the type H or type A curves of figure 2 would be
suitable for simulating the soil-water relationships* Both
curves stipulate that AE equals PE for soil moisture
contents above 50 percent. Having no other basis for
selection, the type H curve was chosen* This curve is
represented by the H table in table 2*
4*4 Method*
The K— coef f ici ents for each crop were determined by
iterative comparisons between actual soil moisture contents
<
-
*
■
30
anct the Versatile Budget estimated soil moisture contents
prior to each irrigation* Figure 3 shows an example of the
output from the simulation and the corresponding
experimental values as obtained from Hobbs (22)*
The ending dates of the stages of growth, as
represented by each row of the K— coef f ic ie nt matrix, were
determined in accordance with the consumptive use curves
derived by Hobbs et al (24)* The coefficients used for the
periods prior to planting were those suggested by Baier et
al (5) for fallow® They are 0*60, 0*15? 0*05, 0*00f 0*00*
0 « 0 0 • The coefficients used for the period subsequent to
harvest for Wheat and Alfalfa were those recommended for sod
( 0*5 0* 0®20g 0® 15* 0*10 1 0*03, 0*02 )* The coefficients
recommended for fallow were employed for Potatoes and Sugar
Be etso
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Figure 3. A sample output of the Versatile Budget simulation for Sugar Beets
during 1960. ( Note: all units are in inches. )
\
■
3 <.
'
Slm. The Weather Model,
5-a.l M9Pte„Car„lo gaflpllttfli*
The Monte Carlo saMipling technique is a method by
which a sample of an independant variable can be
synthetically generated, in a sequential fashion, with a
given frequency distribution. This involves transforming a
random independant number from a uniform probability
distribution and, by use of the graphical method, producing
a sample from the desired frequency distribution* A. number
between, but not including, 0*0 and 1*0 is generated by a
random number generator and is applied to the cumulative
distribution to obtain a sample of the independent random
variable*
The major advantage of sequential generation is the
ability to create a synthetic record longer than existing
historical records® In this way, most of the possible
combinations of the variable sequences will be included in
the synthetic sample depending on the length of generation®
In the present study, the behavior of the plan t-soi 1-water
relationships under most weather conditions will be
simulated* The amount and frequency of occurrence of both
irrigation and drainage will reflect the soil-crop-water
behavior under varying weather conditions®
5*2 Weather Distributions
Weather includes such variables as rainfall,
temperature, wind, etc® It is common knowledge that such
variables fluctuate randomly from day to day or from hour to
32
<
,
33
hour and also that these variables are a function of the
time of dayf month or year* For instance? temperature is
maximum during the summer months and minimum during the
winter monthsv but the maximum and minimum temperatures ? on
a daily basis? are random* Such a phenomena is known as a
Stochastic process and the values it assumes over time are
known as a time series* Daily monthly and annual values of
rainfall? for example? form a discrete time series* Each
random variable of a time series has associated with it a
certain probability distribution at any particular point in
time* If the distribution remains constant throughout the
process? the variable is said to be stationary* Otherwise?
it is n o n—s tatio nary ® Most hydrologic processes are non¬
stationary over long time periods* They are treated?
therefore? as stationary processes over short time periods*
Three variables are necessary to generate weather on a
daily basis* They are wet and dry day sequences? daily
rainfall and daily potential evapotranspiration. A computer
program was written in FORTRAN to read in daily
precipitation amount s and maximum and minimum temperatures
for the Lethbridge station from the computer tape containing
the daily weather data. The temperatures were used to
calculate potential evapotranspiration (PE) according to
equation 5. The date? precipitation and PE values were then
printed onto a second tape from which subsequent work was to
be performed*
••
.
34
5i.d Wgt aarf„ JBgjL-Bay Prpfcdb.il Hies *
Weather is composed of a series of wet days followed
by a series of dry days* Hopkins and Robillard (28)
performed a statistical analysis of daily rainfall
occurrence for three areas in the Prairie Provinces* They
found events on successive days to be statistically
dependant and that a first— order transitional probability
model would serve to approximate the occurrence of dry days*
However f the model did underestimate slightly the total
number of rainy days in the month* Feyerherm and Dean Bark
(18) stated that where Interest lies in computing
probabilit ies for relatively short sequences of wet and dry
days t the first— order Markov chain appeared to be quite
adequate* In an earlier paper, Feyerherm and Dean Bark ( 17 )
had presented the first order Markov chain for wet and dry
sequences in mathematical form as described below*
P<-Xt’ Xt+1’ Xt+2 * . Xt+n^ p(xt) P^t+liV P ^Xt+2 ^Xt+P
p(xt+3l*t+2> •••• P<x
P )
t+n ' t+n-]/
(8)
where :
and
P(Dt) =
x = the event that day t is wet (W) or dry (D)
t
No. of years the (t) day Is dry
|\|q n-F uoarc r> f rornrrlfi
^ U IUA.A.W V- / J
. of years of records
P (D
t+n
Vn-P
No. of years (t+n) day is dry and (t+n-1) day Is wet
No. of years t+n-1 day is wet
Each probability in the expression is dependant on the
events of the previous day* Because simulation by the first
*
s
.
35
order Markov chain is on a daily basis* the conditional
probabilities of a wet day preceded by a dry day and a wet
day preceeded by a wet day need only to be determined*
Jones et at (31) used the Markov chain principle to
calculate a series of conditional probabilities for each
week of the year* They assumed that the probabilities
remained constant over a seven day period* A polynomial
equation was then fitted to the probabilities and a
reasonably good fit was obtained* The two polynomial curves
showed that the conditional probabilities followed definite
seasonal trends* Hence* the method used by Jones was
applied to the Lethbridge data to determine if a similar
seasonal trend existed in the data*
Daily rainfall records spanning a period of 45 years
( 1922 to 1966 ) were used to calculate the rainfall model
parameters* The data for Lethbridge and five other Alberta
stations were available on magnetic tape « The conditional
probabilities for rainfall were calculated as follows;
p(w|d)1
£ wet day following a dry day (i)
total days following a dry day (i)
(9)
P(w|w)i
£ wet days following a wet day (i)
total days following a wet day (i)
(10)
p( W | D ) . represents the probability that any day during the
ith period was wet given that the preceding day was dry*
P( W ! W ) ^ is the probability that any day during the ith
period was wet given that the preceding day was wet* Both
P(w|d). and P( W ] W L were calculated for each 5— day period
c
'
V
.
36
from April 1st to October 31st staking a total of 43 time
periods in ail* It was assumed for the purposes of this
study that the probabilities did not change considerably
over any 5—day period*
A further assumption was made regarding the definition
of a wet day* If the amount of rainfall received was equal
to or greater than 0*01 inch* the day was considered to be
wet® A base level of 0*01 inch was used because of the fact
that the top soil zone of the Versatile Budget has the
capacity of holding only 5% of the total soil moisture*
This value can be small* Hence, a rainfall of 0®01 inch
will influence the moisture content of the top soil zone
sufficient to warrent the use of this amount as the basis
for a wet day* Furthermore, it could not be assumed that
daily consumptive use never reached values of zero inches
during the spring and fall months* Therefore, 0*01 inches
could affect the top soil zone on days experiencing zero
inches of consumptive use* As well, days on which "traces"
were recorded were designated as dry days*
In order to determine if the probabilities followed a
seasonal trend, the probabilities were plotted against their
corresponding period number and a 6th degree polynomial
equation was fi tted to both the P( W | D ) and P(WjW) dat a * An
F-test was performed on both plots to test the equations for
significance* It was found that both polynomials were
significantly different at the 95% level of probability*
Figure 4 shows the actual values plotted against the
<
.
90
37
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38
predicted values using the 6th order polynomial equations*
The equations are:
P(W|D) = 0.32542 - (9.6446 x 10~2)X + (2.1051 x 10”2)X2
- (1.77 x 10"3)X3 + (7.0055 x 10~5)X4 - (1.3067 x 10_6)X5
+ (9.3216 x 10"9)X6 (11)
P(W|W) = 0.46017 - (4.8552 x 10~2)X + (1.3869 x 10_2)X2
- (1.2516 x 10‘3)X3 + (4.878 x 10"5)X4 - (8.5935 x 10_7)X5
+ (5.5955 x 10“9)X6 (12)
where X represents the 5— day period number*
The coefficients of determination were 0*67 and 0*45
for equations 11 and 12 respectively* Figure 4 indicates
that both P( W | D } and P( W J W ) have definite seasonal trends*
Also indicated is the fact that there is & strong tendency,
especially In the latter half of the growing season, for a
dry day to follow a dry day as suggested by the relatively
low values of P(w|d>® Furthermore, the values of P( W j W ) , as
the season progresses, decrease thereby increasing the
probability of dry days to occur* This partly shows why the
average monthly precipitation from July to September, as
illustrated in figure 1, is less than May and June* The
sixth order polynomial equations were used to determine wet
and dry day sequences in the Monte Carlo model®
-5 .* 4 The Rainfall Model*
The next step involved in the simulation of daily
rainfall is to select an appropriate distribution function
which wiil characterize precipitation on a daily basis*
Some investigators {7,14,15,20,52,53,61) have suggested that
rainfall can be characterized by the gamma function*
The
*
.
.
.
39
cumulative gamma, distribution function is
given by the
following expression*
F (x) = - i—
ear(a)
(13)
where : F( x )
x
6
a
r ( a )
cumulative distribution function
precipitation amount in inches
shape parameter dependant on the variability
of rainfall amounts
scale parameter dependant on the magnitude of
the rainfall amounts
complete gamma function
Thom (53) used the concept of mixed distributions to
illustrate the use of the inverse gamma distribution tables*
It was realized by Thom that the nonoccurrence of
precipitation was caused by a set of meteorological
variables different from those causing a measurable amount
of precipitation* Therefore, the distribution must be
broken up into two parts as described below*
G(x) = (1 - p) + pF(x)
(14)
where: G( x ) = the precipitation distribution
F( x ) = the precipitation distribution of measurable
amounts (as described above)
p — the probability of occurrence of a measurable
amount of precipitation
Equation 14 considers both the probability of a day
being wet or dry as well as the probability of receiving x
inches should it be a wet day* The parameters, a and 0,
were determined by the maximum likelihood method, equations
15 and 16, which follow*
*. :
a
1_ + + 4/ 3A
4A
Ae
(15)
(16)
where: <ar and 0 are the g&ciia parameters
™ 1 n
A = In x ■ . 2 In x.
N 1=1
Ae = correction factors given in table 82 of
Yevjevich (62)*
x - average rainfall within a given time interval
N = number of days of rainfall
x ~ amount of rainfall for day i
From the weather records available on magnetic tape* a
computer program was written in FORTRAN to calculate the a
and 0 parameters for days following a wet day and for days
following a dry day® Since the cumulative distribution can
not be easily calculated from equation 13* an expansion
equation; as given by Thom (53)* was used® The equation is
as follows ®
a
F (t ; a) =
T (a 4- 1) e
[1 +
+
a+1 (a+l)(a+2)
+
(17)
where: F(t;<a) = gamma distribution function
t = X/a
X = precipitation Cinches)
a = scale parameter
The parameters were calculated over 15 and 16 day intervals*
depending on whether the month had 30 or 31 days® This made
a total of 14 intervals in the season starting from April
1st® It was assumed that seasonal variation in
1 •
.
41
precipitation amounts would vary little over 15 day periods*
A second program was written to construct the cumulative
frequency distribution of precipitation following both wet
and dry days using the actual data* The actual
distributions were plotted on log probability paper against
the theoretical function for each of the 28 time intervals*
Figure 5 represents a sample plot of actual versus
theoretical cumulative rainfall distribution following a dry
day* The Chi— squared test was used on a random sample of
ten plots in order to determine if the actual distribution
followed the gamma function* Table 3a lists the Chi— squared
values and their respective degrees of freedom for each
distribution chosen* Nine of the ten samples chosen were
found not to be significantly different from the theoretical
distribution at the 90 percent level of probability*
Therefore , the incomplete gamma function was used to
describe the daily rainfall occurrences for the entire
growing season* The a and 0 parameters are listed in
table 4®
5 » 5 The Potential Eva p o 1 gansulg &_t 1 oq .MP.del*.
A computer program was written to calculate daily
potential evapotranspira t ion via equation 5 between the
dates of April 1st to October 31st for each of the 45 years
of records available on magnetic tape* The term Qo (solar
radiation recieved at the top of the atmosphere) was
obtained from Smithsonian tables (37), while WIND (monthly
average wind velocities) were taken from table 7 of
«
~
'
• \ - 1
42
Figure 5. Comparison of actual and theoretical cumulative distribution
of precipitation following a non-rainy day: May 15 - 30.
43
'"'ABLE 3. CHI-SQUARED TEST - PRECIPITATION AND POTENTIAL
EVAPO TRANSPIRATION.
a ) PRECIPITATION
I nterval
Type of
Day
Degrees of
F reedom
Chi— Squared
Values
Apr
1-15
Dry
3
7.365
*
Apr
16-30
Dry
4
7.531
n • s •
J ul
16-31
Dry
3
1.209
n.s«
Aug
16-31
Dry
4
2.384
n . s •
Oct
1-15
Dry
2
2.984
n • s •
Apr
1-15
Wet
3
3.562
n . s .
May
16-31
Wet
5
6.036
n . s .
J ul
1-15
Wet
4
4.868
n. s •
Sep
1-15
Wet
4
6.583
n . s •
Oct
15-31
Wet
3
3.215
n . s •
b ) POTENTIAL
EVAPOTRANSPIRATION
I n t erva 1
Type of Degrees of
Day Freedom
Chi— Squared
Values
Apr 1—15
Wet
2
9.703
J un 1 — 15
Wet
4
4.267
n.s.
J ul 1-15
Wet
5
4.797
n.s.
Aug 16—31
Wet
4
3.329
n.s.
Oct 1-15
Wet
2
6.676
**
Apr 16—30
Dry
4
11.211
❖ $
May 16—31
Dry
5
7.889
n.s.
Jun 1—15
Dry
3
7.005
*
Jul 16-31
Dry
4
13.176
**
Sep 1—15
Dry
4
12.847
**
* significant at the
significant at the
significant at the
n.s. not sigif leant •
0.10 level •
0.05 1 evel •
0.01 level.
\
■
■
TABLE 4. A LIST OF THE a AND 6 PARAMETERS OF THE INCOMPLETE
GAMMA FUNCTION FOR PRECIPITATION.
44
ID
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45
Rutledge (48)®
Because of the increase in relative humidity during
rainfall? potential evapo transpiration? o rt the average? will
be lower on wet days than on dry days* Hence? it was
decided to create two sets of distributions? one to describe
daily PE on wet days and one to describe daily PE on dry
days* Each set of PE distributions would therefore
characterize the daily temperature? solar radiation and
cloud cover® A program was written in FORTRAN to read in
the daily PE values from magnetic tape and to construct
cumulative distributions on a bimonthly basis for PE on dry
days and wet days* A total of 28 sets of data were then
plotted on normal probability paper* The concept of mixed
distributions? as discussed earlier? was again employed in
the construction of the PE distributions* Only those PE
values greater than zero were used to create the
distribution while those values equal to zero were used to
determine the probability of the occurrence of a measurable
amount of PE* These probabilities are presented in table 5*
Because most of the data plotted as straight lines on
normal probability paper? the normal distribution was
assumed to apply* The straight lines were fitted to the
data according to the mean and standard deviation of their
respective dis tribution® A Chi— squared test was performed
on a random sample of ten plots to determine if the normal
distribution applied* A list of the Chi— squared values and
their respective degrees of freedom are given in table 3*
«
.
* .
46
TABLE 5.
BIMONTHLY PROBABILITIES OF POTENTIAL
EVAPOTRANSPI RA TION ON WET AND DRY DAYS
I n te rva L
P( PE D )
PIPE W)
Apr
1-15
0*8180
0.4520
Apr
16-30
0.9267
0.6022
M ay
1-15
0.9810
0.8079
May
16-31
1 .0000
0.9336
J un
1-15
1.0000
0 . 9665
Jun
16-30
1.0000
0 .9957
J ul
1-15
1.0000
1.0000
Jul
16-31
1.0000
0.9932
Aug
1-15
1.0000
0.9935
Aug
16-31
1 .0000
0.9268
Sep
1-15
0.9882
0.7821
Sep
16-30
0.9059
0.5269
Oct
1-15
0.8569
0.5455
Oc t
16-31
0.7221
0.2810
TABLE 6.
SUMMARY
OF THE SMIRNOV-KGLMOGOROV STAr
DAILY PE
VALUES OCCURRING
ON DRY DAYS
I nterva I
S i ze
Statistic
Apr
1-15
408
0.10 **
Apr
16-30
454
0.065 *
May
1-15
464
0 • 05 n. s •
May
16-31
480
0 .040 n • s •
J un
1-15
407
0.05 n.s.
J un
16-30
444
0.06 n.s.
Jul
1-15
466
0.04 n.s.
J ul
16-31
573
0.03 n.s.
Aug
1-15
521
0 • 025 n.s.
Aug
16-31
553
0.04 n.s.
Sep
1-15
500
0.04 n.s.
S ep
1 6—30
461
0.06 n.s.
Oct
1-15
466
0.08 *
Oc t
16-31
433
0.10 **
*
significant
at t he 0 • 05
level
**
significant
at t he 0.01
1 e v e 1
n * s •
not
signi f ican t •
FOR
*
'
Or'O
47
(59HDNI) NOIlVdldSNVdiOdVAS 1VUN3iOd
Figure 6. Comparison of actual and theoretical cumulative distribution
of daily PE occurring on a non-rainy day: July 16 - 31.
48
j
<i
M
H
2
W
H
O
a,
2
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49
For PE occurring on wet days y only two of the five
distrib tut ions were significantly different* These
distributions occurred during the spring and fait months
when weather conditions are unstable and in a state of
change* During the summer months, the distributions were
not significantly different from the theroetical
distributions# In the case of PE occurring on dry days , the
situation was quite different* Only the distribution
representing the latter half of May was non— sign! f icant •
The distribution representing the first half of June was
significant at the 0*01 percent level and all other
distributions were significantly different at the 0*05
percent level® Therefore, it was assumed that the PE values
occurring on dry days did not follow the normal
distribution® However, because the straight lines, as
depicted by the mean and standard deviation of the data, in
most cases, fitted the plotted points extremely well, it was
decided to perform a non— parametric test with the use of the
Smirnov— Kolmogorov statistic. This test assumes that the
distribution is continuous and that the fitted straight line
to the data is distribution free* Potential
evapo transpiration , because it is measured to the nearest
0*01 inch, can be considered to be a continuous event. The
Smirnov— Kolmogorov test indicated that ten of the 14
distributions were not significantly different at the 95
percent level* A list of the Sm i rnov— Kolmogorov statistic
is presented in table 6* The normal distribution was
.
50
accepted as characteristic of daily potential
evapotranspiration amounts. A sample distribution for the
period July 16—31 is given in figure 6 • The means and
standard deviations are listed in table 7 and were used to
simulate daily PE events*
5.* 6. Th^_ Overwinter Percipitat ion Mmtftl*.
The last parameter of the weather model which remains
to be discussed is that of precipitation during the winter
months# There are essentially two directives which can be
taken in the matter* One is to develop the rainfall and the
PE models for the entire year thereby providing a means of
simulating weather for all twelve months of the year* The
main objective^ however, in developing a weather model is to
simulate actual soil moisture conditions on a daily basis*
This can be done satisfactorily and with sufficient ease
during the summer months, but it is extremely difficult to
simulate water movement in frozen soil*
VanSchaik and Rapp C55) performed lysimeter
experiments in which soil moisture contents and water tables
were monitored during two winters for both bare and grass
covered soils with a shallow water table* Two major points
were concluded from their research* The water table showed
a general downward movement during the winter but this
sometimes was nullified by warm Chinook periods* As well,
the soil moisture content of a soil with a shallow water
table increased substantially due to upward capillary
movement of water* However, the moisture content of the
-
'
• .
51
upper 10 inches of soil could only be increased by snowmelt
or fall irrigation*
Further research by Hobbs and Krogaan C2S) indicated
that the fall soil moisture was linearly related to
overwinter precipitation storage* Experiments were
performed on four crops with four irrigation treatments*
Overwinter changes in the root zone soil moisture were
recorded for eight seasons from the harvesting date to the
planting date of each crop* It was found that the crop
species did not significantly affect the soil moisture
content at the harvest date not did the amount of
precipitation stored in the root zone during the winter
months* The storage of overwinter precipitation was found
to be inversely proportional to the fall soli moisture and
was expressed by a linear regression model as follows*
Am = 6® 6 — 0«46M|r
where: = fall soil moisture
AM — overwinter increase in soil moisture
The correlation between storage and precipitation showed
that the storage was more dependent upon spring
precipitation than on fall or winter precipitation*
JRutledge ( 48 ) had assumed that the amount of
overwinter precipitation which was stored in the soil was 35
percent of the total overwinter precipitation for the
Lethbridge area* This estimate was based on experimental
work performed at Swift Current by Staple and Lehane* Since
.
'
'
- ■ .
*
52
this method was based upon actual values of overwinter
preipitation, the method, as used by Rutledge, was adopted
into the model® A program was written to construct a
frequency distribution of the overwinter totai
precipitation* The mean precipitation was found to be 4*35
inches with a standard deviation of 1 « 24 inches* A Chi —
squared test yielded a value of 2»1559 with 5 degrees of
freedom® This value was not significantly different from
the normal function at the 90% level of probability® The
Monte Carlo sampling technique was used to select at the end
of each season a value of overwinter precipitation, 35
percent of which was added to the soil to arrive at a soil
moisture content for April 1st of the next season® The
first year of the simulation run was assumed to be 75
percent of the total available capacity®
'
V
'
£.«. Pr.ggg&mming»
Several points ol Interest in the construction of the
cropping model should be indicated before proceeding any
further® It was the initial intent of the author to write
the program in GPSS (General Purpose Simulation System)®
This language has the ability to perform Monte Carlo
sampling of distributions with the least amount of
experience required on the part of the programmer® Only two
statements are required to simulate a day of rainfall and
likewise only two statements are required to construct a
cumulative frequency distribution from the output variables®
Hence, a cropping model was built using GPSS in which daily
rainfall and PE amounts were deteneined by the Monte Carlo
sampling technique® The daily soil moisture contents for
the four crops were calculated using the Versatile Budget®
The model was built and a dry run was performed® It was
found that 4 seconds of computing time were required to
simulate one day of crop growth® This was far too slow if a
total of 200 years of 214 days each (April 1st to October
31st ) were to be simulated® This would have amounted to
approximately 171*200 seconds or 47 hours of computing time®
The cost would have been astronomical® Hence* it was
decided to rewrite the program in FORTRAN — G® Rewriting
the Monte Carlo model in FORTRAN proved to be much more
difficult and time consuming than in GPSS® One subroutine
each had to be devoted to the rainfall and PE models while
construction of the desired frequency distributions of the
53
.
' , •
54
output variables required three subroutines*
The programi when completed, was run for a period of
one year* The model, this time, required only 4 seconds of
computing time to simulate one season of crop growth*
Hence, to complete 200 seasons of simulation, a maximum of
13 minutes computing time would be required* This was a
considerable reduction in time and more in keeping with the
current financial situation® After considerable editing,
the efficiency of the program was increased and the model
actually took 10 minutes to execute*
The model was divided into eleven parts: a main
program and ten subroutines* A listing of the program and
flow charts of the major subroutines is presented in
Appendix A® Some of the minor things which had to be
considered in the construction of the model will now be
di cussed at this point*
During the course of each day of simulation, two
variables, rainfall and potential evapotranspiration, had to
be simulated® Therefore, two random numbers per day were
required making a total of 428 numbers per season® Also, a
random number was required to determine whether or not March
31st, at th© beginning of each season, was to be a wet or a
dry day* This information was then used to determine the
precipitation functions to be used in calculating daily
rainfall on April 1st* Furthermore, a random number was
required to determine the amount of overwinter precipitation
*
H . 1
55
so that the soil moisture condition at the start of each
season could he calculated* Hence a total of 430 uniformly
distributed random numbers were required for one year of
simulation* This made a total of 86»000 numbers for the
entire 200 years* A random number generator had to be
selected so that it could produce up to 10Qy0Q0 numbers
without exhibiting circularity* Also* it had to have the
capability of producing the same sequence of random numbers
during different runs in order that comparisons of drainage
distributions could be made with and without irrigation* A
pseud ©“random number generator called GGUl from the IMSL
package (International Mathematical Statistical Languagey
29) was found to be suitable for the task* Statistical Chi-
squared tests had shown that 126y000 numbers could be
generated without circularity occurring* The random numbers
were stored in a two dimensional array t RND( 2y214)y where
the columns represented the day number of the season and the
rows represented the random numbers used to calculate
precipitation and potential evapo t ranspir at i on f
re spec ti ve 1 y •
6 * 2 .-SfijmnlJjiiLs.
The application of the random numbers described above
to the precipitation and the PE distributions were carried
out in two different manners worthy of a brief discussion*
6*2*1 Pr^g_l_pi t a tlP-D a
Because calculating the precipitation with the use of
equation 11 involves a great deal of iteration, computer
«
v
56
time would have been increased substantially* Insteady the
values for the gamma distribution for a = 0*5* 1*0* and 1*5,
as given in table II y p 29, of Thom (53) and in the tables
of Pearson ( 42 ) $ were stored in the array, GAM(29,4)* The
Lagrange interpolating polynomial, as described by Stark
(51), was used to perform a two-way interpolation of the
tables* The basic equation is of the form
(x - X ) (x - X )
1 o' (xq- xp 1' (xx- Xq)
such that Pj(x) = f(x©) and Pi ( x i ) = f( xi ) at the two
tabulated points Xq and x a » Tests performed by hand
calculation showed that interpolated values were in close
agreement with the theoretical distribution of both the low
and high probability ranges*
6#2*2 Potential Evago.transpir at ion*
A subroutine, MDNPIS, from the IMSJL statistical
coiapu ter package (29), was used to determine daily PE
values® A random number was selected from the array RND and
it was then transformed into a standard normal deviate z =
( x— u )/s using the above mentioned subroutine
For each
bimonthly period, a regression equation of the type
y = az + b
was used to calculate daily PE amounts* The z term refers
to the standard normal deviate corresponding to the
cumulative probability, y stands for the associated daily PE
value, and a and b stand for the standard deviation and the
mean, respectively, of the PE distribution (table 7)*
#
57
Decision to Irrigate.
Irrigation was performed when the total soil moisture
content had been depleted to 50% of its total capacity to
hold moisture* The decision to irrigate Wheat and Alfalfa
was based upon the total moisture within all six zones* The
decision to irrigate Potatoes and Sugar Beets* on the other
hand* was based upon the total moisture only within those
soil zones from which the roots were actively extracting
water* In other words* If the K - coefficient for a
particular zone during a particular crop stage was zero, the
moisture within that zone was not included in the total sum
of soil moisture® In this way* excessive irrigation during
the early crop growth stages could be avoided* Wheat and
Alfalfa* however* do not require careful irrigation
practices as do Potatoes and Sugar Beets* The generally
recommended practice for Wheat is to give the crop one
thorough irrigation prior to the time of peak consumptive
use during the middle of July* For Alfalfa* 3 ^ six inch
Irrigations are recommended during the season® Hence* it
was decided that all six zones would be used to determine
total soil moisture for Wheat and Alfalfa*
Hobbs et al C 23) had reported on the response of
various crops to several minimum allowable soil moisture
levels* Yield data* for like crops irrigated by three
different treatments, were compared* Irrigation was
performed when the soil moisture content became 1) 25%, 2)
50%, 3) 75% of the total available soil moisture* The
V
.
58
results are tabulated in table 8 lor the lour crops under
study •
TABLE 8. SUMMARY OF THE MINIMUM IRRIGATION LEVELS FOR FOUR
DIFFERENT CROPS (Hobbs et alf 23)*
Crop
Ir r igat i on
Level ( % )
Soft Wheat
50
Potatoes
75
Sugar Beets
25
Alfalfa ( 1st
year
stand )
75
Alfalfa (2nd
year
stand )
50
Ten years of crop growth was simulated with the above
criteria used to determine the irrigation day* The results
indicated that Wheat averaged about 4 irrigations per
season; Potatoes and Altalfa averaged 14 , and Sugar Beets ,
3 irrigations per season* An examination of the Irrigation
Gauge data lor the years 1869 to 1873 indicated that many
faraefs were irrigating approximately when the soil moisture
con t ent
was 50 percent ol
the
total moisture
capacity lor
all crops
« Furthermore, the
I rrigation
Gauge
recommended
1 rota 3
to 4 irrigations
per
season
lor
Wheat, 3 to 4
irrigations lor Potatoes; 3 to 5 irrigations lor Sugar Beets
and from 5 to 6 irrigations lor Allalla* Hence* the
irrigation levels lor all crops were adjusted to the 50
percent level and the model was run again* This time the
average number ol irrigations corresponded to the
recommended number*
\
'
2s. — Result a__ And Conclusions*
■2jLl_ALC±ual.. vs Simulated Data.
Before any meaningful data could be gathered from the
model, it was necessary to perform a check on the program to
verify the accuracy of both the rainfall and the potential
evapotraaspiration models* Such a check is necessary if the
soil moisture content, and thus irrigation and drainage, is
to be simulated with reasonable accuracy under weather
conditions typical of the Lethbridge area* Both the
simulated and the actual sets of data were compared by
examining averages, lengths of dry day sequences and their
respective Aj. and parameters® refers to the rate
occurrence of an event while A2 signifies the yield density
of the event* These two parameters will be explained in a
later section®
The average total simulated rainfall of 45 years for
the period from April 1st to October 31st was 11*96 inches
compared to the actual average of 12*43 inches computed from
1922 to 1966 for Lethbridge* Table 9 lists the bimonthly
averages of rainfall and potential e vapo transpi r at ion •
The author attempted to find a statistical test which
could be applied to the data to show that the actual average
monthly values did not differ significantly from the
simulated monthly values* However, because the actual
values were not derived from a theoretical formula, no
statistical test could be found* Instead, the correlation
coefficient ( r ) and the standard error of estimate ( Sxy ) of
59
1
,
60
TABLE 9. SUMMARY OF SIMULATED AND ACTUAL WEATHER DATA -
45 YEARS.
Precipitation
Actual Simulated
Interval
Mean
( inches )
St • Dev •
( i nches )
M ean
( inches )
St. Dev •
( i nches )
Apr
1-15
0.54
0.4235
0.48
0 .3086
Apr
16-30
0.85
0.7665
0.64
0.5523
May
1-15
0.88
0.8837
1.04
0.7378
May
16-31
1 . 14
1.2348
1 . 19
0.8495
J un
1-15
1 • 57
1.2158
1.45
0 .7771
Jun
16-30
1.65
1. 3676
1.43
1.0948
J uL
1-15
1 .03
1.0205
0.76
0.6166
Jul
16-31
0 . 66
0. 7900
0.82
0.6231
Aug
1-15
0 • 66
0.6534
0.66
0.4913
Aug
16-31
0.86
0. 8165
0.93
0.8943
S ep
1-15
0.83
0. 8009
0.70
0 • 6006
Sep
16-30
0.77
0.7535
0.69
0.5434
Oct
1-15
0.48
0.4791
0.56
0.5364
Oct
16-31
0.52
0.6976
0.63
0 . 563 0
Potential Evapo t ranspir a t i on
Ac t ua l
Interval Mean St. Dev.
( inches ) ( Inches )
Si muL ated
Mean St. Dev.
( inches ) ( inches )
Apr
1-15
0.93
0.4057
0.94
0 .2552
Apr
16-30
1 .35
0.5287
1.43
0.3024
May
1-15
1.75
0.4801
1.73
0.2461
May
16-31
2.25
0.4589
2.13
0.2795
J un
1-15
2.18
0.4105
2.20
0 .2128
J un
16-30
2.40
0.4009
2.40
0.2293
Jul
1-15
2.80
0.3692
2.84
0.1824
J ul
16-31
3.11
0.3869
3.07
0.2633
Aug
1-15
2.73
0.3163
2.74
0.2396
Aug
16-31
2.46
0.4430
2.49
0.2451
S ep
1-15
1.78
0. 4267
1.82
0.2585
Sep
16-30
1 .31
0.5173
1.31
0.2809
Oc t
1-15
1.14
0.4468
1.22
0.2552
Oct
16-31
0.77
0.4224
0.79
0.1805
\
•
61
the data were used to describe the disparity between the two
sets of data. The correlation coefficient is a one a sure of
the degree to which the variables vary together or a measure
of the intensity of association® The standard error of
estimate is measure of the variability of the estimated data
about the actual data® In essence? it is the standard
deviation of ¥ holding X constant®
Agreement between actual and simulated rainfall was
found to be quite goo d® The correlation coefficient was
0*81 7 7 and the standard error of estimate was 0 ® 1 1 92 ® The
standard deviations of the simulated data? in general? were
slightly lower than those of the actual data® This probably
can be attributed to the fact that the continuous functions
estimating the conditional probabilities of rainy and non—
rainy days (figure 4) were used in lieu of the actual
probabilities© The actual probabilities have more variation
than do the functions and therefore would effect higher
standard deviations in the average binmonthly rainfall of
the simulated data®
In conjunction with the total amount of bimonthly
rainfall is the distribution of consecutive periods of dry
days throughout the entire season® Figure 7 represents the
actual versus the simulated relative frequencies of the
number of consecutive days separating wet days for the
entire season® The total number of simulated dry days for
45 years was 1 ? 4 4 8 compared to the actual number of dry days
of 1?442® The longest simulated dry run was 34 days while
.
-
1
.*
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63
the longest actual dry run vas 45 days. When the model was
run for 200 years, the longest simulated dry run was found
to be 40 days. The actual data showed that dry day runs of
44 and 45 days occurred once. It was thought that had the
actual dai ly rainfall conditional probabilities (figure 4)
been employed instead of the probabilities depicted by the
polynomial equations 11 and 12, more actual values of dry
day runs and therefore average rainfall amounts would have
been obtained from the simulation model. However, this
possibility was not tested.
An alternative method of describing the rainfall
pattern was employed to compare actual and simulated data®
The season from April 1st to October 31st was divided into
43 — five day intervals® Within each time interval the
number of wet days and the total amount of precipitation
were summed over the 45 years of both the simulated and the
actual data® Figure 8a and 8b show plots of the average
number of wet days per day and the average amount of
precipitation yield per wet day for the actual and simulated
data. Good agreement exists between the actual and the
generated number of wet days per day except for the month of
May in which the simulated number of wet days slightly
overestimates the actual data. The correlation coefficient
and the standard error of estimate for figure 8a were found
to be 0*7191 and 0.0460 respectively. This Indicates that
the distribution of wet days follows the actual distribution
resonably close. The amount of simulated precipitation
• i
1/X2 (PCPN/STORM) A, (STORMS/DAY)
64
Figure 8a, Actual and simulated A^ values: - 45 years.
. Actual and simulated 1/A^ values: -
Figure 8b
45 years
65
which each storm yields* according to figure 8b, also
estimates fairly well the actual data for the entire season*
The r and the Sxy values foi this case were calculated to be
0*6533 and 0*0367 respectively* Although the simulated and
the actual data do not correlate very well* the dispersion
is very small*
Based on these comparisons it can be concluded that
the Markov Chain model combined with the incomplete gamma
function can be effectively used to simulate daily rainfall
data by way of the Monte Carlo sampling technique for the
Lethbridge area*
The bimonthly average values of potential
evapotranspiration from the simulation compares very
favorably with the actual values in Table 9* The average
total simulated PE for the entire season was 27*11 inches
compared to the actual value of 26 * 97 : a difference of 0*14
inch* The r value and the Sxy value were found to be 0*9359
and 0* 2703 respectively* The maximum discrepancy which
occurs during the periods of April 16"* 3 0 and Sept 1—15, is
0*08 inch* Since the actual PE bimonthly averages were
computed from the daily values estimated by equation 5, the
actual PE values are only estimates. Because the
theoretical distributions of PE are closer to the actual
data than the theoretical distributions of rainfall, the
di screpanc ies of the mean PE values are much less* However,
the variation of PE in the actual data is substantially
greater than the variation of PE in the simulated data as
-
.
.
.
66
noted by their respective standard deviations* Since the
conditional probability functions, as employed in the
incomplete gamma distributions of rainfall, were continuous,
the discrepancy between the standard deviations of the
simulated data and the actual data were small* The
conditional probabilities for the PE distributions ( table 5)
were calculated on a 15 day Interval basis and therefore
were discreet* This might have caused much lower dispersion
in the simulated values and therefore much lower values of
standard deviations were realized* However, this did not
seem to affect the mean values of PE*
The outputs from the weather model have shown to
compare very favorably with the actual weather data for the
Lethbridge area*
A further refinement of the K-coefficients was carried
out at this point* Ten years of simulated crop growth was
performed for each crop* The simulation season was divided
into 43 time intervals of 5 days each* Daily consumptive
use values were summed for each time interval over the 10
years of simulation* Average daily consumptive use values
for each time interval were then plotted against the
experimental curves* The K—coef f icients were adjusted until
the curves showed a good fit* Figures 9 to 12 represent the
simulated versus actual consumptive use curves and Table 10
lists the coefficient matrix for each crop*
The years 1960 to 1963 were in general warmer and
dryer than usual* Hence, the crop consumptive use values
-
'
67
TABLE 10. K - COEFFICIENTS FOR FOUR CROPS.
A ) Wheat
Dates
Ending
Soi l
Zones
1
2
3
4
5
6
May
4
• 60
.15
.05
May
24
. 55
. 30
.10
J une
12
.50
.40
.20
.10
July
5
.40
. 35
. 20
. 20
. 10
July
12
.40
. 30
.25
.20
.10
.05
July
20
.40
.30
.25
.20
.10
. 10
Aug
1
.40
.30
.25
.15
. 10
. 10
Aug
10
.45
.30
.20
.10
.05
. 05
Aug
20
.45
.30
.20
. 1 0
.05
. 05
Oct
31
.50
. 20
.15
. 1 0
.03
.02
B)
Potatoes
Dates
Soi 1
Zones
Ending
1
2
3
4
5
6
May
10
.60
. 15
. 05
J une
4
. 15
.10
.03
.02
June
25
.30
.20
. 10
.03
.02
July
10
.45
.30
.20
.1 0
. 03
. 02
Aug
1
.40
.35
. 25
.15
.10
.05
Aug
12
.45
. 35
.25
. 1 5
.05
.05
Sept
18
.40
.30
.20
. 1 0
.05
. 03
Oct
31
.60
. 15
.05
C )
Sugar Beets
Da tes
Ending
Soil
Zo nes
1
2
3
4
5
6
Apr
25
.60
.10
.05
J une
5
. 15
. 10
.05
.03
.02
J une
26
.20
.15
« 10
. 1 0
.05
.02
July
10
.25
. 20
. 15
. 10
• 10
.05
Aug
1
.35
.25
.20
.1 5
.10
. 05
Sept
1
.35
.25
. 25
.20
.10
. 10
Sept
15
.45
. 25
.20
.20
.15
. 1 0
Oc t
10
.30
. 25
.25
.20
.20
. 10
Oct
31
.60
. 15
.05
. ■ ,
TABLE 10.
con t • d
D ) Alfalfa
Dates
Ending
Soil
Zo nes
1
2
3
4
5
6
Apr
17
.60
. 15
. 05
May
24
.50
.20
. 15
.12
.08
. 05
June
18
. 50
. 25
.23
.22
.15
.10
July
3
.50
.25
.15
. 1 5
.10
.10
July
26
.50
. 25
. 15
. 15
. 10
. 10
Aug
25
.40
. 20
. 18
.15
.12
.05
Sept
17
.35
.25
.20
.15
.15
.10
Oc t
31
.50
. 20
.15
.10
.03
.02
69
were greater than the average values as presented by Hobbs
(24)» An attempt to bring the average consumptive use
values down to a more general level was made* However*
because the values were greatly unaffected by any large
change in the K— coefficients* it was extremely difficult to
force the simulated and actual consumptive use curves to
coincide perfectly without drastically changing the entire
coefficient matrices* Thus, discrepancies exist in figures
9 to 12* However, it is felt that the simulated curves
assume values between the average values and those of the
dryer years of i960 to 1963® Inevitably* the power of the
Versatile Budget to simulate daily consumptive use could
greatly be enhanced if better coefficients had been selected
both during the growing season and during the spring and
fall seasons and had there been more accurate consumptive
use curves available for each crop*
7* 2 Intermittent Processes*
A few researchers (54*63) have regarded daily rainfall
as an interasittent stochastic process® A stochastic process
is a random variable, defined in a probability space* and
dependent on time® If the random variable assumes zero
values for some positions along the time scale and greater
than zero values for all other positions, the process is
said to be intermittent# Rainfall, evaporation* runoff, and
floods are intermittent processes® Similarly, Irrigation
dates and drainage can be considered as intermittent
stochastic processes® They are both dependent on the soil
-
\
■
'
70
moisture level which in turn is a derived variable
influenced by the two stochastic variables of precipitation
and consumptive use. The amount and occurrence of drainage
are stochastic whereas only the irrigation frequencies are
stochastic. The amount of irrigation water applied to the
field is that amount required to replenish the soil moisture
deficit to field capacity at the 50 percent level. It is
therefore a fixed quantity and has no need to be considered
in this study. Because irrigation water replenishes the
soil to exactly field capacity in the atodelt any drainage
which does occur will be due to the combined effect of the
amount and the occurrence of rainfall. The definition of
drainage, therefore? as employed in this study? is that
amount of water which is in excess of field capacity on
day ( i ) •
Yevjevich (63) describes two basic parameters of an
intermittent process. They are:
A| = average number of bursts per unit time interval
A 2 = average number of bursts per unit yield
The Aa and A2 parameters are periodic functions of time with
the year as the period. The term Ag is best described by
its inverse: the average water yield per burst. Because of
dally and seasonal variations? Ax and A2 will vary with
time® However? if the time interval is very small? they can
be considered as constants within that time interval.
The two parameters were calculated according to the
following formulae.
--
*•
' -
71
X
N
2 e (i)
y=i y
5 N
X
N
Z e (i)
y=i y
N
Z x (i)
y=i y
where : e ( i )
y
X ( i )
y
N
y
i
the number of bursts within the ith time
interval and the yth year
the total water yield during the ith time
interval and the yth year
total number of years
the yth year
the ith time interval in the yth year
The interval of time over which the parameters were
calculated was chosen as 5 days as it was felt that the
parameters would vary little over this time span* The
parameters were calculated for both irrigation and drainage
as well as the actual and simulated rainfall*
7.2*1 Dral_n_ajgeJL_ XJ>| lejr.s*
Figures 13 through to 16 present the Xj and the I/X2
curves for three va riabies , two of which are drainage and
one irrigation. Drainage a, represented by the solid line*
depicts the seasonal trend of drainage when irrigation water
has been applied to the soil for the entire simulation run*
Drainage bf represented by the dotted line, depicts the
behaviour of drainage when no Irrigation water at all has
been applied to the soil for the 200 years of simulation*
The dashed line represents the behavior of the ^ 1 parameter
'
: ■ ,1 , ,
72
for irrigation* The 1/ A2 irrigation parameters maintained a
constant value of 3*5 inches for the entire season lor each
of the four crops* Therefore, they were not presented in
the figures and will not be discussed to any great length*
Figures 13 to 16 also show the seasonal behavior of the
average densities of the standard deviations for the A| and
1/ A2 curves for each crop® The average densities are simply
the standard deviations for each interval divided by the
number of days within the interval* This value, then,
represents the average standard deviation on a daily basis*
Figures 13a to 13d represent the A4 curves of drainage
for Soft Wheat, Potatoes, Sugar Beets and Alfalfa
respectively* An examination of the A4 curves for all four
crops indicate that there are two general trends, one for
Wheat and Alfalfa and one for Potatoes and Sugar Beets* The
trends are as follows*
Wheat and Alfalfa:
Potatoes
1® The maximum value of A4 occurs during the month
of June*
2* A secondary maximum occurs during September*
3m Minimum values extend through July and August*
4* There is a sharp decline at the beginning of
July •
and Sugar Beets:
1 • The peak Aj values occur at the beginning of
June and the end of May*
2* High values prevail during May and June*
.
\
. 1
.
73
3# Minimum values occur during July and August.
4. There is a gradual decrease in A4 during June.
Two trends mentioned above are common to alt four
crops. The maximum value of the curves occur during
June, and the value of A* during April 1—15 and from July
onwards are approximately equal.
The average densities of the standard deviations of
the Aa curves (figures 14a and 14b) follow the same seasonal
trends as do their respective A* curves. In other words, on
a long term basis, as the average rate of occurrence of
drainage increases, the range of the rate of occurrence
increases® It is also noted that the \ curves and their
respective standard deviations are almost identical
throughout the entire season for Wheat and Alfalfa as well
as for Potatoes and Sugar Beets. Yet, during May and June,
figures 9 and 12 show that the average consumptive use rate
of Alfalfa is much higher than for Wheat. A similar
situation exists for Potatoes and Sugar Beets during August
and September (figure 10 and 11). The At curve and their
standard deviations are almost identical, yet the
consumptive use curve for Sugar Beets shows that its average
consumptive use is higher than Potatoes® However, in both
cases, it Is noted that the slopes of the curves or the rate
of increase of CU from one day to the next is approximately
equal. This suggests that the drainage frequency is
influenced by the dai ly rate of increase of CU rather than
the absolute daily amount of CU • This fact is further
~
'
■
74
exemplified by the differences which exist between the
shallow rooted crops and the other crops© The slope of the
CU curves are much shallower for Potatoes and Sugar Beets
(figures 10 and 11) than for Wheat and Alfalfa (figures 9
and 12) during the months of May and June© Drainage,
therefore, has a much greater rate of occurrence for the
crops showing the lower rate of daily increase of CU©
The conclusions drawn from the above analyses are
listed below©
1© The daily amounts of consumptive use affect the
average rates of drainage slightly© Crops
which have higher daily consumptive use values
but equal rates of increase, will not
experience any appreciable difference in their
average drainage rates©
2® It follows from the above that drainage rates
are not influenced by the cumulative amount of
consumptive use over a period of time©
3.
The slope
or
the
rate of increase of
daily
consumptive
use
affects the drainage
ra tes
greatly©
Low
rates of increase cause
high
rates of drainage while high rates of increase
cause low drainage rates© Therefore, a crop
will not experience very many drainage problems
if its rate of daily increase in water use is
high during the early crop growth stages©
-
SIMULATED DATA
I - 1 - - — t— - 1 - 1 - t
in
o
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in
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d
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(S3HDNI) 3Sfl 3 A I id W CIS NOD AHVd
oc
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Figure 9. Comparison of actual and simulated daily consumptive use averages for wheat.
- —
0.3 5 t
76
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Figure 10. Comparison of actual and simulated daily consumptive use averages for Potatoes.
0.30
77
U
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Figure 11, Comparison of actual and simulated daily consumptive use averages for Sugar Beets.
78
<
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Figure 12, Comparison of actual and simulated daily consumptive averages for Alfalfa.
.
0.15
0.10-
D RAIN AGE o
- IRRIGATION
Figure 13a. A^ curves for Wheat.
Figure 13b. A^ curves for Alfalfa.
VI (BURSTS /DAY) Xi (BURSTS /DAY)
Figure 13c. A^ curves for Potatoes.
Figure 13d
A^ curves for Sugar Beets
(BURSTS/DAY) X, (BURSTS / DAY )
81
Figure 14a. Standard deviation of the A^ curves
for Wheat and Alfalfa.
Figure 14b. Standard deviation of the A^ curves
for Potatoes and Sugar Beets.
V\, (INCHES/BURST) '/X, (, NCHES /BURST)
82
Figure 15a. l/X^ curve for Wheat.
l/A^ curve for Alfalfa.
Figure 15b.
'
YXa (INCHES/BURST) V \2 {INCHES /BU RST)
0.6
DRAINAGE a
Figure 15c. I/A2 curves for Potatoes.
. 1/^2 curves for Sugar Beets.
Figure 15d
■
1/A, (inches /burst) i/a2 (inches /burst)
0.1 Or
0.08
- WHEAl
- AIFALFA
0.06
0.04 ■
0.021-
0.00* -
APR
MAY JUN JUL AUG SEPT OCT
Figure 16a. Standard deviation of the 1/^ curves
for Wheat and Alfalfa.
Figure 16b.
Standard deviation of the 1 / A 2 curves
for Potatoes and Sugar Beets.
'
85
7»2*2 Drainage; A? Parameters.
An examination of the 1 / a 2 curves (figures 15a to 1 5d )
indicate that the amount of drainage was much more variable
than the occurrence of drainage. No distinct seasonal
trends prevailed! however*
The 1/^2 curves maintained constant average values of
approximate ly 0*25 inches per burst throughout the months of
May and June and then gradually decreased to 0*20 inches
from July to October® During the month of June? however,
the yield per burst appears to reach average values of
between 0 « 30 and 0 » 35 inches for most of the crops except
Wheat® This apparently is the result of the fact that the
1/A2 curve fox* rainfall peaks during the same month and
therefore effects a small increase in the amount of
drai nage •
The variability of the drainage yields between the
values of 0®20 and 0. 30 inches for all of the four crops
corresponds to the average values of rainfall yield as
illustrated in figure 8b. In other words, since the amount
of drainage apparently is unaffected by consumptive use
rates, it may be assumed* therefore* that it is affected by
the amount of rainfall the soil receives® An examination of
all the 1/ A g curves yields the speculation that the drainage
curves follow the same general trend as do the precipitation
curves®
Figures 16a and 16b show the seasonal behavior of the
standard deviation for the 1/ A2 curves for all four crops®
-
\
-
.
86
Except for the months of May and June* the standard
deviations approximate each other fairly closely* A
compari sion of the average dally consumptive use curves for
Potatoes and Sugar Beets (figures 10 and 11 ) shows that the
values are approximately identical from April to June*
Consequent ly 5 it can he expected that the mean and the
standard deviations of the amount of drainage to be
approximately identical. A similar comparison for Wheat and
Alfalfa ( figures 9 and 12) shows that although there is a
large discrepancy in the consumptive use curves during May
and June, there is relatively little discrepancy in their
respective 1/ A2 curves® The discrepancy, however, does show
up in the standard deviations curves. The difference
between the consumptive use curves for Wheat and Alfalfa and
Potatoes and Sugar Beets is quite marked during May and
June. However, this difference is not reflected to any
great degree in the 1 / A 2 curves but is very pronounced in
the standard deviation curves.
From the above compari s ions , it can be concluded that
the daily consumptive use rates have much more influence in
determining the daily variability rather than the mean
drainage yields. The daily consumptive use rates determine
the variability of the drainage amounts whereas the daily
rainfall amounts will determine the upper limit of the
amount of daily drainage® Therefore, a shallow rooted crop,
because it exhibits lower consumptive use rates during May
and June,
will not exhibit higher average drainage yields
-
'
.
.
.
87
but will exhibit a higher range over which the drainage
yields can vary. In general; the long terra drainage yield
will correspond to the average rainfall amount whereas the
variability of individual drainage bursts will be determined
by the daily consumptive use rates of the crop in question.
_?*2.3 Irrigation Parameters.
The X| curves for irrigation are plotted as dashed
lines in figures 13a to 13d so that comparisons between
drainage and irrigation can be made. Examination of the
irrigation ^ curves indicate that the maximum concentration
of irrigation occurs during July and August for most of the
crops® Alfalfa? however? shows that irrigation is more or
less constant from June to September. This is probably due
to the fact that Alfalfa has the highest total consumptive
use over the entire growing season. Wheat; Potatoes? and
Sugar Beets are irrigated mainly during July and August when
the amount and the occurrence of precipitation is low? the
consumptive use rates are maximum and the chance of drainage
is minimal.
7*2.4 Drainage on Unlrri&ated Soil?.
Figures 13 and 15 also show the behaviour of the X i
and the 1/^2 parameters of drainage for crops which have not
been irrigated. No drainage problems for both Wheat and
Alfalfa existed whereas Potatoes and Sugar Beets did show
slight problems during June and part of July. The amount of
drainage water tended to average about the same with or
without irrigation. This is shown by the variation in the
.
88
1/A2 curves* Hence, it can be concluded that irrigation
water, even though it is applied at the exact instance the
soil deficit reaches the 50 percent level, contributes
subst ant ial ly to the drainage problems of irrigated soils*
?_*_3 Irrigation Lapse Times*
The probability curves presented in figures 17 to 20
represent the cumulative probability distribution of the
irrigation lapse times for each individual irrigation and
crop® An irrigation lapse time is defined as that interval
of time, in days, between the beginning of an interval to an
irrigation day® The beginning of the interval, in this
case, was selected as April 1st® The difference between the
nth irrigation and April 1st is called the lapse time®
The curves were derived in the usual manner of
constructing frequency distributions* The dates for each
individual irrigation and for each crop were stored in a
frequency table from which cumulative probabilities were
calculated according to the following plotting position®
k
2
i=l
N + 1
where:
p , _ = cumulative probability of the kth item
n = absolute frequency of the i t h item
N = total sum of all absolute frequencies
The cumulative probabilities for irrigation dates were
calculated and tabulated during the simulation run and then
plotted on normal probability paper as shown in figures 17
to 20.
-
' •
..
200
89
Figure 17. Cumulative distribution of irrigation lapse dates for Wheat.
ozz
90
(SAVCI) 3 Wl i 3SdV1
Figure 18. Cumulative distribution of irrigation lapse dates for Potatoes.
91
o
cs
CN
(SAVO) 3WII 3SdV1
Figure 19. Cumulative distribution of irrigation lapse dates for Sugar Beets.
2C0
92
o
o'
o
. o
- O
_ o
Np
O 0s
- o >*
_ o
<
CO
o
QL
O
<5
Figure 20. Cumulative distribution of irrigation lapse dates for Alfalfa,
93
TABLE 11. DESCRIPTION OF THE IRRIGATION PROBABILITY CURVES.
Irrigation Mean St. Dev.
Crop
Numbe r
N
Prob •
Date
of Date
Whea t
1
20 0
100.0
J une
25
8.9
2
20 0
100.0
July
13
6.8
3
200
100.0
J uly
26
8.2
4
194
97.0
Aug
13
14.3
5
138
69.0
Sept
12
22.1
6
21
10.5
S ept
26
21.0
7
1
0.5
Oct
8
0.0
Potatoes — 1
5
2.5
May
10
1 .2
_2
1
0.5
May
13
0.0
_ 3
1
0.5
J une
23
0.0
1
193
96.5
J uly
15
4.1
2
200
100.0
July
29
o*8
3
195
97.5
Aug
17
10.0
4
92
46 . 0
Sept
6
11 .2
5
3
1.5
Sept
15
0.6
Sugar Beets — 1
1
0.5
Apr
25
0.0
_2
1
0.5
May
28
0.0
1
198
99.0
July
16
6.3
2
200
100.0
Aug
2
5.9
3
20 0
100.0
Aug
18
7.4
4
196
98.0
Sept
6
11.6
5
129
64.5
S ept
23
11.0
6
24
12.0
Oct
4
7.6
Alfalfa 1
200
100.0
May
30
8.6
2
200
100.0
J une
20
10.6
3
20 0
100.0
J uly
9
10.0
4
200
100.0
J uly
25
10.4
5
199
99.5
Aug
12
13.9
6
178
89.0
Sept
1
17.4
7
98
49. 0
S ept
19
19.9
8
23
11.5
Sept
26
14.0
1 preseason irrigation
2 irrigation during emergence
3 irrigation between emergence and flowering
- N too small for a distribution (curve not shown)
■
f-
94
TABLE 12.
SUMMARY OF THE SM IRNOV-KOLM ORGO RO V STATISTIC FOR
THE IRRIGATION DISTRIBUTIONS.
Irri gat ion
C rop
Number
N
St at is t ic
Whea t
1
200
0.06 5
n. s •
2
200
0. 080
n. s •
3
200
0.130
*
4
194
0. 140
*
5
138
0.070
n. s •
6
21
0.155
n • s •
-
1
—
Potatoes
__ i
5
—
_ 2
1
-
_ 3
1
—
1
193
0.120
2
200
0.080
n • s •
3
195
0.075
n . s •
4
92
0.090
n. s •
-
3
—
Sugar Beets
1
„2
1
-
1
198
0.100
**
2
200
0.045
n. s •
3
200
0.070
n . s .
4
196
0.100
5
129
0.050
n. s •
6
24
0.115
n • s •
Alfalfa
1
200
0.115
**
2
200
0. 100
**
3
200
0.075
n . s .
4
200
0.070
n • s ®
5
199
0. 085
n» s •
6
178
0.125
*
7
98
0.115
n. s .
8
23
0.150
n • s •
1 preseason irrigation
2 irrigation during emergence
3 irrigation between emergence and flowering
— N too small for a distribution (curve not shown)
■ ;
’
95
With each distribution curve there is associated a
probability© For instance, for 200 of the 200 simulated
years, Wheat received at least one irrigation each year,
whereas, a total of five irrigations were performed for only
28 years* Therefore, the probability associated with the
first and the fifth irrigation are 1.0 and 0©14
respectively© Table 11 lists the curve numbers with their
respective probabilities© The table indicates that Wheat
had at least three irrigations per season. Potatoes had two
irrigations, Sugar Beets had three, and Alfalfa had four
irrigations© In the case of Potatoes and Sugar Beets, the
probabilities associated with the first irrigations are not
1©0 because of the fact that the conditions (i©e« the
number of soil zones) upon which the irrigation dates were
based were different during the early stages of growth than
in the later stages of growth© In the drier years the first
irrigation might have occurred when the roots occupied only
the first four soil zones, whereas, in the wetter seasons,
sufficient rainfall had permitted the roots to extend into
the sixth zone prior to the first irrigation© Table 11
lists the total number of irrigations, N, the irrigation
probability and the mean and standard deviation of the
irrigation dates©
According to the probabilities, most of the first
irrigations had occurred after the roots had entered the
sixth zone© This corresponds to the approximate dates of
June 25 and June 5 for Potatoes and Sugar Beets
.
-
' ' .
IRRIGATION DATES WITH PROBABILITY EQUAL OR LESS THAN - WHEAT
96
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Sept 14 Sept 19 Sept 22 Sept 26 Sept 29
98
respectively* These dates are taken from table 10* Because
there were so few irrigations prior to these dates (Potatoes
— 7 and Sugar Beets — 2) these irrigations were not plotted*
As can be seen from Figures 17 to 20, the plotted
points followed fairly straight lines on normal probability
paper* Thus, a Chi— squared test was performed to test the
assumption that the irrigation dates followed a normal
function* All were found to be highly significant*
Therefore, it was decided to perform a Stai rnov-Ko Imogorov
distribution free test on the data* Only seven of the 24
distributions were found to be significantly different*
Table 12 lists the Smi rnov— Kolmogorov statistic*
Because of the fact that an irrigator considers the
type of theoretical distribution to be irrelevant, it was
felt that the lines, as depicted by the means and standard
deviation, would serve the purpose of characterizing the
irrigation distributions* Tables 13 to 16 list the
cumulative probabilities and their respective irrigation
dates In tabular form* A broad spectrum of probability
levels was used in an attempt to consider as many different
types of weather patterns to which these computations might
be relevant* For instance, the low levels of irrigation
probabilities may be relevant during years in which the
season is exceptionally dry, whereas, the high levels may be
of greater interest during excessively wet seasons*
.
99
7*4 Summary gf results.
A summary of the sneauits are Listed below*
1* Irrigation contributes significantly to
drainage problems* Wheat and Alfalfa
experienced peak drainage rates of 0*05 and
0*03 bursts per day with irrigation and zero
drainage rates without irrigation*
Sirni liari ly f Potatoes and Sugar Beefs exhibited
peak drainage rates of 0*125 and 0*12 bursts
per day with irrigation compared to only 0*01
bursts per day without irrigation*
2* Irrigation water is mainly applied during July
and August* Dry seasons will require post¬
season irrigations* Irrigation should not be
performed during May and June for the shallow
rooted crops®
3* Drainage problems are more critical for shallow
rooted crops during the early growth stages
than during later stages* May and June have
the highest drainage rates of approximately
0*125 bursts per day with a standard deviation
of 0*20 bursts per day* In other words*
drainage problems can occur every 3 to 13 days
with an average of an 8 day return period* The
varibility of rainfall plus low consumptive use
rates during these months are the major causes
of drainage problems*
V
'
100
4* The amount of daily rainfall determines the
upper limit of the daily drainage amounts*
5* The daily consumptive use rates determine the
actual daily amounts of drainage* High
consumptive use rates will decrease drainage
yields whereas low consumptive use rates will
increase drainage yields*
6* The daily rate of increase of consumptive use
has a, profound influence on the rate of
occurrence of drainage* Wheat and Alfalfa
averaged a daily rate of increase of 0*004
inches and had a peak drainage rate of 0*05
bursts per day while Potatoes and Sugar Beets
averaged 0*003 inches but had a peak drainage
rate of 0*125 bursts per day during May and
June •
7* The average rate of drainage is affected only
slightly by the individual daily rates of
consumptive use*
8* The rate of occurrence of drainage is highest
during May and June for shallow rooted crops*
9* All crops experienced the least drainage
problems during the latter half of July* The
occurrence of drainage averaged 0*01 burst per
day (100 days per burst) with an average
deviation of 0*05 bursts per day (20 days per
burst)* The yield per drainage was about 0*20
■
.
101
Inches per burst plus or minus 0*01 inches
burst*
pe r
-2-i. C.qo civ signs*
The main objective of this study was to develop an
irrigation and a crop growth simulation model which could be
used as a tool to obtain information regarding the behaviour
of soil drainage to weather and to different crops*
Incorporated into the model were theoretical distributions
of rainfall and potential evapotranspiration and conditional
probabilities of rainy and non- rainy days* A model of
consumptive use was employed to determine crop water use
according to the water extraction patterns of the roots and
the dryness cruves of the soil* Soil moisture conditions
under four crops were thus simulated over a period of 200
years •
Actual weather records for Lehtbridge, Alberta, were
used to develope the weather model for the simulation* It
was found that both the rainfall amounts and the rainfall
probabilities were dependent upon the time of the year*
Furthermore, rainfall amounts of less than 0*10 inch
constituted & significant portion of each rainfall
distribution during the season* The rainfall probabilities
showed definate seasonal trends and were considered to be
important in simulating weather*
The weather model was run on the computer and 45 years
of simulated data were shown to compare favorably with
actual data for Lethbridge* It was concluded that the best
method of comparing actual and simulated rainfall was to
compare their and 1/A2 parameters* Although the
102
V
,
103
correlation between the actual and simulated was not
substantially hight the standard error of estimate was very
small indicating that the average fluctuation between the
actual and the simulated values was insignificant*
The Versatile Soil Moisture Budget was used to
calculate daily consumptive use* The accurracy of this
model was found to be mainly dependant upon the selection of
the K—coef f ici ents * Manipulation of the K— coefficients in
order that the proper average consumptive use curves might
be assumed proved to be ext re men ly difficult and time
consuming* On the other handy to adjust the coefficients so
that the simulated soil moisture content conincided with
actual field data proved to be rather easy* However y it was
felt that this latter method would not be sufficien tly
accurate in a Monte Carlo model which requires long term
average values* Thereforef it was concluded that the
Versatile Soil Moisture Budget can be used in a Monte Carlo
model to provide the basic crop variables provided that the
K-coeff ici ents are selected so that local long term average
consumptive use curves are simulated*
Probability distributions of irrigation lapse dates
were obtained from the model for each Irrigation and each
crop* From the slopes of the distributions, it was
concluded that at least the first two irrigation dates for
each crop were relatively uninfluenced by wet and dry years*
This is illustrated by the shallow slopes of the
distribution lines* The dates of the latter most
-
\
.
104
irrigations were stab slant! a. 1 1 y influenced by wet and dry
years® In these cases? steeper slopes indicating larger
variability are prevelent* Due to the high consumptive use
rates* the variability of irrigations and thus the slopes of
the distribution lines are minimum during June and July* In
September and October* when consumptive use Is low* rainfall
contributes more to the soil moisture thereby increasing the
variability of irrigation dates and increasing the slopes of
the distributions® An Irrigator* through the use of such
probability curves* could decide the approximate date of
irrigation provided he knows the cumulative amount of
rainfall from April 1st to the present date*
The A* and 1/ A2 curves and their respective standard
deviations provided a means of investigating the behavior of
soil drainage under the influence of irrigation* consumptive
use and rainfalls Moreover* it was shown that drainage was
a direct result of irrigation practices and not rainfall*
Little ©r no drainage was observed when irrigation practices
were not simulated* These curves also suggested that the
shallow rooted crops are more susceptable to over-irrigation
than deep rooted crops during the early growth stages* As
the crop matures the risk of damaging a crop decreases*
Furthermore, the standard dec i a lion of the 1 / A2 curves
suggest that the amount of water which drains from the soil
is dependant on crop consumptive use during the early growth
stages* It therefore was concluded that the A* and 1 / A2
curves are a valuable method of viewing the trend of both
-
■
105
drainage and rainfall®
fLa. JRec.oimngnda t i on s »
1* The accuracy of the dai iy consumptive use model
could undoubtedly to© i aprov cd with the use of K—coef f icl ent s
which could better approximate the average consumptive use
curves for each crop© Selection of the K-coefficients
should toe based upon more up to date experimentally
determined consumptive use curves* Hence , research
regarding water use for various crops is needed*
2® A better method of determining planting dates
based on rainfall, temperature , and soil moisture conditions
should toe developed in order to make the length of the
growing season a variable in accordance with the weather*
3© The length of each crop growth stage is, in
reality, affected toy the soil moisture conditions and the
weather* A method of varying each stage of growth according
to the amount of rainfall received and the potential
evapo transpiration should be developed* This ability would
enhance the effectiveness of the K— coefficients to simulate
dally consumptive use*
4® The possibility of obtaining probabilities of
the number of rainy days and the number of drainage periods
within a given time interval should be investigated* As
wellf the probability of the total amount of rainfall and
drainage within a given time period should also be obtained*
5* The simulation model should be extended to
include other major crops, different soil moisture
capacities, different soil types and different localities*
106
IQ* REFERENCES
1© Alien* I®H« and J®fi« Lambert® 1969® Dependance of
Supplemental Irrigation Scheduling on Weather
Probability and Plant Response to Soil Moisture
Regime® ASAE Paper No® 69—943#
2® flaier * VI » and G® W« Robertson® 1965# Estimation of
Latent Evaporation From Simple Weather
Observations# Can# J# Plant Sci • 43:276—284®
3® Baler* W® and G#W# Robertson# 1965# A New Versatile
Soil Moisture Budget# Can# J# Plant Sci# 46:299—
315.
4# Baler, W# 1969# Concepts of Soil Moisture
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A ton® 1971® Dynamic Simulation of Vertical
Infiltration into Unsaturated Soils® Water
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8® Bowser, W#E®, T#W® Peters and A# A* Kjcarsgaard. 1963®
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Milk Rivers Development Irrigation Project®
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9# Buras , N®, M® D« Nir and E# Alperovits# 1973#
Planning and Updating Farm Irrigation Schedules#
ASCE( IR ) 99:43-51#
10®
Campbell, W«D« 1971® Harvest Simulation
Decision Making® Unpublished M®Sc#
University of Alberta, Edmonton, Alberta,
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Thesis ,
Canada •
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107
I
! I
108
12* Co ligado , M®C» , W® Baler and W® S. Sly. 1968. Risk
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' i »
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t
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Downsview, Ontario.
39.
Morey, R.V. and J.R. Gilley. 1972. A Simulation Model
for Evaluating Irrigation Management Practises.
ASAE Paper No. 72-774.
9
O
Nielson, G.L. 1971. Hydrogeology of the Irrigation
Study Basin, Oldman River Drainage, Alberta,
Canada. Water Resources Division, Alberta Dept.
b
of Agr« , Edmonton, Alberta.
41.
Nimmer, G.L. and G.D. Bubenzer. 1972. Determining
Irrigation Potential — A Computer Model. ASAE
Paper No. 72—726.
42.
Pearson, K. 1922. Tables of the Incomplete T
Function® His Majesty's Stationary Office,
London, England.
43.
Rapp, E. and J.C. van Schaik. 1971. Water Table
Fluctuations in Glacial Till Soils as Influenced
by Irrigation® Can. Agr. Eng. 13:8—12.
44.
Rasheed, H.R* , L.G. King and J® Keller© 1970®
Sprinkler Irrigation Scheduling Based on Water and
Salt Budget. ASAE Paper No. 70—736.
45.
Richardson, C.W® and «J.T. Ritchie. 1973. Soil Water
Balance For Small Watersheds. ASAE Trans. 16:72—
77.
46.
Robertson, G.W. and R. M. Holmes. 1959. Estimating
Irrigation Water Requirements From Meteorological
Data. Publ. No. 1054, Research Branch, Can. Dept.
Agr., Ottawa, Canada.
47.
Rochester, E.W. and C.D. Busch. 1972. An Irrigation
Scheduling Model Which Incorporates Rainfall
Predictions. American Water Resources Association,
Water Resources Bull. No. 8( 3 ) : 608— 6 1 3 .
-
Ill
48® JJutledgef P.L® 1968® The Influence of the Weather on
Field Tract atoll Ity In Alberta® Unpublished M®Sc®
Thesis* University of Alberta, Edmonton, Alberta®
49® Rutledge, P®L® and D®G® Russell® 1971® Work Day
Probabilities for Tillage Operations in Alberta®
Agr® Eng® Res® Bull® No® 71—1, University of
Alberta® Edmonton, Alberta®
50® Selirio, I®S® and D®M® Brown® 1972® Estimation of
Spring Workdays from Climatological Records® Can®
Agr® Eng® 14:79—81®
51® Stark, P«A® 1970® Introduction to Numerical Methods®
Collier— MacMillian Canada Ltd®, Toronto, Ontario,
Canada® pp« 284—288®
52c Thom, B®C® 1958® A Note on the Gamma Distribution®
Mon® Weath® Rev® 68:117—122®
53® Thom, H»C« 1968® Direct and Inverse Tables of the
Gamma Distribution® Environment Data Service EDS—
2, Technical Report, Silver Spring, Maryland, USA®
54® To do i*o vie 9 P® and V® Yevjevich® 1969® Stochastic
Processes of Precipitation® Hydrology Paper No.
35, Colorado State University, Fort Collins,
Colorado, USA®
55® VanSchaik, J»C® and E® Rapp® 1970® Water Table
Behavior and Soil Moisture Content During the
Winter® Can® J® Soil Sci® 50:361—366®
56® Weaver, C®R® 1967® A Computer Algorithm for Pierce's
Soil Moisture Deficit® Ohio Research and
Development Center, Research Circular 156,
Wooster, Ohio®
57® Wilcox, J.C* 1962® Rate of Soli Drainage Following an
Irrigation® III® A New Concept of the Upper
Limit of Available Moisture® Can J® Soil Sci®
42: 122-128.
58® Wi 1 lardson , L.S® and W.L. Pope® 1963® Separation of
Evapo tr anspirat ion and Deep Percolation® ASCE( IR )
89:77-89.
59. Windsor, J.S® and V*T® Chow. 1970. A Programming
Model for Farm Irrigation Systems® Hydraulic
Engineering Series No® 23, Dept. of Civil
Engineering, University of Illinois, Illinois,
USA.
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60* Windsorf J . S • and V*T* Chow* 1971* Model lor Farm
Irrigation in Humid Areas* ASCE( IR ) 972369—385*
61* Wiser* E*H* 1966* Monte Carlo Methods Applied to
Precipitation Frequency Analysis* ASAE Trans*
9:538-542.
62* Yevjevich, V* 1972* Probability and Statistics in
Hydrology* Water Resources Publication* Fort
Collins* Colorado*
63* Yevjevich, V* 1972* Stochastic Processes in
Hydrology* Water Resources Publi cation * Fort
Collins* Colorado*
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Appendix A
The cropping model was written in FCJRTRAN — G
language. It consists of a main program and ten
subroutines* One subroutine each is devoted to the rainfall
and the P* E® models* one to the overwinter precipitation
model* and one to the cropping model* Two subroutines are
devoted to frequency tabulations while two other subroutines
initialise the constants for the entire model and set
several variables to their initial values at the start of
each year*
A listing of the
pages* Flow charts
also presented*
program is given on the following
of the more important subroutines are
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MAIN PROGRAM
Initialize the ending dates of each month and
each total monthly PE value.
Input total number of years to be simulated and
crop specifications.
Initialize summers to zero.
Do for each year to be simulated.
Initialize summers and counters to zero.
Initialize month, bimonth and week numbers to 1.
Generate 430 pseudo-random numbers for the entire
season.
If first random number is less than the probability
of rainfall for March 31st, R = 2 otherwise R - 1.
Update the number of the month.
Update the number of the current week.
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Update the bimonthly number.
Calculate today's rainfall amount.
Calculate today's potential evapotranspiration.
Sum total rainfall and PE for each month.
Calculate the difference between today's PE
and the monthly average daily PE value.
Do for each crop.
If today is
update crop
equal to the last day of a crop stage,
stage number for the crop.
Calculate crop consumptive use for today and update
the soil moisture content.
Sum total crop data for each crop each month.
Sum crop data values for this season.
Sum the total rainfall and PE for this season.
Sum and sum the squares of the total monthly
and seasonal rainfall and PE values for each season.
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Sum and sum the
data values for
squares of
each year.
the total seusonal crop
Output monthly and seasonal totals of rainfall, PE
and crop data for the current year.
Sum Lhe total monthly values of the crop data for
each year.
Calculate oversintcr precipitation and drainage for
each crop and update the soil moisture content.
Calculate mean and standard deviation of the monthly
and seasonal rainfall and PE amounts.
Calculate the mean and the standard deviation of the
total annual crop data values.
Output the mean and the standard deviation of the
monthly and annual values of rainfall and PE and the
annual values of the crop data.
Calculate the total monthly means of the crop data.
Output total monthly values of the crop data for each
crop.
Calculate and output the 1 and 2 parameters for
rainfall and for each crop.
Calculate and output a frequency table of the dates of
each individual irrigation ( 1 to 14 ) for each crop.
Calculate and output a frequency table of irrigation
dates for each crop.
117
Calculate and output a frequency table of drainage
dates for each crop.
Calculate and output a frequency table of runoff
dates for each crop.
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suuroutine
RAIN
Subroutine to determine daily rainfall values.
Reset day number of year in relation to April 1st.
Calculate probability of a non-rainy day occurring today.
Select the next sequential random number.
Is today dry?
Adjust RN for a mixed distribution.
Select a and g values of the theoretical gamma distribution
for rainfall.
Set R = 2 indicating rain today.
If alfa is less than 1.0.
Select maximum column number 4 and maximum alfa value
of 1.0
Select column number 3 and alfa value 0.5
Do for each row of the gamma table.
Select row number of the gamma table by comparing F to
the probability in the gamma table.
If F is less than GAM, exit the do loop.
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Select the. row and column which lie on the opposite
side of the F probability and the alfa value respectively.
Calculate rainfall by a 2-way interpolation of the rows
and columns selected above. (Legrange method. Stark, 51)
Is rainfall less than or equal to zero?
If length of consequtive dry days is greater than zero,
tabulate the frequency of N.
Set length of dry runs to zero.
Sum rainfall amounts on a bimonthly basis.
Sum the total number of rainy days and the total amount
of rain on a weekly basis.
Set rdinfall to zero.
Set R to 1 indicating no rain today.
Sum the number of consequtive non-rainy days.
If today is not October 31st.
Tabulate frequency of last dry run
Reset dry run to zero
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If today is not the last day of the current bimonthly
period.
Sum and sum the squares of the total monthly rainfall
amounts .
Reset summation to zero.
If today is not the last day of the current 5-day period.
Sum and sum the squares of the number of rainy days in the
last 5-day period.
Sum and sum the squares of the total amount of rainfall
in the last 5-day period.
Reset summers to zero.
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subroutine
EVAPO
Subroutine to determine daily potential evapotrnnspiration.
Reset day number o£ year with respect to April 1st.
Probability of zero incites of PE occurring today.
Select next sequential random number.
Does today experience zero inches of PE?
Adjust RN for a mixed distribution.
Calculate standard deviate (X) of probability F
IMSL statistical package (29).
Calculate today's PE value given the mean and the standard
deviation of the frequency distribution of the current
bimonthly period.
If today's PE is zero or less.
Sum daily PE amounts on a bimonthly basis.
Set today's PE to zero.
If today is not the last day of the current bimonthly
period.
Sum and sum the squares of the total PE amount in the
last bimonthly period.
Reset summer to zero.
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SUBROUTINE
WINTER
Subroutine to calculate overwinter precipitation, overwinter drainage
and to update the soil moisture content for April 1st cf tin* next year. The
subroutine also outputs statistics for overwinter drainage.
Select last random number generated for this year.
Calculate standard deviate X of F.
IMSL statistical package (29).
Calculate overwinter precipitation.
Do for crops 1 to 4.
Set summer to zero.
Set drainage equal to precipitation.
Do for each soil zone.
Add drainage from zone 1-1 to zone I.
No drainage into zone I + 1.
Calculate drainage into zone I + 1.
Set zone I to capacity.
Sum water conLent in all 6 zones.
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Note total water content in all 6 zones.
Select minimum and maximum values of drainage.
Sum and sum the squares of overwinter drainage.
If the current year is not the last year to be
simulated - return.
Output table headings.
Do for each crop.
Calculate mean value of overwinter drainage.
Calculate standard deviation of overwinter drainage.
124
SUUROUTINK
SOIL
Subroutine which utilizes the Versatile Soil Moisture Budget to
1) calculate daily consumptive use values
2) update the soil moisture status for each soil zone
3) make irrigation decisions
Reset crop data to zero.
Today's precipitation = infiltration into the soil.
Note current crop growth stage number II.
Do for each soil zone.
Calculate soil moisture content (in %) for zone I.
Calculate the W term in the VB model.
Note the K - coefficient for zone I, crop growth stage II,
and crop IC.
If today occurs during 1st or 2nd crop growth stage
or if current soil zone I is 1 (top zone).
Adjust K - coefficient for soil dryness in the above layers.
Set values to zero.
Select coefficient from Z - table according to the
soil moisture content (in %) in zone I.
Calculate consumptive use from zone I.
Store consumptive use values.
Note total moisture in all 6 soil zones.
If crop is Wheat or Alfalfa.
If Potatoes and Sugar Beet roots have penetrated into
the 6th soil zone.
Note deepest zone into which roots have penetrated.
Do for zones no. 1 to LSTG.
Sum moisture in zones 1 to IT,.
126
Calculate soil moisture percent of only those zones
where roots exist.
If soil moisture content is less than 50%.
No irrigation water today.
Calculate amount of irrigation water to be applied.
Update current irrigation count.
If rainfall is less than 1.0 inch.
Calculate water infiltration into the soil.
Runoff = rainfai 1 - infiltration
Total infiltration = irrigation + infiltration
Do for each soil zone.
Add drainage from zone 1-1 and subtract consumptive
use from zone I.
127
No drainage from zone I.
Calculate drainage into zone I-fl.
Set zone I to capacity.
Sum water content in all zones.
Note total water content in all 6 zones.
Store all crop data in array AMOUNT.
Update frequencies of occurrences of the dates of
each crop data.
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END
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MAIN program:
200 YEARS SIMULATION OF WEATHER AND CROP GROWTH
VARIABLE DESC
CRPSTG
WEEK
FREQ
STAGE
WK
MO
MONTH
R
MODAY
COEF
CONTNT
CAPAC
YEAR
DAY
PEMEAN
IYR
PRECIP
CRMSUM
CRASUM
AMOUNT
PPT
PPE
IC
IT
TSUMPT
ATOTAL
MSUM
RIPTION
ARRAY CONTAINING ENDING DATES FOR EACH CROPSTAGE
VECTOR OF ENDING DATES OF CONSEQUTIVE 5-DAY PERIODS
VECTOR OF ENDING DATES OF EACH BIMONTHLY PERIOD
CROP STAGE NUMBER
WEEK NUMBER
BIMONTHLY NUMBER
MONTH NUMBER
PREVIOUS DAY INDICATOR (1 - DRY, 2 - WET)
VECTOR OF ENDING DATES CF EACH MONTH
ARRAY CONTAINING K-COEFF IC I ENT MATRIX FOR EACH CROP
CURRENT SOIL MOISTURE CONTENT FOR EACH SOIL ZONE
SOIL SOI STURE CAPACITY OF EACH ZONE
YEAR NUMBER
DAY NUMBER IN THE YEAR (91 TO 304)
VECTOR CONTAINING AVERAGE DAILY PE FOR EACH MONTH
TOTAL NUMBER OF YEARS TO BE SIMULATED
MONTHLY AND ANNUAL TOTALS OF RAINFALL AND PE
SUMMATION OF MONTHLY CROP DATA
SUMMATION OF A'NNUAL CROP DATA
VECTOR CONTAINING CROP DATA VALUES
DAILY RAINFALL VALUE (IN.)
DAILY PE VALUE (IN)
CROP NUMBER
1. WHEAT
2. POTATOES
3. SUGAR BEETS
4. ALFALFA
CROP DATA ITEM NUMBER
1. IRRIGATION QUANTITY
2. DRAINAGE
3. DEFICIT
4. CU
5. RUNOFF
MEAN AND ST. DEV. OF MONTHLY AND ANNUAL RAINFALL AND PE TOTALS
MEAN AND ST. DEV. OF ANNUAL CROP DATA VALUES
TOTAL SUM OF CROP DATA VALUES FOR EACH MONTH
REAL MSUM( 5 , 4 , 7 ) , TSUMPT( 8,2,2 ) , ATOTAL( 5,4,2) ,PEMEAN( 7 ),AVG< 5 ),CROP
1*8! 4)
INTEGER CRPSTG, WEEK, FREQ, STAGE, DAY, WK,R, YEAR, MODAY( 7 )
COMMON / BUDG/ COEF( 6,10,4 ) ,TABLE( 100 ) ,CRMSUM( 5,4,7), WEEK( 43 ) , CRASUM
1(5,4 ) , CCNTNT( 7,4 ) ,CAPAC( 7 ) ,CRPSTG( 10,4), PRECIP( 8,2 ) , FREQ( 14 ), STAGE
2( 4 ), AMOUNT! 5), IRRNO( 4 ) , PPT, PPE , DA Y , WK ,MO , PED I F , R , Y EAR , I C
DATA CROP/ 'WHEAT* , ‘POTATOES* , • SUG BE ET* ,• ALFALFA • /
DATA MODAY/ 120, 151 ,181 ,212,2 43 ,273, 304/ , ASTRI K/ ' ****•/
DATA PEMEAN/ 0.076, 0.1 29, 0.1 53, 0.191, 0.167,0. 103,0.062/
C INPUT NUMBER OF YEARS TO BE SIMULATED
READ! 4,1 ) IYR
1 FORMAT! 13)
C INPUT CROP SPECIFICATIONS
READ( 5,2) TABLE, COEF, CONTNT, CAPAC
FORMAT! 10( 10F5.2/ ),40( 6F4.2/ ),( 7F5.2 ) )
INITIALIZE SYSTEM COUNTERS
CALL INTIAL
C SET ANNUAL SUMMATIONS TO ZERO
DO 100 K=l,7
DO 100 J=l , 4
DO 100 1=1,5
100 MSUM! I ,J ,K ) = 0.00
DO 101 1=1,2
DO 101 J=1 ,4
DO 101 K=1 , 5
101 ATOTAL! K»J , I ) = 0.00
DO 102 1=1,2
DO 102 J=l,2
DO 102 K= 1,8
102 TSUMPT! K,J , I )=0.00
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C BEGIN SIMULATION CF SEASON
C
DO 3000 YEAR= 1 , IYB
C RESET ANNUAL COUNTERS AND SUMMATIONS
CALL BEGIN
MONTH= 1
MO=l
' WK=1
C OBTAIN PSEUDO-RANDOM NUMBERS FOR ENTIRE YEAR
CALL RANDOM! RN)
R=1
C IF 1ST RANDOM NUMBER LESS THAN THE PROBABILITY OF
C RAINFALL ON MARCH 31ST
IF( RN.LE. 0.2444 )R=2
C
C BEGIN DAILY SIMULATION
C
DO 2000 DAY=91,304
C UPDATE MONTHLY, WEEKLY AND BIMONTHLY COUNTERS
IF( DAY • GT. MOD A Y( MONTH) )MONTH=MONTH+ 1
IF( DAY. GT. WEEK( WK ) )WK=WK+1
I F( DAY .GT.FREQ! MO ) )MO=MO+l
C CALCULATE RAINFALL AND PE FOR TODAY
CALL RAIN
CALL EVAPO
C SUM DAILY RAINFALL AND PE FOR EACH MONTH
IF! PPT.GT. 0.0 0 )PRECIP( MONTH, 1 >=PRECIP( MONTH, 1 )+PPT
PREC I P( MONTH, 2 )=PREC IP( MONTH ,2 >+P PE
PEDI F=PP E— PEMEAN( MONTH )
C
C CALCULATE CU AND SOIL MOISTURE FOR EACH CROP
C
DO 2000 IC=1 , 4
C UPDATE CROP STAGE NUMBER
I F( DAY. GE. CRPSTG( STAGE( IC ), IC ) )STAGE( IC )=STAGE! IC )+l
C CALCULATE CU AND UPDATE SOIL M.CM FOR TODAY
CALL SOIL
C SUM DAILY CROP DATA FOR EACH MONTH
DO 1200 IT= 1,5
1200 CRMSUM! IT, IC, MONTH )=CRMSUM( I T, IC , MONTH )+ AMOUNT! IT )
2000 CONTINUE
C SUM MONTHLY CROP DATA FOR EACH SEASON
DO 200 1=1,7
DO 200 IC=1 , 4
DO 200 IT= 1,5
200 CRASUM! IT , IC ) =CR ASUM( IT, IC )+CRMSUM! IT, IC, I )
C SUM DAILY RAINFALL AND PE OVER ENTIRE SEASON
DO 201 11=1,2
DO 201 1=1,7
201 PREC I P! 8,11 )=PREC IPC 8,11 >+PRECIP( 1, 1 1 )
C SUM TOTAL MONTHLY RAINFALL AND PE FOR EACH SEASON
DO 205 J=1 ,2
DO 205 1=1,8
TSUMPT! I , J , 1 )=TSUMPT( I , J, 1 )+PRECIP( I , J)
205 TSUMPT! I , J,2 )=TSUMPT( I , J ,2 )+PRECIP( I , J )*PRECIP( I , J )
C SUM ANNUAL CROP DATA FOR EACH SEASON
DO 206 IC= 1,4
DO 206 IT=1 , 5
ATOTAL! IT, IC, 1 )=ATOTALC IT,IC, 1 )+CRASUM( IT, IC )
206 ATOTAL! IT, IC,2 )=ATOTAL( IT , IC , 2 )+CRAS UM( IT, IC )*CRASUM! IT, IC)
C OUTPUT TOTAL MONTHLY RAINFALL AND PE
WRITE! 1,3) ! ! PREC IP! I , J ) , 1 = 1 , 8 ), J=l, 2 )
3 FORMAT! 7F6 .2, F8.2, • - ' , 7F6 • 2 , F8 • 2 )
C OUTPUT TOTAL ANNUAL CROP DATA
WRITE! 2, 4 ) CRASUM
4 FORMAT! 20F7. 2 )
C SUM MONTHLY CROP DATA FOR EACH SEASON
DO 260 MO= 1,7
DO 260 IC=1,4
DO 260 IT= 1,5
MS UM ! IT , IC , MO )=MSUM! IT , I C , MO )+CRMSUM! IT, IC.MO)
260
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C CALCULATE OVERWINTER PRECIPITATION
CALL WINTER( IYR)
3000 CONTINUE
Y=FLOAT( YEAR )
C CALCULATE MEAN AND ST. DEV. FOR RAINFALL AND PE
DO 310 I T= 1,2
DO 310 M=l,8
SS=TSUMPT( M, IT , 1 )*TSUMPTC It, IT, 1 )
TSUMPTC M,IT, 2 )=SQRTC ( TSUMPT( M,IT,2 )-SS/Y )/C Y-1.00 ) )
310 TSUMPT( M , IT , 1 )=TSUMPT( M,IT, 1 )/Y
C CALCULATE MEAN AND ST. DEV. FOR CROP DATA
DO 320 IC= 1,4
DO 320 IT=1,5
SS= ATOTALC IT, IC, 1 )*ATOTALC IT, IC, 1 )
ATOT AL( IT , IC, 2 )=SQRT( ( ATOTALC IT, IC, 2 )-SS/Y )/C Y-1.00 ) )
320 ATOTALC IT, IC, 1 )= ATOT AL( IT, IC, 1 )/Y
C OUTPUT MEANS AND ST. DEV.
WRITEC 1,6 ) < ASTRIK,K=1 , 103 >,TSUMPT
6 FORMATC 103A1/C 7F6.2,F8.2, • - • , 7F6. 2 , F8. 2 ) )
WR I TEC 2,7) C ASTRIK,K = 1, 140),ATOTAL
7 FORMATC 140A1/C 20F7 .2 ) )
C OUTPUT MONTHLY AVERAGES FOR CROP DATA
WRITEC 6, 9 )
9 FORMATC ' 1* ,30X , 'MONTHLY AVERAGES FOR!-*)
DO 360 IC=1 ,4
WRITEC 6,10) CROPCIC)
10 FORMATC , 12X, • CROP . • , A8, 5X, ' MO' , 1 OX , ' I RR * , 6X, 'DR* ,5X,
1,4X,'C.U. RUNOFF')
350
360
11
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DO 360 MO= 1,7
DO 35 0 I T= 1 , 5
AVGC IT)=MSUMC IT,IC,MO)/Y
WRITEC 6,11) MO, A V G
FORMATC' ' ,36X,I2,5X,5F8.2)
CALCULATE Yl AND Y2 PARAMETERS
CALL PARMTRC YEAR )
CALCULATE FREQUENCY DISTRIBUTIONS
1. DATES OF EACH IRRIGATION CIST,
2. IRRIGATION DATES COLLECTIVELY
3. DRAINAGE DATES
4. RUNOFF DATES
CALL ITABLEC 1 , 14, ' DATES * , YEAR )
CALL ITABLEC 15,15, * IR DATES', YEAR)
CALL ITABLEC 16,16, 'DR DATES', YEAR)
CALL ITABLEC 17 , 17 , 'RUNOFF • , YEAR )
STOP
END
2ND,
3RD,
ETC. )
'DEF*
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SUBROUTINE INTIAL
SUBROUTINE TO INITIALIZE SUMMERS AND COUNTERS TO ZERO
VARIABLE DESCRIPTION
F< I , J » K }
AMT( I ,J ,K )
NUMBER
IRRNO
STAGE
CRASUM
CRMSUM
PRECIP
FREQUENCY
DATES AND
I = 1-14
= 15
= 16
= 17
1-4
1 -
TABULATION OF IRRIGATION DATES,
RUNOFF DATES FOR EACH CROP
DRAINAGE
J -
I =
WEEKLY
K =
43
IRRIGATION NUMBER DURING A SEASON
IRRIGATION DATES TAKEN COLLECTIVELY
DRAINAGE DATES
RUNOFF DATES
CROP NUMBER
WEEK NUMBER
SUMMATION OF IRRIGATION AND DRAINGE
1
2-5
6-10
10-13
1
2
1-43
NUMBER
RAINFALL
DRAINAGE FOR EACH WEEK AND CROP
IRRIGATION FOR EACH WEEK AND CROP
CU FOR EACH WEEK AND CROP
SUM
SUM OF SQUARES
WEEK NUMBER
OF OCCURRENCES OF IRRIGATION AND DRAINAGE
J =
I =
TOTAL
FOR EACH WEEKLY PERIOD (SUBSCRIPTS SAME AS ABOVE)
IRRIGATION NUMBER
NUMBER OF CURRENT CROP GROWTH STAGE
SUMMATION OF ANNUAL CROP DATA
SUMMATION OF MONTHLY CROP DATA
MONTHLY AND ANNUAL TOTALS OF RAINFALL AND PE
INTEGER CRPSTG , WEEK, FREQ , STAGE, DA Y , WK ,R , YEAR , F*2 ,SEQ
COMMON / BUD G/ COEF< 6,10,4), TABLE( 100), CRMSUM( 5,4,7), WEEK( 43 ) , CRASUM
1(5,4 ) , CCNTNT( 7 ,4 ) ,CAPAC( 7 ),CRFSTG( 10,4), PRECIP( 8,2 ) ,FREQ( 14 ), STAGE
2( 4 ) , AMCUNT( 5 ), IR fi NC( 4 ) , PPT, PPE , DAY , WK ,MO , PED I F , R , Y EAR , I C
COMMON /PARM/ AMT! 43,2,13 ) , NUMBER( 43 , 2 , 9 ) , PT( 1 4 , 2 , 2 ),SEQ( 100 )
COMMON F( 214,4,17 )
RESET SIMULATION COUNTERS
DO 1 1=1,17
DO 1 J= 1,4
DO 1 K = 1 ,214
F( K , J , I )=0 0
DO 7 1=1,13
DO 7 J= 1 , 2
DO 7 K= 1 , 43
AMT( K,J, I )=0 • 00
DO 8 1=1,9
DO 8 J= 1 , 2
DO 8 K= 1 ,43
NUMBER! K, J , I )=000
DO 9 1=1,2
DO 9 J= 1 , 2
DO 9 K=1 ,14
PT( K , J , I )=0.00
DO 10 1=1,100
SEQ( I )=00
RETURN
RESET SEASONAL COUNTERS
ENTRY BEGIN
DO 5 1=1,4
IRKNO( I )=00
STAGE! I )*1
DO 5 J = 1 , 5
CRASUM! J, I )=0 .00
DO 5 K= 1,7
CRMSUM! J, I , K )=0.00
DO 6 J= 1 ,2
DO 6 1=1,8
PRECIP! I ,J ) = 0 • 00
RETURN
END
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SUBROUTINE RAIN
C
C SUBROUTINE TO DETERMINE DAILY RAINFALL
C
C VARIABLE DESCRIPTION
C PWW
c
C QWW
C GAM
C ALFA
C BETA
C PT
C RSUH
C NSOMWK
C ASUMWK
C SEQ
C RN
C
INTEGER CRPSTG, WEEK, FREQ, STAGE , DA Y , WK , R , YEAR » SEQ
COMMON / BUDG/COEF( 6, 10, 4 I, TABLE! 100 ) , CRMSUMC 5,4,7), WEEK! 43 ) ,CRASUM
1( 5, 4 ), CONTNT! 7,4 ) ,CAPAC( 7 ), CRPSTG! 10, 4), PR EC IP! 8,2 ) , FREQ! 1 4 ), STAGE
2! 4 ), AMOUNT! 5 ) , IR RNO! 4 ) , PPT, PPE , DAY , WK , MO , PEDI F , R , Y EAR , I C
COMMON / PRO B/ PWW! 43,2 ) ,PE! 14, 2, 2 ), GAM! 29,4 ), ALFA! 14,2 ) , BETA! 14,2 ),
1PP! 14,2 )
COMMON /PARM/AMT! 43,2, 13), NUMBER! 43, 2,9 ) , PT! 14,2,2 ) , SEQ! 100 )
COMMON /RNDM/RDUM, RND! 2,214),RNW
DATA RSUM, ASUMWK, NSUMWK/2*0. 00, 00/,N/00/
ID Y=DAY— 90
C PROB. OF NON— RAINY DAY OCCURRING TODAY
QWW=1. 00000-PWW! WK,R)
C SELECT RANDOM NUMBER
RN = RND{ 1 , IDY )
C IF TODAY IS DRY
IF! RN.LE.QWW)GO TO 1
C ADJUST RN FOR MIXED DISTRIBUTION
F=! RN-QWW )/PWW( WK,R )
C SELECT ALFA AND BETA VALUES
A= ALFA! MO,R )
B=BETA! MO, R )
R=2
C SELECT COLUMNS TO BE INTERPOLATED
IF! A.LT. 1.0 )GO TO 2
JJ=4
AL=1 .0
GO TO 3
2 JJ = 3
AL= 0 . 5
CALCULATE TODAYS RAINFALL - LEGRANGE INTERPOLATION, STARK !51)
DO 4 11 = 1,29
IF! F. LT. GAM! 1 1 , 1 ) >GO TO 5
CONTINUE
1=11-1
J=J J-l
Y2=( A-AL )*2.00
Y 1 = 1 . 0-Y2
X2=! F-GAM! 1,1))/! GAM! 11,1 )— GAM! 1,1))
X 1 = 1 ® 0— X2
PPT=( ! GAM! I,J >*Xl+GAM! 1 1 , J )* X2 )* Y 1+! GAM! I , J J ) *X 1+G AM! II , JJ )*X2 )*Y2
1 )*B
IF! PPT. LE. 0.00 )GO TO 1
C TABULATE LENGTH OF CONSEQUTIVE DRY DAY RUNS
IF! N.GT.00 >SEQ! N ) = £EQ< N )+l
N = 00
RSUM=RSUM+PPT
NSUMWK=NSUMWK+1
AS UMWK= ASUMWK +PPT
GO TO 6
C IF NO RAINFALL
1 PPT=0 • 0
R= 1
N=N+1
IF! DAY. LT. 304 )GO TO 6
CONDITIONAL PROBABILITY OF RAINFALL FOR EACH WEEK
GIVEN THAT THE PREVIOUS DAY WAS DRY! R=1 } OR WET! R=2 )
PROBABILITY OF A NON— RAIN Y DAY
INVERSE GAMMA VALUES AS PER TABLE II, THOM !53)
ALFA VALUES OF THE ESTIMATED GAMMA FUNCTION FOR RAINFALL
BETA VALUES OF THE ESTIMATED GAMMA FUNCTION FOR RAINFALL
BIMONTHLY SUM AND SUM OF SQUARES FOR PRECIPITATION AND PE
BIMONTHLY SUMMATION OF RAINFALL
WEEKLY SUMMATION OF THE NUMBER OF RAINY DAYS
WEEKLY SUMMATION OF RAINFALL AMOUNTS
TABULATION OF CONSEQUTIVE NON- RAINY DAY RUNS
PSEUDO-RANDOM NUMBER
-
;>
■
...
134
SEQ! N > “ SECH N) + l
N=QO
C SUM BIMONTHLY RAINFALL
6 IF( DAY* NE» FREQ( MO ) )GO TO 10
PT( MO , 1 , 1 » = PT( MO , 1 , 1 l+RSUM
PT( MO, 2, 1 )=PT ( MO, 2,1 )+RSUM*RSUM
RSUM=0.00
C SUM WEEKLY RAINFALL AMOUNTS AND OCCURRENCES
10 IF( DAY.NE. WEEK( WK ) ) RETURN
NUMBER! WK, 1 , 1 )=NUMBER( WK , 1 1 J+NSUMWK
NUMBER! WK, 2,1 ) = NUMBER! WK , 2, 1 )+NSUMWK *NSUMWK
AMT! WK ,1,1 )=AMT! WK,1 ,1 )+ASUMWK
AMT! WK , 2 , 1 )=AMT! WK,2, 1 )+ASUMWK*ASUMWK
NSUMWK=0
ASUMWK=0.00
RETURN
END
"
135
SUBROUT INE EVAPO
C
C
C
C
c
c
c
c
c
SUBROUTINE TO DETERMINE DAILY POTENTIAL EVAPOTHANSP IR ATION
PP
QWW
RN
PE
PSUM
SUMMATION OF DAILY PE
CONDITIONAL PROBABILITIES OF PE OCCURRING
PROBABILITY OF NO PE OCCURRING
RANDOM NUMBER
MEAN AND STANDARD DEVIATION FOR EACH PE DISTRIBUTION
INTEGER CRPSTG , WEEK, FREQ , STAGE , DAY , WK , R , YEAR , SEQ
COMMON /BUDG/COEFI 6,10,4 ) , TABLE! 100 ) , CRMSUMt 5,4,7), WEEK( 43 ) „CRASUM
1(5,4 ),CCNTNT( 7,4) ,CAPAC( 7 >,CRPSTG( 1 0 , 4 ) , PSEC I P( 8,2 ) ,FREQ( 14 ), STAGE
2( 4 ), AMOUNT! 5 ) , IRRNG< 4 ) , PPT, PPE , DAY, WK, MO , PEDI F , R , Y EAR , IC
COMMON /PROB/PWW( 43,2 ) ,PE( 14, 2, 2 ),GAM( 29,4 ), ALFA( 14,2 ), BETA! 14,2 ),
1PP( 14,2 )
COMMON / PARM/ AMT( 43,2, 13),NUMBER( 43, 2,9 ),PT( 14,2,2 ) , SEQ{ 100 )
COMMON /RNDM/RDUM, RND( 2,214), RNW
DATA PSUM/ 0.00/
IDY=DAY— 90
C PROBABILITY OF NO PE OCCURRING TODAY
QWW=1.G000— PP( MO,R)
RN = RND( 2, IDY )
C IF NO PE OCCURS TODAY
IF< RN. LE.QWW )GO TO 7
C ADJUST RN FOR MIXED DISTRIBUTION
F=( RN-QWW )/PP( MO, R )
C CALCULATE STANDARD VARIATE AND PE FOR TODAY
CALL MDNRIS( F,X, IER)
PPE=PE( MO, 2, R )*X+PE( MO, 1 ,R )
C SUM DAILY PE
IF( PPE. LE. 0.00 )GO TO 7
PSUM=PSUM-*-PPE
GO TO 8
7 PPE=0 .00
C SUM DAILY PE FOR EACH WEEK
8 IF( DAY.NE • FREQ( MO ) ) RETURN
PT( MO, 1 , 2 ) = PT( MO, 1 ,2 )+ PSUM
PT( MO, 2, 2 )=PT( MO, 2, 2 )+PSUM*PSUM
PSUM=0 .00
RETURN
END
-
\
.
136
SUBROUTINE WINTER! IYK)
C
C
C
SUBROUTINE TO CALCULATE TOTAL OVERWINTER PRECIPITATION
RANDOM NUMBER FOR OVERWINTER PRECIPITATION
OVERWINTER PRECIPITATION
MINIMUM DRAINAGE OVER 200 YEARS
MAXIMUM DRAINAGE OVER 200 YEARS
OVERWINTER DRAINAGE DUE TO WPPT
SUM AND SUM OF SQUARES OF OVERWINTER PRECIPITATION
c
VARIABLE
c
RNW
c
WPPT
c
MIN
c
MAX
c
DR
c
MEAN
c
INTEGER CRPSTG, WEEK, FREQ, STAGE , DAY , WK , R , YEAR
REAL MEAN! 4,2) ,MAX( 4), MINI 4)
COMMON /BUDG/COEF! 6, 10,4 ), TABLE! 100 ) ,CRMSUM( 5,4,7 ) ,WEEK( 43),CRASUM
1( 5,4) ,CONTNT( 7,4), CAPAC( 7 ),CRPSTG( 10,4), PRECIP! 8,2 ) , F REQ( 14), STAGE
2( 4 ), AMOUNT! 5 ) , IRRNO! 4 ) , PPT , PPE , DAY , WK, MO, PE DIF, R, YEAR, IC
COMMON / RNDM/ RDUM , RND! 2,214 ) , RNW
.DATA MEAN, M AX , MIN/ 12*0.00,4* 1000. 0/
C CALCULATE OVERWINTER PRECIPITATION ! MON TE CARLO SAMPLING)
F=RNW
CALL MDNRIS! F , X , IER )
WFPT=! 1.242474*X+4. 350465 >*0.350000
IF! WPPT.LE.0. 00 ) WPPT-0. 00
C CALCULATE SOIL MOISTURE CONTENT FOR EACH CROP NEXT SPRING
DO 32 ICP=1,4
SUM=0.00
DR = WPPT
DO 30 1=1,6
CONTNT! I , I CP )=CONTNT! I , ICP >+DR
IF! CONTNT! I,ICP).GT .CAP AC! I ) )GO TO 3 1
DR=0 .00
GO TO 30
31 DR=CONTNT! I , ICP )—CAPAC! I )
CONTNT! I, ICP )=CAPAC! I )
30 SUM=SUM+CONTNT! I , ICP )
CONTNT! 7 , ICP )=SUM
IF! DR .LT .MIN! ICP ) )MIN! ICP )=DR
IF! DR. GT. MAX! ICP ) )MAX! ICP )=DR
MEAN! ICP , 1 )=MEAN! ICP , 1 )+DR
MEAN! ICP ,2 ) = MEAN! ICP , 2 )+DR*DR
32 CONTINUE
C OUTPUT MEAN AND ST. DEV. OF OVERWINTER DRAINAGE
IF! YEAR. NE. IYR )RETURN
WRITE! 6,1)
WRITE! 6, 2 )
1 FORMAT! *1 OVERWINTER DRAINAGE FOR EACH CROP')
2 FORMAT! ,30X, ' CROP MAXIMUM MINIMUM MEAN ST DEV')
Y=FLO AT! YEAR )
DO 40 1=1,4
XM=MEAN! 1 , 1 )/ Y
VAR = ! MEAN! 1,2 )— MEAN! 1,1 )*MEAN! 1,1 )/Y )/! Y-1.0 )
SD=0 .00
IF! VAR.GT.0.0 0 )SD=SQRT! VAR )
40 WR I TE! 6,3) I , MAX! I), MIN! I ),XM,SD
3 FORMAT! *0' ,30X,I5,3F10.2,F10.6>
RETURN
END
.
■ • .
137
c
c
c
c
c
c
c
c
c
c
SUBROUTINE RANDOM(BN)
SUBROUTINE TO OBTAIN PSEUDO-RANDOM NUMBERS
VARIABLE DESCRIPTION
RR VECTOR CONTAINING 430 PSEUDO-RANDOM NUMBERS FOR ONE SEASON
SDUM RANDOM NUMBER FOR MARCH 31 ST. OF EACH SEASON
RND ARRAY OF RANDOM NUMBERS FOR PRECIPITATION (t) AND PE (2)
RNW RANDOM NUMBER FOR OVERWINTER PRECIPITATION
REAL SEED*8,RR( 430 )
COMMON /RNDM/RBUNt»KND( 2,214 )»RNW
EQUIVALENCE ( RND( 1 ),RR< 2) )
THE SEED NUMBER JS THAT VALUE RECOMMENDED BY IMSL PACKAGE (29)
DATA SEED/ 0.1 23457D0/
CALL GGUM SEED, 430 ,RR )
RN=RRC 1 )
RETURN
END
.
'.'I «c
\
'
138
SUBROUTINE SOIL
C
C SUBROUTINE TO CALCULATE DAILY CU AND SOIL MOISTURE CONTENT
C FOR EACH CROP (BASED ON THE VERSATILE SOIL MOISTURE BUDGET)
C
C VARIABLE DESCRIPTION
CURRENT SOIL MOISTURE IN EACH SOIL ZONE (IN)
POTENTIAL SOIL MOISTURE IN EACH SOIL ZONE (IN)
SOIL MOISTURE RATIO
AS PER VERSATILE BUDGET
K-COEFFJCIENT, ZONES 1-6, CROP STAGES 1-10, CROP 1-4
Z-TABLE OF 100 COEFFICIENTS DEPICTING SOIL DRYNESS CURVES
K— COEFFICIENT ADJUSTED FOR DRYNESS IN LOWER ZONES
ACTUAL EVAPOTRANSPIRATION FOR EACH SOIL ZONE
DIFFERENCE BETWEEN DAILY PE AND MONTHLY AVERAGE PB
DAILY CONSUMPTIVE USE
SOIL ZONE NUMBER INTO WHICH ROOTS HAVE PENETRATED
IRRIGATION AMOUNT
DRAINAGE
RUNOFF
TOTAL MOISTURE IN ZONES INTO WHICH ROOTS HAVE PENETRATED
SOIL MOISTURE RATIO OF SOIL ZONES INTO WHICH ROOTS HAVE PENETRATED
TOTAL WATER CAPACITY FROM TOP ZONE TO ZONE I
NATURAL LOGRITEM OF DAILY RAINFALL
WATER INFILTRATION INTO SOIL
VECTOR STORING CROP DATA VALUES
VECTOR STORING AE FOR EACH SOIL ZONE
REAL CCF( 6 ),DEL( 6 ), SUMCAP( 6 )
INTEGER CRPSTG, WEEK, FREQ, STAGE, DAY, WK,R, YEAR, SMR , LSTG( 10,4)
COMMON / BUDG/ COEF( 6,10,4 ) , TABLE( 100 ) ,CRMSUM( 5,4,7) ,WEEK( 43 ),CRASUM
1( 5,4 ),CONTNT( 7,4 ),CAPAC( 7 )»CRPSTG( 1 0 , 4 ) , PREC IP( 8 , 2 ) , FREQ( 14 ), STAGE
2( 4 ), AMOUNT! 5 ) , IRRNO( 4 ) ,PPT, PPE, DAY, WK, MO, PED IF, R, YEAR, IC
DATA LSTG/6,3 , 4, 5, 7* 6, 4, 5, 8* 6, 5, 18*6/
DATA SUMCAP/ 0.35, 0.87,1.75,3.50,5.25,7.00/
C RESET CROP DATA TO ZERO
J?R=Q » 0
DR=0.0
CU=0.0
RUN=0.00
ain=ppt
c
C CALCULATION OF A.E. FOR EACH SOIL ZONE
C
C SELECT CROP STAGE
II=STAGE( IC)
C DO FOR EACH SOIL ZONE
DO 100 1=1,6
C CALCULATE SOIL MOISTURE RATIO
SNC=CONTNT( I , IC)/CAPAC( I )
C CALCULATE W TERM
W= 7. 9 1-0.1 1*SMC* 100.0
IF( W.LT.0.0 )W=0.
C SELECT K— COEFFICIENTS
COF( I )=COEF( I , II , IC )
C IF II LESS THAN 3RD CROP GROWTH STAGE OR I EQUALS 1ST SOIL ZONE
IF(II .LT.3.0R . I.EQ. 1 )GO TO 2
C ADJUST K— COEFF IC I ENT FOR DRYNESS IN ABOVE LAYERS
DO 1 J=2 , I
K=J-1
1 COF( I )=COF( I )+COF( I )*COF( K )*( l.-CONTNT( K, IC )/CAPAC( K) )
2 IT=SMC* 1 00 .
IF( IT.GT.O )GO TO 3
C IF SOIL MOISTURE RATIO IS ZERO
WORK=0.
W=0 •
GO TO 4
C SELECT SOIL DRYNESS COEFFICIENT FROM Z-TABLE
3 WORK=TAELE( IT )
C CALCULATE AE FOR ZONE I
4 AE=COF( I )*WORK*PPE*SMC*EXP( — W*PEDIF )
C
CONTNT
c
CAPAC
c
SMC
c
w
c
COEF
c
TABLE
c
COF
c
AE
c
PEDIF
c
CU
c
LSTG
c
RR
c
DR
c
RUN
c
SUMCON
c
SMR
c
SUMCAP
c
OGER
c
AIN
c
AMOUNT
c
DEL
a
■ \j
'
■ • .
IF! AE.GT.CONTNT! I , IC ) )AE=CONTNT! I ,IC )
C STORE AE VALUES FOR EACH ZONE
DEL! I )=AE
C CALCULATE TOTAL CU
CU=CU+AE
100 CONTINUE
C
C DECISION TO IRRIGATE
C
IL=6
SUMCON=CONTNT! 7, IC )
IF( IC.EQ,1.0R.IC.EQ.4)G0 TO 10
IF( LSTG( II r IC )®EQ.6 )GO TO 10
IL=LSTG( II , IC )
SUMCON=0.00
DO 11 ISTG=?1,IL
11 SUMCON=SUMCON+CONTNT! ISTG,IC )
10 SMR = SUMCON/SUMCAP! IL )* 100.0
IF( SMR.LE.50 ) GO TO 20
RR=0.
GO TO 28
20 RR=SUMCAP( IL )/2.00
IRRNO( IC )=IRRNO! IC )+l
C
C APPLYING PRECIPITATION TO EACH ZONE
C
28 IF( PPT.LE, 1 .00 )GO TO 29
C CALCULATE AMOUNT CF RUNOFF
OGEH = ALOG< PPT )
AIN=0. 91770+1. 81100*OGER-0.97300*OGER*CONTNT! 1,IC)/CAPAC( 1)
I F( AIN. CT. PPT >AIN=PPT
RUN=PPT-AIN
29 DR=RR+AIN
SUM=0 •
C UPDATE TODAY'S SOIL MOISTURE CONTENT
DO 30 1=1,6
CONTNT( I ,IC )=CONTNT( I, IC )+DR-DEL! I )
IF( CONTNT( I , IC ).LT .0, )CONTNT( I , IC )=0.
IF( CONTNT( I , IC ).GT .CAPACC I ) )GO TO 31
DR=0 • 00
GO TO 32
31 DR=CCNTNT( I , IC )-CAPAC( I )
CONTNTI I , IC )=CAPAC< I )
32 SUM=SUM+CONTNTC I , IC )
30 CONTINUE
CONTNTI 7 , IC )=SUM
C STORE CROP DATA
AMOUNT! 1 )=RR
AMOUNT! 2 )=DR
AMOUNT! 4 )=CU
AMOUNT! 3 )=CAPAC! 7 )— CONTNT! 7 , IC )
AMOUNT! 5 )=RUN
C TABULATE FREQUENCIES OF IRRIGATION
CALL TAB
RETURN
END
■
'
n © o oo o«j o o o'
140
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
SUBROUTINE TAB
SUBROUTINE TO TABULATE IRRIGATION FREQUENCIES AND TO SUM IRRIGATION
AND DRAINAGE WEEKLY
VARIABLE DESCRIPTION
IRSUM
DRSUM
NIRSUM
NDRSUM
AMT
NUMBER
AMOUNT
F( I T J , K )
WEEKLY
WEEKLY
WEEKLY
WEEKLY
SUMMATION AND SUM
SUMMATION AND SUM
VECTOR CONTAINING
SUMMATION
SUMMATION
SUMMATION
SUMMATION
OF
OF
OF
OF
OF SQUARES
OF SQUARES
CROP DATA
IRRIGATION
DRAINAGE
IRRIGATION OCCURRENCES
DRAINAGE OCCURRENCES
OF IRRIGATION
OF IRRIGATION
AND
AND
DRAI NAGE
DARINAGE
AMOUNTS
OCCURRENCES
ARRAY CONTAINING FREQUENCIES FOR IRRIGATION, DRAINAGE AND RUNOFF DATE.'
I = DAY OF YEAR (1-214)
J = CROP ( 1-4 )
K = 1-rl 4 (NUMBER OF IRRIGATIONS IN TEE SEASON)
= 15 (COMBINED IRRIGATION DATES IN SEASON)
= 16 (DRAINAGE DATES)
- 17 (RUNOFF DATES)
REAL IRSUM( 4 ) ,DRSUM( 4 )
INTEGER NIRSUM! 4),NDRSUM( 4)
INTEGER CRPSTG , WEEK, FREQ , STAGE, DA Y , WK , R , YEAR , F*2 ,SEQ
COMMON / BU DG/ COEF( 6, 10,4 ), TABLE! 100 ),CRMSUM( 5,4,7 ) , WEEK! 43),CRASUM
1(5,4 ) , CGNTNT( 7,4), CAPAC( 7 ),CRPSTG( 10 , 4 ) , PRECIP( 8,2 ) , FREQ( 14 ), STAGE
2( 4 ), AMOUNT! 5 ), IRRNC! 4 ) , PPT, PPE , DA Y , WK , MO , PEDI F , R , Y EAR , I C
COMMON /PARM/ AMT( 43,2,13 ), NUMBER! 43, 2,9 ) , PT( 14,2,2 ) , SEQ( 100 )
COMMON F( 214, 4,17 )
DATA IRSUM , DRSUM, NIRSUM, NDRSUM/ 8*0.0 0,8*00/
I D=DA Y— 90
IF! AMOUNT! 1 ).LE. 0.00 )GO TO 6
UPDATE FREQUENCY OF IRRIGATION DATES
F( ID, IC, IRRNO! IC ) )=F( ID, IC, IRRNO< IC ) )+l
F( ID,IC,15)=F( ID , IC , 1 5 ) + l
SUM IRRIGATION AMOUNT AND OCCURRENCES
IRSUM( IC)=IRSUM( I C )+ AMOUNT! 1 )
NIRSUM( IC )=NIKSUM( IC )+l
IF( AMOUNT! 2 )»LE.0.00 )GO TO 7
UPDATE FREQUENCY OF DRAINAGE DATES
F< ID, IC, 16 )— F ( ID, IC, 16 ) + l
SUM DRAINAGE AMOUNT AND OCCURRENCES
DRSUM! IC ) = DRSUM( IC )* AMOUNT! 2 )
NDRSUM! IC )= NDRSUM! IC )+l
IF! AMOUNT! 4 ) . LE. 0 .00 )GO TO 8
ITC=IC+9
SUM AND SUM OF SQUARES OF CU
AMT! WK, 1 »ITC )=AMT( WK, 1 , ITC ) + AMOUNT( 4 )
AMT! WK, 2 , I TC )=AMT( WK,2,ITC )+AMOUNT( 4 )*AMOUNT( 4)
IF! AMOUNT! 5 ).LE. 0.00 )GO TO 9
UPDATE FREQUENCY OF RUNOFF DATES
F( ID, IC, 17 ) = F( ID, IC ,17 >+l
IF! IC .LT.4 )RETURN
IF IC REPRESENTS LAST OF THE 4 CROPS
IF! DAY. NE. WEEK! WK ) )RETURN
IF DAY IS LAST DAY IN WEEK (WK)
DO 5 1=1,4
J = I + 1
SUM AND SUM OF SQUARES OF DRAINAGE AMOUNT AND OCCURRENCES
AMT! WK , 1 , J ) = AMT! WK,1,J )+DRSUM( I )
AMT! WK, 2, J )=AMT( WK,2,J )+DRSUM( I )*DRSUM( I )
NUMBER! WK,1,J )=NUMBER( WK , 1 , J )+NDRSUM! I )
NUMBER! WK, 2, J )=NUMBER( WK , 2 , J )+NDRSUM( I )*NDRSUM( I >
AMOUNT AND OCCURRENCES
J=I + 5 . „
4 AND 5UM OF SQUARES OF IRRIGATION
AMT! WK , 1 ,3 )=AMT! WK , 1 , J )+ IRSUM! I )
AMT! WK,2, J )=AMT! WK,2,J ) + I RSUM! I )*IRSUM( I )
NUMBER( VK* 1 • J )=NU MBER( WK f 1 * J )+NI RSUM ( I )
NUMBEH( WK#2 » J >=NU.BEK( UfKf 2» J )+"Nr ESUM( I )+NIBSUll(
■
. V;
.
-
141
C BESET SUMMATIONS TO ZERO
DRSUM{ I >=0.00
NDRSUMC I >=00
IRSUM( I ) = 0.00
5 NIRSUU(X)=00
RETURN
END
oo -j O' cn ik (J w
SUBROUTINE PARMTRt YEAR )
C
SUBROUTINE
TO CALCULATE AND OUTPUT LAMDAl
AND
LAMDA2 PARAMETERS
C
VARIABLE DESCRIPTION
c
LAM 1
OCCURRENCE PER DAY
c
LAM 2
YIELD PER OCCURRENCE
c
VARl
DENSITY CF VARIANCE OF LAM 1
c
V2
VARIANCE OF LAM2
c
RATIO
VARl /LA Ml
c
PROD
PRODUCT OF LAM I AND LAM2
c
SD1
DENSITY OF STANDARD DEVIATION
OF
LAM 1
c
SD2
DENSITY OF STANDARD DEVIATION
OF
LA M2
c
MEAN
MEAN OF WEEKLY VALUES OF CU,
PRECIPITATION AND
PE
c
VAR
WEEKLY STANDARD DEVIATION OF
CU,
PRECIPITATION
AND
c
c
SEQ
FREQUENCY OF CONSEQUTIVE DRY
DAY
RUNS
INTEGER WK,YEAR,SEQ
RE AX. LAN 1 , LAN 2 ,MEAN( 4)fSD( 4)»CROP*8t 4 )
COMMON / PARM/ AMTt 43,2,13 ) , NUMBER! 43,2,9) ,PT( 14,2,2 ),SEQC 100 )
DATA CROP/ 'WHEAT' , 'POTATOES' , »SUG BEET' ,• ALFALFA' /
C
Y=FLOAT< YEAR )
DO 50 IC=1 , 9
C OUTPUT Yl AND Y2 STATISTICS FOR RAINFALL, IRRIGATION AND DRAINAGE
IF( IC.EQ. 1 ) WRITEt 6,1 )
IFt IC.GE.2. AND. IC.LE.5 ) WRITE! 6,2 ) CROP( IC-1 )
I Ft IC.GE.6.AND.IC.LE.9 )WRITEt 6,3 ) CROPt IC-5 )
WRITE(fc,4 )
YY= YEAR* 5.0
DAY S=5 • 0
DO 50 WK=1 , 43
IFt WK.LT.43 )GO TO 10
YY = 4 • 0* YEAR
DAY S=4 • 0
C CALCULATE Yl STATISTICS
10 X=NUMBERt WK, 1 , IC )
LAMl=X/YY
Vl=( NUMBER t WK , 2, IC )— X*X/ Y )/( Y-1.0 )
SD1=0 .00
IF ( VI. GT. 0.00 )SD1=SQRT( VI )/DAYS
VAR1=V1/ DAYS
IF( X.EO.O.OO )X=1.0
RATIO = ( V1*Y )/X
Xl^AMTt WK, 1 , IC )
C CALCULATE Y2 STATISTICS
LAM2=X1/X
V2=( AMTt WK, 2, IC )-Xl*Xl/Y )/( Y-l .0 )
SD2=0 .00
IF( V2.GT.0 .00 )SD2=SQRT( V2 J/DAYS
PROD=LAMl*LAM2
50 WR ITE( 6 , 55 ) W K ,LAM 1 , VARl , SD1 , RAT IO, LAM2 , SD2 , PROD
55 FORMATt • *,I3,7F10.4)
1 FORMATt ' 1 RAINFALL PARAMETERS' )
FORMATt '1 DRAINAGE PARAMETERS. A8 )
FORMATt • 1 IRRIGATION PARAMETERS A 8 )
FORMATt , 9X , 9 LAUl ' ,6X, 'VARl* , 3X , • ST DEVI' ,5X,' RATIO* ,6X,'LAM2* ,3
IX, 'ST DEV2' ,3X ,' PRODUCT* )
FORMATt '1 CONSUMPTIVE USE STATISTICS: MEAN AND STANDARD
IDE VI AT ION ' )
FORMATt /////I18,A8,T36,A8 , T56 , A8 ,T77 , A8 )
FORMATt *0 WEEK' ,4t6X,' MEAN ST DEV'))
FORMATt • 1 PRECIPITATION AND POTENTIAL EVAPOTRA NSPIRATION • // *
1 MEANS AND STANDARD DEVIATIONS'/////' MONTH RAINFALL
2 ST. DEV. POT.EVAPO. ST. DEV.' )
C CALCULATE AND OUTPUT CU STATISTICS
WRITEt 6, 5 )
WR I TEt fc , 6 ) t CROPt I ) , I*l» 4 )
WRITEt 6,7)
YY=YEAK*5. 0
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DO 105 WK=1,43
IF( WK.EQ.43 ) Y Y =Y E A R * 4 . 0
DO 100 IC=10, 13
ITC=IC-9
X— AMT( WK , 1 , 1C )
MEAN ( ITC )=X/YY
VAR = ( AMT!WK,2, IC)-X*X/YY )/!YY-1.0 )
SD! ITC ) = 0.00
IF( VAR.QT.0.C0 )SD! ITC )=SQRT! VAR)
100 CONTINUE
105 WRITE(6,9) WK,!MEAN! I ),SD( I )fI = lf4)
FORMAT! • • , 15,4! F10.2, F10.6 ) )
CALCULATE AND OUTPUT RAINFALL AND PE STATISTICS
WRITE! 6,8)
DO 150 1=1,14
DO 140 J = 1 , 2
MEAN! J )=PT! I , 1 , J )/ Y
VAR=! PT! 1 , 2 , J )— PT ( I, 1,J )*PT( 1,1, J)/Y)/( Y-1.0)
SD! J )=0 .00
IF! VAR-GT.0.0 0 )SD! J )=SQRT! VAR )
140 CONTINUE
150 WR ITE! 6,151 ) I,! MEAN! J ), SD( J ),J = 1 ,2 )
151 FORMAT! »0« ,6X , 14, 9X, F5.2 , 10X , F7. 4, 16X,F5.2, 10X,F7. 4 )
C CALCULATE AND OUTPUT CONSEQUTIVE DRY DAY RUN STATISTICS
WRITE! 6, 160 )
160 FORMAT! • 1 RELATIVE FREQUENCIES OF DRY DAY RUNS.*////)
ISUM=00
DO 70 1=1,100
70 ISUM=ISUM+SEQ! I )
SUM=ISUM
WRITE! 6, 161) ISUM
161 FORMAT! '-* ,30X,* RUN LENGTH FREQUENCY PERCENT TOTAL FREQUE
1 NCY * , 18 )
DO 80 1=1,100
IF! SEQ! I ). EQ. 00 )GO TO 80
PER=SEQ! 1)4 10 0.0/ SUM
WRITE! 6 , 102 ) I, SEQ! I), PER
102 FORMAT!* * , 30X , I 6 , 8X , 16 , 7X , F6. 2 )
80 CONTINUE
RETURN
END
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SUBROUTINE ITABLE( K1 , K2, X, I Y )
SUBROUTINE TO CALCULATE AND OUTPUT CUMULATIVE FREQUENCY DISTRIBUTIONS
VARIABLE DESCRIPTION
K COUNTER SPECIFYING IRRIGATION NUMBERS K1 TO K2
NZ NUMBER OF CROPS HAVING NO KTH IRRIGATION
SUM TOTAL SUM OF DAY NUMBER OF THE YEAR FOR KTH IRRIGATION
SUM2 SUM OF SQUARES OF SUM
IN DAY NUMBER OF THE YEAR
N TOTAL NUMBER OF KTH IRRIGATIONS
MAX LATEST DAY ON WHICH KTH IRRIGATION WAS PERFORMED
AVG AVERAGE DAY OF THE KTH IRRIGATION
SD STANDARD DEVIATION OF DAY NUMBER OF THE KTH IRRIGATION
XI UPPER LIMIT OF DAY IN FREQUENCY DISTRIBUTION
F OBSERVED FREQUENCY OF IRRIGATION FOR EACH DAY, IRRIGATION AND CROP
DF PERCENT OF TOTAL OBSERVED FREQUENCY
AF CUMULATIVE PERCENTAGE OF TOTAL OF EACH OBSERVED FREQUENCY
R CUMULATIVE REMAINDER OF TOTAL OF EACH OBSERVED FREQUENCY
XM MULTIPLE OF MEAN
DEV PERCENT OF 200 Y^ARS OF EACH FREQUENCY
THE ABOVE CODES ALSO APPLY FOR DRAINAGE AND RUNOFF
DIMENSION X( 2 )
REAL* 8 SUM, SUM2,CSOP(4 )
INTEGER*2 F
COMMON F( 214, 4,17 )
DATA CROP/ 'WHEAT' , 'POTATOES' ,' SUG BEET' ,' ALFALFA' /
NZ=0
DO 50 K=K1,K2
IF NZ=4, NO MORE IRRIGATIONS TO CONSIDER
IF( NZ.EQ.4 JRETURN
NZ=0
DO FOR EACH CROP
DO 50 J=1 , 4
SUM=0.00
SUM2=0.00
N=0
SUM AND SUM OF SQUARES OF VARIATE
DO 2 1=1,214
IF! F! I, J, K >.EQ.00 )GO TO 2
IN=I+90
SUM=SUM+F( I , J , K)*IN
SU M2= SUM2+F! I ,J,K )*IN*IN
N=N+F! I , J , K )
CONTINUE
MAX= IN-90
IF( N. GT.OO )GO TO 3
IF TOTAL FREQUENCY OF KTH IRRIGATION FOR CROP J IS ZERO
NZ=NZ+1
GO TO 50
Y=FLOAT( N )
MEAN AND ST. DEV. OF VARIATE
AVG=SUM/Y
IF( N. NE. 1 ) SD2 =( SUM2-Y*AVG*AVG )/FLOAT( N-l )
SD=0 .00
IF( SD2.GT. 0.0 0 )SD=SQRT( SD2 )
C OUTPUT HEADINGS
WRITE! 6 , 100 ) CROP! J ), K, X
100 FORMAT! ' 1ENTR IES IN TABLE ', 1 OX ,' MEAN ARGUMENT' , 10X ,' STANDARD DEV I A
1TION' , 10X, ' CROP NO. . . • , A8 , 10X, • ITEM NO. . . • , 13 , 5X, 2 A4 )
WR ITE! 6 , 10 1 ) N , AVG , SD
101 FORMAT! • • , 12X,I4, 13X,F10.4, 18X,F10.4 I
WRITE! 6, 150 )
150 FORMAT! 1 IX, 'UPPER* , 7X , • OBSERVED' , 6X, • PER CENT' ,2! 6X, • CUMULATIVE* ),
16X, ' MULTIPLE' ,6X, ' PER CENT')
WRITE! 6, 151 )
151 FORMAT! 11X, 'LIMIT* , 6X ,' FREQUENCY ', 6X ,' OF TOTAL' , 6 X , ' PERCENTAGE • ,7X
1 ,' REMAINDER* , 7X, ' OF MEAN',6X,' OF 200* )
C CALCULATE FREQUENCY STATISTICS
Y=FLOAT! IY )
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D=FLOAT( N+l )
IFC N • LT . 30 )D=FLOAT(N )
AF=0 .
DO 51 1=1, MAX
I F ( F ( I , J , X ) • EQ • 0 )GO TO 51
DF=F< I, J,K)*100./D
AF= AF+DF
R= 100 »-AF
XI =FLOAT( I + 90 )
XM=0 .00
IF( AVG.GT.0.00 )XM=XI/AVG
DEV=F( I ,J, K )* 100. 0/Y
WRITEC 6,152) X I , F( I , J , K > , DF , AF , R , XM, DEV
152 FORM AT( • ' , 9X , F6 . 2 , 9X , 16 , 8X , F6. 2 , 2( 10X,F6.2 ), 8X, F6 .3, 8X,F7.3 )
51 CONTINUE
WRITEC 6, 153)
153 FORMAT! • REMAINING FREQUENCIES ARE ALL ZERO*)
50 CONTINUE
RETURN
END
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