iMWl.
NAVAL POSTGRADUATE SCHOOL
Monterey, California
THESIS
DETERMINATION OF LAND ELEVATION
USING TIDAL DATA
CHANGES
by
Francisco Antonio Torres Vidal
Abreu
September 1980
Th
Th
esis Advisor: W. C.
esis Co-Advisor: D. P.
Thompson
Gaver, Jr.
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4 TITLE r«<d Su*<ir(«)
Determination of Land Elevation Changes
Using Tidal Data
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Master's Thesis;
September 1980
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Francisco Antonio Torres Vidal Abreu
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Naval Postgraduate School
Monterey, California 93940
I I CnNTWOLLINC OFFICE NAME anO AODMCIS
Naval Postgraduate School
Monterey, California 93940
12. NEPOUT DATE
September 1980
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124
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SUPPLEMENTAHY NOTES
It KEY WOnOt (Ctmilnu* on r»w»r»» tid» II nacaaaarr "4 Identity kr bloek number)
Vertical Crustal Movements
Land Elevation Changes
Tide Time Series Trend Analysis
Tide Station Elevation Changes
Pacific Coast Tide Stations
Cumulative Analysis
20 ABSTRACT (Conlinu* an rmvaraa aMa II nacaaa^rr 1>^ Idmnlllr kr *!•«* maa*«0
The purpose of this thesis is to study the temporal pattern
of vertical land movements at selected Pacific Coast tide sta-
tions. The relative motion of the land at these stations is
indicated by the relationship between monthly mean sea levels
measured at pairs of stations. Examination of historical monthly
mean sea level data by means of graphical and spectral analysis
led to the use of an anomaly filter which adjusts for mean
DO /,°r,, 1473
(Page 1)
EDITION OF 1 MOV 8* IS OaSOUCTE
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monthly differences. A cumulative analysis procedure of Wyss
[1977] was adopted to the study of the relative movement of
seven tide stations. Determination of the type of vertical
movement between pairs of stations, the date of sudden move-
ment, and the station responsible can be determined from analysis
of the cumulative curve of monthly sea level difference. Results
of the cumulative analysis show that tide stations, whether
separated by short or long distances, experience frequent changes
in relative elevation. Whether these are caused by land move-
ments or changes in station datum, or both, is not known.
DD Form 1473
, 1 Jan 73
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Approved for public release; distribution unlimited
Determination of Land Elevation Changes
Using Tidal Data
by
Francisco Antonio Torres Vidal .Abreu
Lieutenant Commander, Portuguese Navy
Portuguese Naval Academy, 1965
Submitted in partial fulfillment of the
requirements for the degree of
MASTER OF SCIENCE IN OCEANOGRAPHY (HYDROGRAPHY)
from the
NAVAL POSTGRADUATE SCHOOL
September 1980
ABSTRACT
The purpose of this thesis is to study the temporal
pattern of vertical land movements at selected Pacific
Coast tide stations. The relative motion of the land at
these stations is indicated by the relationship between
monthly mean sea levels measured at pairs of stations.
Examination of historical monthly mean sea level data by
means of graphical and spectral analysis led to the use of
an anomaly filter which adjusts for mean monthly differences,
A cumulative analysis procedure of Wyss [1977] was adopted
to the study of the relative movement of seven tide stations.
Determination of the type of vertical movement between pairs
of stations, the date of sudden movement, and the station
responsible can be determined from analysis of the cumula-
tive curve of monthly sea level difference. Results of the
cumulative analysis show that tide stations, whether separ-
ated by short or long distances, experience frequent changes
in relative elevation. Whether these are caused by land
movements or changes in station datum, or both, is not known.
TABLE OF CONTENTS
I. INTRODUCTION 11
II. ALTERNATIVE ANALYSIS OF MEAN MONTHLY SEA LEVEL
DATA 21
III. THE CUMULATIVE PROCEDURE 31
IV. SUMMARY AND CONCLUSIONS 44
TABLES 48
FIGURES 54
APPENDIX A: COMPUTER PROGRAMS 99
1. Program Used to Perform Spectral Analysis 99
2. Program Used to Perform Least Squares Best Fit -10 2
3. Programs Used to Perform Filtering Action 104
4. Program Used to Perform Cumulative Analysis 108
5. Other Subroutines Called -HO
BIBLIOGRAPHY -118
INITIAL DISTRIBUTION LIST--- - -120
LIST OF TABLES
I. Time Series Used for Each Tidal Station 48
II. Differences Obtained for Long Term Trends
When Using Raw and Filtered Data 49
III. Comparison of Spectra for Seattle Derived
from the Raw Data, 12 Month Running Means,
and Anomalies (16 Degrees of Freedom) 50
IV. Sea Level Trends Obtained for Three Stations
Using Different Lengths of Time Series 51
V. Trends of the Difference Data for Pairs of
Tide Stations for Specific Time Series 52
VI. Trends of Elevation Change Relative to
San Francisco 53
LIST OF FIGURES
1. Mean monthly sea level for Seattle (SE) ,
Crescent City (CC) , and differences (SE-CC) 54
2. iMean monthly sea level for San Francisco (SF) ,
San Diego (SD) , and differences (SF-SD) 55
3. Mean monthly sea level for Santa Monica (SM) ,
Los Angeles (LA), and differences (SM-LA) 56
4. Power density function of the spectra of Seattle
data (2 degrees of freedom) 57
5. Power density function of the spectra of Seattle
data (8 degrees of freedom) 58
6. Power density function of the spectra of Seattle
data (16 degrees of freedom) 59
7. Power density function of the spectra of
San Francisco data (2 degrees of freedom) 60
8. Power density function of the spectra of
San Francisco data (8 degrees of freedom) 61
9. Power density function of the spectra of
San Francisco data (16 degrees of freedom) 62
10. Raw data, 12-month running means, and anomalies
for Seattle 63
11. Raw data, 12-month running means, and anomalies
for San Francisco 64
12. Raw data, 12-month running means, and anomalies
for Santa Monica 65
13. Raw data, 12-month running means, and anomalies
for Los Angeles 66
14. Power density function of the spectra of 12-month
running means --Seattle (2 degrees of freedom) 67
15. Power density function of the spectra of
anomalies-- Seattle (2 degrees of freedom) 68
16. Power density function of the spectra of 12-month
running means- -San Francisco (2 degrees of
freedom) 69
17. Power density function of the spectra of
anomalies- -San Francisco (2 degrees of freedom) 70
18. Trends of the raw data for SM, LA, and (SM-LA)
using a 10-year window 71
19. Trends of 12-month running means for SM, LA,
and (SM-LA) using a 10-year window 72
20. Trends of anomalies for SM, LA, and (SM-LA)
using a 10-year window 73
21. Trends of anomalies for SE , SF, and (SE-SF)
using a 20-year window '^4
22. Trends of anomalies for SE, SF , and (SE-SF)
using a 30-year window "-
23. Trends of anomalies for SE , SF, and (SE-SF)
using a 40-year window ^^
24. Cumulative differences (SE-LA) using raw data
and matching the station means ^^
'to
25. Cumulative differences (SE-LA) using anomalies
and matching the station means 7'
'to
26. Cumulative differences (SE-LA) using 12-month
running means and matching the station means ^^
27. Cumulative differences (SE-LA) using raw data
and matching the first data points ^0
28. Cumulative differences (SE-LA) using anomalies
and matching the first data points SI
29. Cumulative differences (SE-LA) using 12-month
running means and matching the first data points 82
30. Residuals for a second degree best fit curve
for SE-LA for 1924-1974 (using raw data) 83
31. Residuals for a first degree best fit curve
for SE-LA for 1924-1947 (using raw data)
32. Residuals for a first degree best fit curve
for SE-LA for 1948-1965 (using raw data)---
84
85
33. Residuals for a first degree best fit curve
for SE-LA for 1966-1974 (using raw data) 86
34. Cumulative differences (SF-CC) using anomalies
and matching the first data points 87
35. Cumulative differences (SF-LA) using anomalies
and matching the first data points 88
36. Mean monthly sea level for San Francisco (SF) ,
Los Angeles (LA), and SF-LA (1944-1951) 89
37. Mean monthly sea level for San Francisco (SF) ,
Los Angeles (LA), and SF-LA (1957-1964) 90
38. Cumulative differences (LA-CC) using 12-month
running means and matching the first data points 91
39. Cumulative differences (AV-SM) using 12-month
running means and matching the first data points 92
40. Correlogram for Seattle data (anomalies) 93
41. Correlogram for San Francisco data (anomalies) 94
42. Correlogram for Santa Monica data (anomalies) 95
43. Correlogram for Los Angeles data (anomalies) 96
44. Correlogram for SE-SF (anomalies) 97
45. Correlogram for SM-LA (anomalies) 98
ACKNOWLEDGEMENTS
The author wishes to express his appreciation to
Dr. Warren C. Thompson as thesis advisor for his guidance,
knowledge, method and systematic assistance in the prepar-
ation of this study and to Dr. Donald Gaver as co-advisor
for his useful comments and collaboration.
The assistance of Lt. Dale E. Bre tschneider , NOAA
Corps, of the National Oceanic and Atmospheric Administra-
tion, Pacific Environmental Group, in the timely procurement
and transmission of data is greatfully recognized.
Special appreciation is given to my loving wife and
children whose understanding and patience enabled me to
complete the necessary research.
10
I. INTRODUCTION
The determination of relative rates of land elevation
change can be made using the resultant information from
geodetic work. However, geodetic levelling is repeated very
infrequently because it is time consuming and represents a
high cost. In coastal areas tidal data, available for many
sites, has been used for the same purpose. Its use is not
an innovative idea; books and technical papers dealing with
this subject have been written by Lisitzin [1974], Roden
[1963], Hicks and Shofnos [1965a], Hicks [1972b], and several
others. It was, perhaps, the reading of a paper by Balazs
and Douglas [unpublished] of the National Geodetic Survey of
the National Oceanic and Atmospheric Administration (NOAA)
entitled "Geodetic Levelling and the Sea Level Slope Along
the California Coast" that constituted the challenge point
for the beginning of this thesis. These authors found a
clear and large discrepancy between relative movement rates
from repeated levellings and tidal observations, but were
not able to explain the reasons for that situation.
In this thesis it was decided to use sea level data from
several stations on the Pacific Coast of the United States
of America aiming at possible establishment of a continuous
history of differential vertical land movements among tide
stations along this coast. In order to maximize the resolution
11
of the results it was decided to use mean monthly sea level
data, rather than the mean annual data used in some past
studies. The data, originated by the National Ocean Survey
(NOAA) , was provided by the Pacific Environmental Group of
the National Marine Fisheries Service (NOAA) located at
Monterey, California, and all the computational work was
accomplished on the IBM-360 computer at the W. R. Church
Center of the Naval Postgraduate School. As some of the
received data was not continuous, only periods without
missing data were used to avoid the introduction of an un-
controlled source of errors. Table I shows the time series
used for each station.
The ocean level is constantly changing and its heights
are recorded at long-term tide stations using automatic
(analog or digital) methods. The recorded heights are
related to the zero of a tide staff which is supposed to be
connected by differential levelling to several nearby bench-
marks. If the observations are averaged in a special way in
order to filter out the short-term water level variations, a
value for the mean sea level is obtained. The mean monthly
data used here, which is derived from a direct average of
all the hourly values during an entire month, can be con-
sidered a close approximation to the mean sea level value
for the site where it was collected for that averaging inter-
val. But this mean sea level, so important to the hydrogra-
pher, geophysicist , and geodesist, is not a constant value
12
from month to month. Short-term variations of duration less
than one month, like oscillations in barometric pressure
caused by daily temperature fluctuations, wind effects,
seiches, and tsunamis, are some of the causes that give rise
to the fluctuations or departures from the mean water level.
But exactly because they are transient and of such short-
term duration, when averaged over the period of a month they
are not significant. Some of the causes of longer term water
level variation that account for month-to-month sea level
differences are:
(a) The movement of the axis of rotation of the earth
(Chandlerian motion) with an approximate period of 14 months.
(b) The nodal cycle of the moon with a period of 18.613
years.
(c) The sun spot cycle with an approximate period of
11 years.
(d) Variations from the average meteorological conditions
(e) Variation of the average sea \\^ater density in the
vicinity of the tide station.
(f) Dynamic effect of ocean currents (the last three
causes are meteorologically and/or oceanographically induced,
producing sea level changes with the following characteristics:
(1) Duration- - larger anomalies average about three
months but may be as long as 10-34 months [Roden, 1966].
(2) Amplitude- -monthly variations from the mean
are commonly 100mm but may reach 300mm [Bretschneider , 1980].
13
(5) Coastwise coherence- -very high over distances
less than 200 km; significant over distances of the order of
1200 km for tide stations located in the same macro environ-
ment [Roden, 1966],
(4) Tide station exposure- -within the same macro
environment on the open coast and in bays, exposure is rela-
tively unimportant but stations well inside estuaries and
fjords where there is large fresh water runoff may be
exceptions) .
(g) Eustatic changes in elevation of sea level due to
the melting of the ice accumulated on the continents.
(h) Any kind of crustal movements, including natural
isostatic movements (generally assumed as low and gradual
with rates of change varying from zero on stable coasts to
approximately 40mm per year [e.g., Hicks and Shofnos, 1965a]
in areas of rapid crustal rebound), sudden movements associ-
ated with earthquakes and faulting, and the very rapid
movements induced by man resulting from oil and groundwater
withdrawal, vibrations related with land use, etc.
The presence of long-term vertical movements of both
water and land contained in a tidal data time series makes
it impossible to determine with high precision the absolute
rates of land change. Except for the case of large rates of
crustal rebound, we never know with enough confidence if a
long term increase of mean sea level was due to land sub-
sidence, to a slow rise of mean sea level, or to both causes
14
Lisitzen [1973] and Hicks [1972] tried to estimate
absolute rates of land change from single station tide data.
The subjective assumptions used by both were not strongly
convincing, though with scientific reasons behind them. The
former concluded that the eustatic rise of the sea level
began in 1891, and the latter simply applied the eustatic
sea level rise rate of 1.0mm per year found by Gutenberg
[1941] (using sea level data from 69 stations in 22 different
regions around the earth) . This study deals only with rela-
tive rates of change determined from time series data using
pairs of tide stations.
Since vertical movement that can occur in the earth's
crust is one of the components contained in each tidal time
series, it would be a simple matter to quantify this compon-
ent if all the others could be evaluated and removed from the
data. However, this approach is not possible because of lack
of data on some components, difficulties in quantifying the
effects of other components, and also difficulties in isolat-
ing the effect of astronomic tidal components of known period
(e.g., the nodal cycle of the moon) from the data.
Using a different approach, it can be said that each
monthly time series from a given tide station is composed
of random fluctuations (short-term climatological variations
are an example) , periodical fluctuations (the astronomical
components that force the tides and the annual climatological
cycle), and what can be considered, at least to a first
15
approximation, to be linear trends (the eustatic rise of
sea level and the movements of the earth's crust). As random
and periodic data have no long-term trends in themselves, any
trend detected when analysing the data should reflect the
latter factors.
Various methods have been used to study sea level changes
but only four will be briefly described as a background neces-
sary to introduce and better understand the work done in this
thesis :
-Gutenberg [1941] used mean annual sea-level data col-
lected at long-term tide stations around the earth. At that
time the unavailability of computers made the use of the least
square fitting technique very difficult. The author proposed
and used a short-cut system of averaging the first x and the
last X years, and dividing the difference between the two
means by the number of years between the weighted midpoints
of these intervals. His criterion for the selection of x was
as follows: If the number of years of data was greater than
25, X should be 10, but if t < 25, x = t/3. Gutenberg does not
speak about errors and the purpose of his paper was to deter-
mine the eustatic rate of rise of the sea level.
-Roden [1963], studying sea level variations in Panama,
used anomalies of mean monthly sea-level data obtained by
taking the difference between the monthly sea level and the
long-term mean for the same month. To find the trends, Roden
assumes a record z(t) which is the sum of a deterministic
16
function of time and a stationary random fluctuation,
z(t) = a + bt + x(t)
where b is an unknown constant representing the long term
trend in which we are interested. Here, x(t) is the sta-
tionary random fluctuation with zero mean and known auto-
correlation or spectrum. Roden uses the solution,
b = /J"K(T-t) z(t)dt
where T is the record length, and b is an estimate of the
unknown constant b. The particular shape of the Kernel K(T-t)
depends upon the autocorrelation of x(t) . The mean square
error of the estimate of 6 used was
e' = 12 6VT3
where 5^ refers to the spike at the origin, assuming white
noise.
In a later paper, Roden [1966] uses the expression,
b = ^ t^^^¥- l)^(t)
c^T^ + 6cT + 12 -^
where c is the constant decay of the autocorrelation of the
sea level fluctuations. For the mean square error of the
estimate the following expression was used:
2 4 cm
£2 = O.
TCc^T-" + 6cT + 12)
where m denotes the variance of the data
o
17
-Hicks and Shofnos [1965a, 1965b] and Hicks [1972a,
1972b] used yearly mean sea level values weighted by a tri-
angular array (1,2,3,4,3,2,1) and then computed the trends
by fitting a least- squares line of regression. The formulas
used for the computation of the slope, b, of this line as
well as for the standard error of the slope, Sb, are:
Sxy - il20Slll
b = -— ^
Ex'
(^x)
Sb =
Sy .X
Z.X - -^^ -
n
where n represents the number of yearly values. S^y.x is an
estimate of the error variance around the assumed line
relating y to x; Sy.x is an estimate of the error standard
deviation, and given by:
c /~T~z rWP TTT (Zx) (Zy) ,
Sy.x = / Ey^ - ^-^ - b(Exy - -^^ ^^ ^ ^ )
__
-Merry [1980] also uses a least- squares fit but applies
it to unfiltered daily mean sea levels in a study of secular
sea level changes.
One important statement that must be made is that these
authors, although using different types of sea level data
(daily, monthly, and yearly) , different smoothing techniques.
18
and different methods to determine the trends contained in
their data, all assume that both the eustatic rise of the
sea level and the movements of the earth's crust can be
considered linear to a first approximation. It should also
be noted that all dealt with single-station analysis, that
is, the data from each tide station was analysed independently
of other tide stations to obtain rates of sea level change.
Because of the impossibility of making accurate determin-
ation of absolute rates of land change using single station
analysis, it was decided to eliminate or greatly minimize the
effect of sea level changes in the tide data by comparing the
tidal data from several pairs of stations separated by var-
ious distances [Table I]. Subtracting from each mean monthly
sea level value at station A the corresponding value for the
same month and year at station B, we generate a new set of
time series data which must contain within itself the rela-
tive rate of land elevation change between the two stations,
free of any eustatic related trend. Because of the principle
on which it is based, we will refer to this procedure as
"differential tidal levelling"; the procedure may be thought
of as an analog of the method of differential spirit levelling
used in geodetic surveying.
In this procedure we assume that the glacial -eustatic ,
the climatological , and the oceanographic "long term trends"
do not significantly differ in themselves over distances on
the order of hundreds of kilometers. These are therefore
19
eliminated in theory by the described differencing computa-
tion. The difference between the two series also leaves some
random and periodic information that should not introduce any
trend. This method, besides allowing the detection of rela-
tive rates of land elevation change between any two stations
due to tectonic effects, has the advantage of largely removing
the short term and seasonal variations, especially at stations
separated by short distances. The method also provides a
basis for calibration of an entire coast. Nevertheless, it
should be used with care particularly when comparing widely
separated station pairs, which is a common situation along
the west coast of North and South America.
It was with this background that the computational phase
began. The next section describes successive efforts to
derive rates of land elevation change from sea-level differ-
ence data at paired tide stations. Section III treats in
detail the procedure introduced by Wyss [1977], and further
developed in this study, to accomplish this objective.
20
II. ALTERNATIVE ANALYSIS OF MEAN
MONTHLY SEA LEVEL DATA
The first step taken was to plot the mean monthly sea
level data for all of the stations as well as the differences
between several pairs of stations. For illustration, three
examples are shown on Figures 1, 2, and 3 for the pairs
Seattle-Crescent City, San Francisco-San Diego, and Santa
Monica-Los Angeles, respectively. Only eight years of data
are shown in order to avoid obscuring details. The upper and
middle values on each graph represent the data for the first
and second station of each pair, the lower values being the
resultant difference. Each symbol represents one month
(January through December) related with the year indicated,
and is obtained from an average computation of hourly values
recorded in units of feet. The vertical scale is relative
and is the same for all the curves (1 foot/inch).
These three station pairs were selected to illustrate
the effect of distance between the stations (note Table I) .
Each set of station data shows a distinct annual cycle, and
it is possible to detect visually a long term trend for a
period as short as eight years for some of the stations.
The difference data also show this long term trend; however,
the annual cycle is not so evident and depends upon the dis-
tance between the two stations. The variation of the differ-
ences is smallest for the closest station pair due to similarity
21
of data, which creates a more effective cancellation. The
differences appear to be much more random than the original
data for each station. Taking differences has the effect of
cancelling out systematic variations common to the two series.
The visual evidence of an annual cycle, despite being
disguised in the difference data, oriented this work toward
the need for a spectral analysis of station data in order to
determine if other frequencies with significant energy are
present. The occurrence of annual and other cycles would
justify a filtering operation before treating the data for
trends. The short length of some of the time series was a
clear temptation to perform the analysis with a small number
of degrees of freedom (with a correspondent lack of confidence
in the results) , but stations like Seattle and San Francisco
with at least 912 data points (76 years) allowed the use of
16 degrees of freedom. The spectral analysis was performed
on several stations for 2, 8, and 16 degrees of freedom with
the subroutine PREPFA, shown in Appendix A-1, using the prin-
ciple of the Fast Fourier Transform (FFT) . Other subroutines
called by PREPFA can be found in Appendix A-5, except PLOTG
and RHARiM which are library subroutines of the W. R. Church
Computer Center.
Figures 4, 5, and 6 show the power density function of
the spectra for Seattle data for 2, 8, and 16 degrees of
freedom, respectively (note different scales). Figures 7,
8, and 9 are the corresponding graphs for San Francisco.
22
For both stations the maximum detectable frequency is 0.5
cycles per month. This frequency, Fn = jj— , is determined
by the time interval At between each data point, where Fn is
the Nyquist frequency and At=l month. In all six figures
noticeable sharp concentrations of energy (peaks) are seen
to occur only in the low frequency part, with almost nothing
at higher frequencies. The typical decrease of noise with
increasing frequency may be seen best in Figure 7. Identifi-
able by their high concentration of energy, only the frequen-
cies of 0.083 and 0.167 cycles per month could be found.
These correspond to the annual and semi-annual cycles. These
prominant peaks, although coincident with the tidal constitu-
ents Sa and Ssa, are essentially of meteorological and ocean-
ographic origin. Thus, with so high a concentration of energy
at these specific frequencies with a periodic origin (but not
absolutely repetitive year after year) , it was decided to
eliminate this interference from the station data before
computing long-term sea level trends.
In order to remove the two prominent cycles appearing in
the spectra, it was decided to experiment with two different
types of filters. A principle to be adhered to was that the
annual and semi-annual cycle should be removed without sig-
nificantly modifying the proportionality of the energy exist-
ing at the other frequencies. This requirement of maintaining
the energy proportionality was adopted to guarantee that only
the periodic components in the station data, which do not
23
contribute to the long-term trend, are removed by the filter-
ing process.
The first filter used is a simple 12-month running mean,
12 months being the length of the averaging window needed to
remove the effects of both the annual and semi-annual cycles.
The operation of this filter is very easy. The first data
point is obtained by averaging the first 12 data points from
the raw data, the second is obtained by averaging the next
12 data points (2 through 13), and so on. It is obvious that
with this method, so often used in practice, we are introduc-
ing a "tail" effect. During the entire averaging process the
first and the last (nth) data points were just called once,
the second and the (n-l)th twice, ..., while all the data
points between and including the 12th and the (n-ll)th were
used twelve times. A subtle consequence is that the filtered
data series is 11 months (11 data points) shorter than the
raw data series, so that the sea level trends computed from
the rai\f and filtered data may be expected to differ slightly.
Nevertheless, it is a very effective filter for special
applications .
The second filter used removes the long-term mean monthly
sea levels from the raw monthly data to produce a time series
of monthly sea level anomalies. This is accomplished by
first averaging the sea level values for each month (i.e.,
first for January, then February, and so on) and then sub-
tracting these 12 means from the respective monthly values
24
in the raw record. The result is a record of monthly sea
level anomalies relative to the long-term mean monthly sea
levels .
For all three sets of monthly station data, i.e., the
unfiltered sea level values termed the raw data, the 12-month
running mean data, and the anomaly data, the trend, the
scatter of the monthly values, and the standard error of the
slope were obtained using the conventional least-square for-
mulas referred to in the introduction. Some comments will be
made later about the errors introduced with the use of these
formulas when the data are correlated. The computer program
used to perform these calculations was subroutine LEASTS,
found in Appendix A-2. Appendix A-3 gives the subroutines
RMEANl and ANOMAL written to perform the two types of filter-
ing just described.
An example, using four stations, of the effects of apply-
ing these filters to the raw data is illustrated in Table II
The table shows that for a given station, the sea level
trend computed by the three methods agrees quite closely.
The values that describe the scatter of the data and the
standard error of the trend are what would be expected from
the three methods, i.e., smaller values for the scatter of
the 12-month running means than for the raw data or anomalies
The reader should be cautioned not to compare the trends
between stations because they are obtained from different
series of years.
25
With regard to the variability of the filtered data
compared to the variability of the raw data, a better feeling
can be obtained through Figures 10, 11, 12, and 13. Each
graph shows from top to bottom the raw data, the 12-month
running means, and the anomalies for the first eight years
of tidal series at stations Seattle, San Francisco, Santa
Monica, and Los Angeles, respectively. These four figures,
in conjunction with Table II, show that the trends and the
variabilities of the anomalies are much closer to those of
the raw data.
It was stated above that it was desired that the filters
used should remove only the annual and semi-annual components
and that the spectra of the filtered data should maintain the
proportionality of the energy distribution with frequency
observed in the spectrum of the raw data. The question of
whether the two filters used satisfy these conditions will
now be addressed. The transfer function for each filter is
different, and so the results of the spectral analysis may
be expected to differ somewhat. This study was done using
four stations with the indicated 2, 8, and 16 degrees of
freedom, but the results for only two stations for 2 degrees
of freedom are presented. Figures 14 and 15 refer to Seattle
and show the spectra resulting from the application of the
12-month running mean and the anomaly filters, respectively;
these should be compared to the spectrum for the raw data
shown in Figure 4. Similar spectra for San Francisco are
26
shown in Figures 16 and 17 and should be compared to Figure 7.
The differences can be seen more quantitatively in Table III,
which is discussed below.
From visual inspection of these figures it can be seen
that both filters very effectively eliminate the annual and
semi-annual components, but while the running window filter
removes practically all the energy contained in frequencies
greater than that of the annual cycle, the anomaly filter
eliminates only the undesirable peaks of one cycle/year, two
cycles/year, and multiples of these frequencies, leaving the
energy at the other frequencies with the same approximate
proportionality. To illustrate the effects of the filters
quantitatively. Table III shows energy density ratios obtained
from comparison of these spectra for Seattle (using 16 degrees
of freedom) . By removing the values closest to the periods
which we intend to eliminate (12 and 6 months) , the quotient
(3)/(5) is seen to vary between the values of 1.272 and 0.888,
while the quotient (3)/(4) varies between 1.330 and infinity.
The same conclusion can be reached also by visual comparison
of the plots.
After this examination the conclusion that all further
work should be prosecuted using the anomalies was reached.
Nevertheless, in order to further compare the filtering pro-
cedures, the raw data and both sets of filtered data were
used in the following investigations. The sea level trends
for three tide stations computed for different time intervals
27
are shown in Table IV. The table shows that the trends are
time dependent. These results were not unexpected because
other authors [Gutenberg, 1941 and Roden, 1966] already-
referred to the problem of the "instability" of the long
term trends.
Although the trends of the sea level at one individual
station are clearly shown not to be constant, there is reason
to believe that the trend of the differences in monthly sea
level between a pair of stations, particularly closely spaced
stations, might be much more nearly constant. In order to
inquire into this question the following experiments were
performed. Programs were run in order to find and plot the
running trends of the differences, where the trend was com-
puted for a selected time interval or window. Thus, for a
given time series of monthly sea level differences between
two stations, the trend for the first x years is computed,
where x is the window length, then the operation is repeated
by stepping the window one month at a time and computing a
new trend. The result is a graphical computation of a time
series for an x-year trend for a given station pair. The
selection of the window length was not arbitrary. Knowing
the existence of long period tidal components expected to be
contained in the tidal series, including the regression of
the moon's node (18.613 years), the revolution of the lunar
perigee (8.847 years), and the revolution of the solar perigee
(20.940 years), the closest numbers of integer months to these
28
values were chosen. The results, which are not presented,
did not show a constant value for the trend of differences
within some reasonable standard error. It was only obvious
that by increasing the length of the window the variability
of the resultant trends was decreasing, which was expected.
In order to go further in this analysis, plots and
computations were made for several pairs of stations using
windows covering 10, 20, 30, and 40 years of data, when pos-
sible. Also, raw and filtered data were used in all experi-
ments. For illustration, Figures 18, 19, and 20 show for
the station pair Santa Monica-Los Angeles (SM-LA) the trends
for the monthly difference data; also shown are the trends
for each station. The length of the window used is 10 years
These three figures refer to the raw data, 12-month running
means, and anomalies, respectively. The pattern presented
using these types of data is almost the same. The yearly
cycles are evident in the raw data computations (Figure 18} ,
and the degrees of smoothing obtained with the filters can
be observed in Figures 19 and 20. An important observation
relates to the variability of trends obtained for a close
pair of stations. For the station pair of SM-LA the trends
of the differences range between -2.5 and +7.0 mm/year for
the period considered. This suggests that for this specific
time-window (10 years) , time periods can be found during
which Santa Monica rose relative to Los Angeles (or Los
Angeles subsided relative to Santa Monica) while other
29
periods reflect the opposite. On the time ordinate in these
figures "mean year" means the central year of the window used
for the calculation of the trends. To show how the variabil-
ity of the trends for each station and for the differences
are smoothed with an increase of the window length, Figures
21, 22, and 23 show the trends of the monthly anomaly data
using 20, 30, and 40-year trend windows. All refer to the
pair Seattle-San Francisco (SE-SF) . The difference curve is
the lowest one on all the graphs.
These demonstrations show that the sea level trend is
determined, in part, by both the width and the mean position
in time of the window used; accordingly, it was determined
that the aim of this work should be modified. Instead of
trying to refine or otherwise improve on the values already
published by several authors using single station tide meas-
urements to estimate rates of land change, the study was
reoriented in order to find the evolution with time of the
relative movement between two stations. This will be the
topic of the next chapter.
30
III. THE CUMULATIVE PROCEDURE
Wyss [1977], in a study of land elevation changes asso-
ciated with earthquake occurrence, introduced a method of
cumulative analysis using monthly sea level differences be-
tween two very close tide stations. In order to apply this
technique, it is desirable to adjust the monthly sea levels
at one station relative to the other so that their means are
equal. Thus, if for stations A and B, all the data values
of B were modified in order to make the average of B equal
to the average of A, the differences between A and B would
have values with a random variation around zero. The dif-
ference values, (A-B), are then cumulated in a time series
beginning with the earliest monthly difference value. The
cumulative curve that results will be a random walk around
the value zero if the sea level history at the two stations
is identical. There will be swings up and down, but eventu-
ally a return to zero will occur. If the elevation of one
station relative to the other is different, this random plot
will soon show an evident and pronounced trend. In this case,
if the monthly differences are cumulated over many years, very
large cumulated differences amounting to many feet may result.
Now, if the relative elevation changes suddenly instead
of cumulating small positive or negative values around zero,
values containing a constant increment (positive or negative)
31
are now added introducing a trend on the random data, and
an inflection point will appear in the cumulative curve. The
difference between slopes on each side of the discontinuity
allows determination of the amount of relative elevation
change. The inflection point, itself, allows the identifica-
tion of the date for relative elevation change. In the case
of a sudden elevation change or jump revealed by a cumulative
curve, the station which is responsible cannot be identified.
However, the latter can be determined when more than one
station pair is used because a change in the slope of the
cumulative curves must occur at the same time in all pairs
of stations containing the common station.
Before examining the results of the cumulative analysis,
further explanation must be given about the kind of curves
that can be expected when using the cumulative procedure.
As was said before, there is strong reason to reject the
assumption of long-term linear sea level changes, but there
is no reason to avoid the assumption of linear relative ele-
vation changes occurring over short periods of time. If we
make this assumption, only two possibilities can occur.
Either the two stations are stable relative to one another
and the trend of the monthly sea level differences between
them is zero, or there is elevation convergence (a negative
trend) or divergence (a positive trend) between both stations
In the first case, a straight line will be shown in the cumu-
lative curve, with a positive or negative slope, depending on
32
whether the elevation of one station is higher or lower than
the other; the linear segment will be horizontal if the means
of the monthly sea levels are equal at the two stations for
the time period represented. In the second case, if the
monthly differences converge or diverge, the amount to be
summed each time is different from the preceding value and
follows a linear law of variation. The cumulative curve will
then show a parabola with a negative or positive slope,
respectively. Subroutine CUMMUL, (Appendix A-4), was written
to perform this type of analysis.
To illustrate the above description of straight line seg-
ments with different slopes, as well as branches of parabolas
with positive and negative slopes, some of the cumulative
graphs produced are presented. Figures 24, 25, and 26 show
the cumulative differences for the pair Seattle-Los Angeles
using raw data, anomalies, and 12 months running means (three
sets of curves were produced for all pairs of stations because
the different degrees of smoothing on each curve were helpful
for identifying the inflection points). One interpretation
of the cumulative curve in these figures is that they consist
of three legs or segments, the first one between 1924 and
1947, and the other two between 1948-1965 and 1966-1974. This
pattern in which the cumulative values become more negative
and suddenly change toward more positive was not expected.
These graphs show that the averaging procedure used to achieve
an initial matching of the data for both stations was not the
33
most suitable because it caused a change in slope from nega-
tive to positive, thereby suggesting a change in sign of the
movement between the two stations. From the three figures
we can see that the Seattle data was negative relative to
the Los Angeles data during the first interval and became
positive during the second and third legs. The absolute
values of the slopes of the first and second legs strongly
suggest that some event occurred about the end of the year
1947. Nevertheless, it must be noted that in 1947 a change
in the sign of the slope also occurred. A change in the sign
of the slope of a cumulative curve means that one data set
crosses the other, so that the sign of the differences re-
verses. If the cumulative curve is made by linear legs, then
a symmetrical picture results in which the slope on either
side of the crossing point has the same magnitude but is
opposite on sign. In Figure 24, the character of the dis-
continuity shows that not only did a "jump" occur between the
two stations but the "jump" caused a reversal in the sign of
the differences. In order to avoid misinterpretation of the
cumulative graphs, the decision was made to recumulate the
monthly difference values after first equating the initial
data point of each station. The graphs presented in Figures
27, 28, and 29 show these results for the same pair of sta-
tions, using raw, anomaly, and 12-month running mean data,
respectively. Their appearance is seen to be markedly differ-
ent from Figures 24-26. This group of six plots was helpful
34
in two ways; first, inflection points are easily identified,
and second, the apparent differential station movements can
be isolated from background noise. This procedure of produc-
ing a set of six plots for each pair of stations was the
routine used for all possible pairs among the seven stations
dealt with.
The cumulative curves in Figures 28 and 29 never cross
the zero line. Their relatively smooth appearance could lead
to an alternative interpretation that instead of three straight
line legs, the curves could be viewed as a long branch of a
parabola. This interpretation could justify the computation
of a single difference trend for the entire data series. From
simple observation, there is no doubt that the long term trend
between the two stations must be positive; the value of 2.13
mm/yr +^0.28 mm/yr is obtained for the entire raw data series
(1924-1974) .
In order to test the hypothesis of these cumulative curves
being composed of three straight line segments versus a branch
of a parabola, two methods were used.
The first method, assuming the curves to be composed of
three linear segments, was to calculate the differential sea
level trends from the monthly difference values for each leg.
The results obtained for SE-LA are given in the upper part of
Table V. The three values for the trends are very close to
zero, indicating similar movement at both stations during
each period, but also that "jumps" occurred between each leg.
35
The measure of a "jump" can be obtained (as already explained)
by computing the difference between the slopes of the adjacent
legs on the cumulative curve.
The second test was to perform a least-squares best fit
to obtain the hypothetical parabola represented by the cumu-
lative data of Figures 24 through 29, and then to compute and
plot the residuals obtained by taking the differences between
the cumulative curve and the best fit parabola. For this
purpose subroutine LSQPL2, a library subroutine of the Com-
puter Center of the Naval Postgraduate School, was used. The
residuals between the cumulative curve of Figure 27 and the
best fit parabola are shown on Figure 30. It is clear that
the variation of the residuals is not random, and some struc-
ture can be observed such as would be expected from fitting
a parabola to straight line segments. The next step was to
fit each one of the three segments of the cumulative distri-
bution with first degree curves (straight lines) and again
compute the residuals. If the hypothesis of three linear
legs is reasonable, the variance of the residuals should show
a significant decrease and should display a more random dis-
tribution around zero. Figures 31, 32, and 33 illustrate
these residuals for the first, second, and third legs, respec-
tively. The randomness around zero is evident, at least for
the periods shown in Figures 32 and 33, and a decrease of the
extreme values can also be observed. Thus, if the cumulative
curve for SE-LA indicates three intervals of constant elevation
36
difference between stations, the jumps indicated about 1947
and 1965 amount to 4.6cm and 4.4cpi , respectively. Another
conclusion can also be extracted from this analysis of the
residuals. Both Figures 31 and 30 clearly show a unique
structure in the data between the end of 1927 and 1934, sug-
gesting that something of geological significance may have
happened between these dates.
Similar tests were performed for several pairs of stations
where the cumulative curve suggested the possibility of a
single long term difference trend. Figure 34 is an example.
This curve was initially chopped into three pieces. The first
branch of parabola included the period from 1950 to 1959, the
second from 1960 to 1964, and the last from 1964 to the end
of the series. The values for the trends of the partial
series as v\^ell as the entire series, computed from the monthly
differences, are shown in the lower part of Table V. The test
of the residuals was also applied, this time with three second-
degree curves for the best fit of each branch. The results
were similar to those for the SE-LA station pair. Again, the
analysis of all pairs containing SF and CC not only allowed
the identification of the inflection points, but also allowed
identification of the station whose movement was responsible
for it.
An example of station identification can be observed in
Figure 35, where a change of movement occurred near the end
of 1960. This figure refers to the pair SF-LA where the
37
movement which occurred in 1947 can also be seen. By compar-
ing the plots for SE-LA (Figures 24-29) with the plot from
SF-LA (Figure 35) , it is seen that an inflection point occurs
on both plots for 1947. Therefore, Los Angeles must be the
station responsible for this inflection. Near the end of
1960 a similar movement appeared on plots SF-CC and SF-LA.
Using the same logic, SF is the responsible station. Figures
36 and 37 show plots similar to Figures 1, 2, and 3, but in-
clude the periods containing the inflections of 1947 and 1960
for the pair SF-LA.
Cumulative analyses were made of the 21 possible station
pairs for the seven stations analysed. Two plots are pre-
sented in Figures 38 and 39 which illustrate some of the
typical situations that were found. By disregarding small
features, Figure 38 shows three distinct periods. The first
one ends in 1959 and has a positive trend. The second one
extends from 1959 to 1963 with a negative trend, and a third
leg (almost linear) has a trend close to zero. Figure 39
shows six distinct straight line segments with different
slopes and a small part of a parabola (between the end of
1951 and the end of 1952) . All of the cumulative distribu-
tions appear to show only segments of straight lines and
branches of parabolas. Therefore, it can be assumed that
the difference data represents dominantly linear sea level
trends .
38
At this point it is of interest to present the results
of cumulative analysis applied to the seven tide stations
studied. The method chosen to do this is to reference the
relative vertical motions to a single station on the coast.
San Francisco was chosen as the reference station because of
its long time series without missing data (1899-1978) and
also because only two vertical movements were ascribed to it
during this long period. Table VI shows, for the time inter-
vals listed, the sea level trends and errors in the trends
for SE, CC, AV, and LA relative to SF. Assuming that San
Francisco experienced discontinuous movements only in 1931
and 1960, all the other movements indicated by the boundaries
between the time intervals shown are attributed to the other
four stations. Although the sea level trend history for each
station relative to the reference station could theoretically
be determined directly from linear and parabolic segments
fitted to the cumulative plots produced, time did not permit
this. Trends shown in the table were instead computed by
fitting linear curves to the station difference data for each
of the many intervals betiveen the identified inflection
points. Identification of the inflection points was found
in some cases to be quite subjective. It should also be
noted that the cumulative curves of nearly all station pairs
that include SF appear to consist of parabolic segments.
Additional comments regarding the data in Table VI are
necessary. The first one is that very short-term trends,
39
which cover a length of time like one year or two, must be
looked upon with suspicion because they are dominated by the
assymetry of the annual cycle still present in the difference
data. Second, the stations of Santa Monica and San Diego
were not included in the table because all cumulative plots
including these two stations show a complicated pattern with
continuous and aperiodic waves. The other cumulative graphs
show generally well defined patterns leading to easy identi-
fication of movements. Some dates of relative sea level rate
change at these stations are suspected, e.g., Santa Monica in
1941, 1945, 1955, and 1956 and San Diego in 1942, 1950, and
1959, but it is impossible to define these discontinuities
with the same confidence as for the other five stations.
Also, it is evident from Table VI that as the length of each
period under analysis decreases, the standard error of the
slope increases. This is a price that must be paid when
using cumulative analysis. The detection of apparent inflec-
tion points results in chopping the entire time series into
shorter segments, with the corresponding reduction in con-
fidence in the trends obtained.
Some comments must be made about the computations of
the trends as well as about the corresponding errors. The
standard least-squares formulas used by other investigators
in this field and also used in this thesis are not the most
appropriate if the data to which a curve is to be fitted is
not independently random. High correlation between the
40
monthly sea level values can be observed in the correlograms
in Figures 40, 41, 42, and 43, prepared for Seattle, San
Francisco, Santa Monica, and Los Angeles, respectively.
Nevertheless, if the length of the series is large and it is
assumed that for each time segment the trend of the data is
smooth, a reasonable approximation is obtained for the value
of the slope. But a poor approximation of the standard error
of the slope results if the time series is short and the data
fluctuations are not independently random. In order to obtain
a representative standard error a corrective factor must be
applied to the conventionally calculated standard error.
This factor [Bloomfield, 1980] is:
1+ 2 Z P^
T=l '■
where x is the time lag in months and p is the autocorrela-
tion function at lag x. This factor is positive, greater
than one, and multiplicative; accordingly, it introduces an
amplification of the value of the standard error of the slope
If, for simplification, a geometric decay of the correlogram
is assumed (and that is not always the case) this formula can
be simplified to
where p, is the value of the autocorrelation function for a
time lag equal to one.
41
Figures 44 and 45 show the correlograms for the differ-
ences (using anomalies) between the pairs SE-SF and SM-LA.
From these two examples it can be seen that the variability
of the trend found with the ordinary least- squares formula
should be increased for the first case by a factor of 1.3,
using p-. = 0.26, and for the second case this factor is 2.17,
using Pt = 0.65. Thus, the calculation of the corrective
factor is different for each pair of stations and must be
computed separately for each case. The values obtained for
SE-SF, CC-SF, SF-AV and SF-LA, are 1.83, 1.69, 1.91, and 1.91,
respectively. As the confidence limits for some of the trends
are relatively large, the associated trends can be largely
disregarded. Thus, care must be taken with the contained
information, even knowing that it represents the most probable
values .
As an example of the interpretation of Table VI, the
positive relative trend shown for SE (2.34 mm/yr) for the
period 1899-1910 means that sea level was rising at Seattle
relative to San Francisco or that the Seattle tide station
was subsiding relative to San Francisco during that period
of time.
The question should be raised as to the cause of the
relative vertical motions between tide stations determined
from the application of the cumulative analysis. There are
two plausible causes for the sea level differences:
42
(1) Occasionally or steadily occurring real land eleva-
tion changes. It should be pointed out here that the occur-
rence of sudden movements as shown on the cumulative plots
agrees with results found from on-going studies using Very
Long Baseline Interferometry to the effect that crustal plate
motions in California may be characteristically jerky at
regional scales of hundreds of kilometers and not continuously
slow and smooth [Whitcomb, 1980]. Differential tidal level-
ling in the Salton Sea reported by Wilson [1980] also indi-
cates jerky crustal movements.
(2) Occasionally occurring changes in the station datum.
These may be due to tide staff displacement, replacement of
the staff or gage, moving the gage, displacement of the in-
strument datum, construction work at the pier, etc. These
kinds of changes are likely to be very small, although they
appear to be detectable by the cumulative analysis technique.
Insufficient time and information precluded this study
from determining which of the above phenomena was responsible
for the movement for each disturbance data at each station.
43
IV. SUMMARY AND CONCLUSIONS
This thesis investigated the problem of determining the
rate of land elevation change from monthly mean sea level
data. The techniques used by different authors are surveyed
as well as the different types of data and methods used for
the calculation of trends. The assumption of linearity of
the sea level trend used by previous authors in estimating
absolute rates of land elevation change is also discussed.
In addition, causes of water level variation contained in
monthly sea level time series are enumerated. Because of
the impossibility of separating the many components, it is
concluded that absolute values for the rates of either sea
level change or land elevation change cannot be found.
Accordingly, the use of a differential tide levelling pro-
cess was proposed in which the elevation of one tide station
relative to another is determined.
In the experimental phase, mean monthly sea levels at
seven tide stations on the Pacific Coast of the United States
were chosen for study. Spectral analysis was first performed,
and this showed peaked concentrations of energy around the
annual and semi-annual cycles for all stations. Except for
these frequencies, the spectrum up to the frequency of 0.5
cycles per month contained only noise, showing a typical decay
with increasing frequency. These concentrations of energy,
44
essentially of meteorological and oceanographic origin, were
then filtered out of the monthly station data using two fil-
tering processes, a 12-month running mean filter and an
anomaly filter. Descriptions of these filters are given in
the text. The effect of each filter on removing the annual
and semi-annual cycles without otherwise altering the prop-
erties of the monthly sea level data was then tested by com-
paring the power density spectrum of the filtered time series
with that of the raw monthly data. The results obtained favor
the use of the anomaly filter for future work because it
eliminates the concentration of undesirable energy in all
other frequencies found in the unfiltered data.
Several experiments were performed in computing sea level
trends from sea level difference data, both filtered and un-
filtered, using running intervals or windows with arbitrary
and non-arbitrary time lengths. It was found that the sea
level trend determined for a given station is dependent on
the window length used, the chronological location of the
window in the time series, and whether the monthly sea level
values used are unfiltered (raw) data, 12-month running mean
data, or anomaly data. Because of these totally time depend-
ent results obtained for sea level trends, attention was
redirected to the use of a new technique introduced by Wyss
[1977] involving cumulative analysis of time series data.
The method, termed here the cumulative analysis procedure,
which is further developed in this thesis, involves cumulation
45
of the monthly values from beginning to end of a tide series.
For this purpose, the monthly sea levels for the seven tide
stations studied (Seattle to San Diego) were combined to
yield 21 station pairs. The cumulative plot for a typical
station pair appears to be composed of straight line segments
and branches of parabolas. The linear segments represent
time intervals during which the elevation of the two stations
remains constant, while parabolic segments indicate a linear
rate of change in elevation occurring between the stations.
The segments vary in length up to 17 years. The fact that
each cumulative curve consists of several segments indicates
that differential vertical movements between tide stations,
whether separated by short or long distances, occur frequently.
Also noted in some of the cumulative curves are inflec-
tion points. These indicate a sudden vertical displacement
of one tide station relative to another. The cumulative
curve containing an inflection does not reveal at which sta-
tion the movement occurred, but the responsible station can
be identified from examination of two or more cumulative
curves containing the station.
The process of chopping the cumulative curve into seg-
ments (and of determining whether each segment is linear or
curvilinear) was found in some cases to be quite subjective.
Accordingly, tests were devised, and applied to selected
cumulative curves, to determine whether each segment iden-
tified is best represented by a linear or a parabolic fit.
46
The first test compares the residuals obtained between a
best fit parabola and the cumulative curve with residuals
obtained between a best fit line segment and the same cumu-
lative curve. The other test compares the sea level differ-
ence trend for each well defined segment of the cumulative
curve with the long term difference trend obtained for the
entire time series.
The cumulative procedure, although limited by subjective
interpretation of the cumulative curve, is sensitive to very
small sudden or continuous changes in the elevation of one
tide station relative to another. In addition, in contrast
to conventional geodetic methods, application of the proce-
dure produces a continuous history, month by month, of dif-
ferential elevation changes between two stations (over the
period of common tide measurements) . It is recommended that
the cumulative procedure be used in future analyses of tidal
data time series, although further investigation should be
done to refine the analysis techniques. It should also be
determined whether, for the tide stations studied, the changes
in elevation of one station relative to another are caused by
real land elevation changes or by changes in station datum,
or both.
47
/
TABLE I
TIME SERIES USED FOR EACH TIDAL STATION
c^ ^- c u 1 Number ~. ^ • Approximate Distance
Station Symbol ^^ ^ears ^^"^^ Series ^^^^^^en Stations
Seattle SE 76 1899-1974
Crescent City CC 25 1950-1974
San Francisco SF 80 1899-1978
Avila AV 14 1946-1959
Santa Monica SM 33 1933-1965
Los Angeles LA 55 1924-1978
San Diego SD 31 1931-1961
68 5 Km
481 Km
333 Km
2 44 Km
38 Km
153 Km
Note: The data format for all the stations was mean
monthly sea level values in feet on computer cards.
48
TABLE II
DIFFERENCES OBTAINED FOR LONG TERM TRENDS
WHEN USING RAW AND FILTERED DATA
Seattle
(1899-1974)
San Francisco
C1899-1974)
Santa Monica
(1933-1965)
Los Angeles
(1924-1978)
Record length: 76 yrs
76 yrs
33 yrs
55 yrs
Raw data: b 1.922609 1.911846 2.814878 0.590955
Sy.x 84.77 59.263117 63.338516 60.829502
Sb 0.127954 0.089446 0.334117 0.149132
12 Month b 1.912255 1.886021 2.541461 0.510632
running Sy.x 30.506815 29.897564 29.283841 26.579253
means Sb 0.046890 0.045953 0.161142 0.066826
Anomalies: b 1.927965 1.905879 2.718758 0.555566
Sy.x 62.328436 50.409106 42.951712 39.328127
Sb 0.094073 0.076083 0.226574 0.096418
where: b is the slope of the long term trend in mm/yr;
positive values indicate a rising sea level
trend for the entire record.
Sy.x is an estimate of the error standard deviation
in mm.
Sb is the standard error of the slope in mm/yr
about the long term trend. It was calculated
as if the fluctuation of the monthly values are
independent. These estimates of the standard
error are almost certainly too small.
49
TABLE III
COMPARISON OF SPECTRA FOR SEATTLE DERIVED FROM
THE RAW DATA, 12 -MONTH, RUNNING MEANS, AND
ANOMALIES (16 DEGREES OF FREEDOM)
(1)
Period
(months)
(2)
Frequency
(eye. /mo.
Energy Density
(feet sq. x month)
(3) (4) (5)
Raw 12 mo. Anomalies
) Data r.m.
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(3)/(4) (3)/(5)
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0.016
0
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0.130
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1.330
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21.333
0.047
0
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0.024
0.
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3.791
1.151
12.800
0.078
1
.324
0.002
0.
112
662.000
11.820
9.143
0.109
0
.214
0.004
0.
098
53.500
2.183
7.111
0.141
0
.103
0.001
0.
067
103.000
1.537
5.818
0.172
0
.188
0.000
0.
070
00
2.685
4.293
0.203
0
.084
0.002
0.
066
42.000
1.272
4.267
0.234
0
.072
0.001
0.
080
72.000
0.900
3.765
0.266
0
.043
0.000
0.
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oo
1.264
3.368
0.297
0
.075
0.001
0.
069
75.000
1.086
3.048
0.328
0
.059
0.000
0.
053
00
1.113
2.783
0.359
0
.052
0.000
0.
053
CO
0.981
2.560
0.391
0
.030
0.000
0.
027
00
1.111
2.370
0.422
0
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0.000
0.
054
00
0.888
2.207
0.453
0
.062
0.000
0.
060
00
1.033
2.065
0.484
0
.033
0.000
0.
030
oo
1.100
2.000
0.500
0
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0.000
0.
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oo
1.041
Note: Only alternative values of frequencies are shown
(Af= 0.0155 cycles/month).
50
TABLE IV
SEA LEVEL TRENDS OBTAINED FOR THREE STATIONS
USING DIFFERENT LENGTHS OF TIME SERIES
Station
Time Period
Tre
Raw data
nd
12
(mm/yr)
mo.r.m.
Anc
)malies
Seattle
1899-
■1974 (
'76
yr)
1.
92261
1.
91226
1.
92796
Seattle
1899-
■1968 (
'70
yr)
1.
80475
1
74807
1.
81080
Seattle
1924-
•1973 (
'50
yr)
2.
73895
2.
71548
2.
74963
Seattle
1924-
■1947 (
'24
yr)
2.
44849
2.
79119
2.
50428
Seattle
1948-
■1965 (
:i8
yr)
1.
83066
1.
77530
1.
89668
San Francisco
1899-
■1978 (
'80
yr)
1.
83336
1
81182
1,
82822
San Francisco
1924-
■1973 (
'50
yr)
2.
36677
2.
348 99
2.
35055
San Francisco
1899-
■1968 (
'70
yr)
1.
87412
1.
84081
1.
86713
San Francisco
1924-
■1978 (
'55
yr)
2.
11264
2.
06595
2.
09943
San Francisco
1899-
■1974 (
76
yr)
1.
91185
1
88602
1.
90588
Los Angeles
1933-
-1962 (
:3o
yr)
1.
06396
0.
85891
0.
94343
Los Angeles
1924-
■1973 (
:5o
yr)
0.
63811
0.
61212
0.
59517
Los Angeles
1924-
■1947 (
:24
yr)
2.
41874
2
39378
2.
23160
Los Angeles
1948-
■1965 (
:i8
yr)
1.
47915
0
90488
1.
14511
Los Angeles
1924-
■1978 (
:55
yr)
0.
59096
0.
51063
0,
55557
51
TABLE V
TRENDS OF THE DIFFERENCE DATA FOR PAIRS
OF TIDE STATIONS FOR SPECIFIC TIME SERIES
c^ ^ • T^ • n • J Trend fmillimeters per year)
Station Time Period ^ .. ^^„ . „ ^^ -.-^ . ^
Partial Series Full Series
SE-LA 1924-1947 (24 yr) 0.0297 2.1375
SE-LA 1948-1965 (18 yr) 0.3513 (1924-1974)
SE-LA 1966-1974 (9yr) -0.0953
SF-CC 1950-1959 (10 yr) 7.1997 3.2682
SF-CC 1960-1964 (5 yr) 12.7439 (1950-1974)
SF-CC 1965-1974 (10 yr) 3.4339
52
TABLE VI
TRENDS OF ELEVATION CHANGE RELATIVE TO SAN FRANCISCO
Stations Time Period Trend (nun/yr) Standard Error of
the Slope (mm/yr)
SE-SF 1899-1910 2.34497 1.993378
1911-1927 1.01102 1.089108
1928-1931 -3.92303 8.252379
1932-1934 3.86056 19.600630
1935-1945 0.27098 2.021985
1946-1952 -7.65468 4.130359
1953-1960 -4.05034 3.501000
1961-1965 -12.63590 6.847633
1966-1974 -0.87969 2.938892
full series 1899-1974 -0.01075 0.119488
CC-SF 1950-1954 1.01228 4.633611
1955-1959 -10.63333 5.191369
1960-1960 -49.87460 46.494390
1961-1963 -15.23740 9.759881
1964-1974 -2.48234 1.627356
full series 1950-1974 -3.26817 0.455462
SF-AV 1946-1947 25.50820 12.868782
1948-1952 -7.61081 2.991260
1953-1959 -0.16552 2.754739
full series 1946-1959 -0.14199 0.862576
SF-LA 1924-1931 -2.19510 1.909139
1932-1947 1.75783 0.758469
1948-1960 1.21958 1.011479
1961-1970 3.62779 1.557846
1971-1978 -3.91764 2.136735
full series 1924-1978 1.52172 0.121347
Notes: 1. The trends were calculated from the raw monthly
data.
2. Discontinuities at San Francisco were found to
have occurred in 1931 and 1960.
53
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98
APPENDIX A-1
PROGRAM USED TC PERFORM SPECTRAL ANALYSIS
SUBROUTINE PREPFA( M, MS ,DT , YYY, Fl , PERI OD, FREQUE,NF )
C
C
C MEANING OF THE PARAMETERS
C
C
C MdNPUT) IS THE INTEGER POWER OF 2 ON THE EXPRESSION
C 2*2**M
C MS(INPUT) IS THE NUMBER OF WINDOWS (ONE HALF OF THE
C NUMBER OF DEGREES OF FREEDOM)
C DT(INPUT) IS THE X-INTERVAL BETWEEN DATA POINTS
C YYY( INPUT) IS THE ARRAY TO BE ANALISED
C Fi (OUTPUT) IS THE ARRAY CONTAINING THE ENERGY VALUES
C PERIOD AND FREQUE ( OUTPUTS ) ARE THE ARRAYS CONTAIMING
C THE VALUES OF THE PERIODS AND FREQUENCI ES, RESPECT I VELY
C NF(OUTPUT) IS THE NUMBER OF FREQUENCIES TO BE ANALISED
C PLUS ONE
C
C
DIMENSION YYY (1250), I NV(12 50) , S(I250) , FIS (1250) ,C( 1250
1),F1(1250)
DIMENSION ART (1250), PERIOD (600), FREQUE (6C0)
240 FORMATCl',' POWER SPECTRUM IS CALCULATED •,/ • TOTAL NU
♦MBER OF SAMPLES=' ,T5,/1X,»THE TIME INCREMENT =',F5.3,/
*1X,«THE NtMBER OF DEGREES CF FREEDOM FOR EACH SPECTRAL
* ESTIMATE =• ,15,///)
250 FORMAT(«0S 'STATISTICS OF SAMPLE NUMBER =SI4)
260 FORMAT ( 'C, 'MEAN VALUE =• , F12.3, 3X, • VARI ANCE =«,F12.3,
*3X,'SKEWNESS =• ,F12. 3 ,3X, • KURTOS IS =• , F8 .3,3X , 'STD DEV
99
♦lATION =• ,F12.3)
270 F0RMAT(«0«,T10,F12.3,T25,F12.3,T40,F12.3)
NM=2*2**M
NM1=NM-1
N=NM*MS
NF=2**M+1
NFREDM=MS*2
T=NM*OT
WRITE(6,240) N, DT, NFREDM
DO 510 1=1, NF
510 F1(I)=0.
IZ=0
K=0
CO 21 127=1, N
ART(I27)=YYY(I27)
21 CONTINUE
CALL AVER/i(ART,N,AMEAN)
DO 22 128=1, N
C(I28)=ART(I28)-A^iEAN
22 CONTINUE
DO 520 MI=1,MS
K=K+1
DO 530 JJ=1,NM
IZ=IZ+1
530 F1S(JJ)=C(IZ)
WRITE (6,250) K
CALL TREND(F1S,NM,DT,U11,U21,U31,U41,URMS1)
CALL SPEC(F1S,M,INV,S,IFERR}
WRITE (6,260) Ul 1 , U2 1 , U31 , U41 ,URMS1
00 540 1=1, NF
540 F1(I)=F1S(I)+F1(I)
520 CONTINUE
DO 550 1 = 1, NF
F1(I)=F1(1)*T/(2.*MS)
100
IFd.EQ.DGC TO 23
PERI0D(I)=FL0AT(NM)/FL0AT(I-1)
FREQUE( I)=1.0/PERI00(I)
GO TO 24
23 PERIOD(I)=FLOAT(NM)
FREQUE(I)*0.0
24 V»RITE(6,270)F1(I) , PERI GD ( I ) , FREQUE { I )
550 CONTIMUE
CALL PL0TG(FREQUE,F1,NF, 1,1, I, 'FREQUENCY IN CYCLES PER
* MONTH* ,29, •
IPOWER DENSITY FUNCT ION ( FEET SQ .MONTH) •, 37, 0 . 0, 0.0, 0.0,
*G.0,7.5,5.0)
CALL PLOKO. 0,0.0, +999)
RETURN
END
101
APPENDIX A-2
PROGRAM USED TO PERFORM LEAST SQUARES BEST FIT
SUBROUTINE LEASTSC LLFIN, XX, YY, TREND, ERREST, ERRTRE )
C
c
C WEANING CF THE PARAMETERS
C
C
C LLFIN( INPUT) IS THE NUMBER OF DATA POINTS
C XX AND YYdNPUT) ARE THE ARRAYS CONTAINING THE X AND Y
C VALUES OF THE DATACY VALUES ASSUMED IN FEET)
C TRENO(OUTPUT) IS THE SLOPE OF THE REGRESSION LIN IN MM
C PER YEAR
C ERREST(OUTPUT) IS THE STANCARO ERROR OF THE ESTIMATE
C IN MM
C £RRTRE(OUTPUT) IS THE STANDARD ERROR OF THE TREND IN
C MM PER YEAR
C
C
DIMENSION XX(999),YY(999)
DOUBLE PRECISION SUMMX, SUMMY, SUKMXX, SUMMYY ,SUMMXY , TREN
*D, ERREST, ERRTRE
210 FORMAT CO*)
220 FORMAT( 1X,3D24,15/)
C0NST1=3657.6
CONST2=30A.8
SUMMX=0.0D0
SUMMY=0.0D0
SUMMXX=0.0D0
SUMMYY=0.0C0
SUMMXY=0.0D0
00 10 LS = 1, LLFIN
102
SUMMX=SLNMX+XX(LS)
SUMMY=SUMMY+YY(LS)
SUMMXX=SLN',MXX+XX( LS) **2
SUMMYY = SlJMMYY+YY( LS ) **2
SUMMXY=S(jMMXY+XX(LS)*YY(LS)
10 CONTINUE
TREND=(SUMMXY-SUMMX*SUMMY/FLOAT(LLFIN) ) / ( SUMMXX-( SUMMX
***2)/FL0AT(LLFIN))
ERREST=DSQRT( {SUMMYY-SUMMY**2/FL0AT( LL FIN)-TREND*( SUMM
*XY-SUMMX«SUMMY/FLOAT( LLFIN)) ) /FLOAT( LLFIN-2 ) )
ERRTRE=ERREST/DSQRT(SUMMXX-SUMMX*«2/FL0AT(LLFIN)»
TREND=TRENC*C0NST1
ERREST=ERREST*C0NST2
ERRTRE=ERRTRE*C0NST1
KRITE(6,220)TREND,ERREST,ERRTRE
WRITE(6,210)
RETURN
END
103
APPENDIX A-3
PROGRAMS USED TO PERFORM FILTERING ACTION
SUBROUTINE RMEANl (NPER, LCOM,LF IM,DXREDT, OYREOT,NDI i,X,
*Y»XB,YB,LCATA)
C
C
C MEANING CF THE PARAMETERS
C
C
C NPER(INPUT) IS THE NUMBER OF PERIODS CF CONTINUOUS DA-
C TACMORE THAN 12 POINTS) WITHOUT MISSING VALUES TO BE
C ANALISED
C LCCM AND LFIM(INPUT) ARE THE ARRAYS CONTAINING THE IN-
C FORMATION CF THE BEGINING AND ENDING POINTS OF EACH
C PERIOD
C DXRECT AND DYREOT( INPUT) ARE THE ARRAYS CONTAINING THE
C X AND Y VALUES TO BE FILTERED
C NDIM(INPUT) IS THE DIMENSION THAT SHOULD BE GIVEN TO
C DXREOT AND DYREOT
C X AND Y( INPUT) ARE DUMMY ARRAYS FOR SCALING PURPOSES
C IF SOME PLOT IS INTENDED TO BE DONE WITH THE OUTPUT
C ARRAYS
C XB AND YB(CUTPUT)ARE THE ARRAYS CONTAINING THE X AMD Y
C VALUES ALREADY FILTERED
C LDAT(OUTPLT) IS THE NUMBER OF DATA POINTS CONTAINED IN
C THE CUTPLT ARRAYS
C
C
DIMENSION X(4),Y(4)
DIMENSION LC0M(20) ,LFIM( 20)
DIMENSION DXREOT { NO I M ), DYREOT (NO I M),XA( 1000) ,YA(1000)
104
OIMENSICN XB(IOOO) ,YB(1000)
LCOUNT^l
DO 62 L=1,NPER
LBEG=LCC^(L)
LSTOP=LFIM(L)
LEND=LBEG+11
SUM=0.0
DO 60 L1=LBEG,LEN0
SUM=SUN+0YRE0T(L1)
60 CONTINUE
N0ATA=LST0P-LBEG-10
00 61 L2=1,NDATA
YA(L2)=SUM/12.0
XA(L2) = DXRE0T(LBEG)-^4.5+FL0AT(L2)
YB(LC0UNT)=YA(L2)
XB{LCCUMI=XA(L2)
LC0UNT=LCCUNT+1
SUM=SUM-DYRE0T(LBEG-1+L2)+DYRE0T(LBEG+11+L2)
61 CONTINUE
62 CONTINUE
LDATA=LCCUNT-1
XB(LDATA+1)=X(3)
XB(LDATA+2)=X(4)
YB(LDATA+1)-Y(3)
YB(LDATA+2)=Y(4)
RETURN
END
105
SUBROUTINE ANOMAL( NY ,NMO, YYY, WWW )
C
C
C MEANING CF THE PARAMETERS
C
C
C NY(INPUT) IS THE NUMBER OF YEARS OF DATA
C NMO(INPUT) IS THE NUMBER OF MONTHS OF DATA
C YYY( INPUT) IS THE ARRAY VALUES TO BE FILTERED
C WWW(OUTPUT) S THE ^RRAY VALUES AFTER BEING FILTERED
C
c
DIMENSION YYY (1250) , YYU 12 50) , WWW( 1250 )
DIMENSION ARRTRA(12)
210 F0RMAT(1X,F12.3)
220 FORMAT(«0S3F12.3)
CO 05 1 = 1, NMC
YYl(I)=yYY(I)
05 CONTINUE
CALL TREND(YY1,NM0,1.0,U11,U21,U31,U41,URMS1)
DO 20 11=1,12
ARRTRA(I1)=0.0
J = I1
DO 10 1 2=1, NY
ARRTRA(I1)=ARRTRA(I1)+YY1(J)
J=J+12
10 CONTINUE
ARRTRA( I1)=ARRTRA( ID /FLOAT(NY)
WRITE(6,210)ARRTRA(I1)
20 CONTINUE
CO 30 13=1, NMO
MM=M00(I3,12)
IF{MM.EQ.C)MM=12
WWW(I3)=YYY(I3)-ARRTRA(MM)
106
30 CONTIMUE
RETURN
END
107
APPENDIX A-4
PROGRAM LSEC TO PERFORM CUMULATIVE ANALYSIS
SUBROUTINE CUMMUL( NY , YEAR IN)
C
C
C MEANING OF THE PARAMETERS
C
c
C NY(INPUT) IS THE NUMBER OF YEARS OF MEAN MONTHLY
C VALUES TO BE ANALISED
C YEARIN(INPUT) IS THE FIRST YEAR OF DATA
C
C
DIMENSION X(4),Y(4)
DIMENSION XNB(IOOO) ,XAS(IOOO)
DIMENSION YNB(IOOO) ,YAS(1000)
DIMENSION DYNBASdOOO) , DXNBAS ( 1000 )
DIMENSION CUMCIOOO)
DIMENSION WNB(IOOO) tWAS(lOOO)
201 F0RMAT(«0«t3F10.2)
NM0=NY*12
CALL PAGE
CALL REACCT(NY,X,Y,YNB,XNB,NP)
CALL READCT(NY,X,Y,YAS,XAS,NP)
DO 05 1=1, NMO
WNB(I) -YNB(I)
WAS(I)=YAS(I)
05 CONTINUE
DO 30 III=lf3
CALL AVERA(YNBtNP,AMEANl)
CALL AV£RA(YAS,NP,AMEAN2)
108
DIFF=AMEAN1-AMEAN2 OR D IFF=YNB ( 1 )-YAS ( 1 )
WRITE(6,201)AMEAN1,AMEAN2,DIFF
CO 10 1=1, NMO
YAS(n=:YAS(I)+DIFF
10 CONTINUE
CALL DIFFER(0,NMO,YNB,YAS,X,Y,DYNBAS,DXNBAS,ND)
DELTAT=1./12.
CUM(li=DYNeAS(l)
IF(III.EQ.3)YEARIN=YEARIN+0.5
XAS( D^YEARIN
DO 20 1=2, NMO
CUM(I)=DYNBAS(I)+CUM{I-I)
XAS^ I ) =XAS( I-1)+DELTAT
20 CONTINUE
CALL PLOTG(XAS,CUM,NMG, 1,1,1, 'TIME ( YEARS )», 12, 'CUMMUL
ISM-SD (FEET) ',30, 0.0, 0,0, 0.0, 0.0, 7. 5, 5.0)
IF(III.EG.2)G0 TO 77
IF(III.EQ.3)G0 TO 30
CALL ANOMAL(NY,NMG,YNB,YNB)
CALL ANCNAL(NY,NMO,YAS,YAS)
GO TO 30
77 CALL RMEAM(01,01,NM0,XNB,WNB,10C0,X,Y,XNB,YNB,NP)
CALL RMEAN1(01,01,NMO,XNB,WAS,1000,X,Y,XAS,YAS,NP)
NMO=NP
30 CONTINUE
CALL PLOT(0. 0,0.0, +999)
RETURN
END
109
APPENDIX A-5
CTHER SUBROUTINES CALLED
SUBROUTINE /SVERA (A,NPTS, AMEAN)
C
C
C MEANING CF THE PARAMETERS
C
c
C A( INPUT) IS THE ARRAY OF Y VALUES TO BE AVERAGED
C NPTS(INPUT) IS THE NUMBER OF DATA POINTS OF A
C AMEAN(OUTPUT) IS THE AVERAGE OF THE Y VALUES OF THE
C INPUT ARRAY
C
C
DIMENSION A(NPTS)
SUM=0.0
CO 100 I=1,NPTS
SUM=SUM+A(I)
100 CONTINUE
AMEAN=SUM/FLOAT(NPTS)
RETURN
END
110
SUBROUTINE TREND( FX ,NTS, DT,FMEAN ,U2, U3 t U4,URM S)
C
C SUBROUTINE TREND EDITS ,CAL IBRATES AND DETRENDS DATA
C
c
C MEANING CF THE PARAMETERS
C
c
C FX(INPUT) IS THE ARRAY OF Y VALUES, (OUTPUT) IS THE DE-
C TRENDED ARRAY OF Y VALUES
C NTS( INPUT) IS THE NUMBER OF DATA POINTS
C DT(INPUT) IS THE X-INTERVAL BETWEEN DATA POINTS
C FMEAN,U2»U3,U4,AND URMSC OUTPUTS) ARE THE MEAN, VAR lANCE
C SKEWNESS,KURTOSIS AND STANCARD DEVIATION OF THE INPUT
C
c
DIMENSION FX(NTS)
C EDITING CATA
C
C
FNTS=NTS
C COMPUTING THE LINEAR TREND
SUMF=0.0
DO 101 1=1, NTS
101 SUMF=SUMF+FX{I)
SUMF1=0.0
CO 102 1=1, NTS
XI = I
102 SUMF1=SUMF1 + XI*FX{ I)
XNM1=NTS-1
XNP1=NTS+1
XM=(1.0/CT)«(12.0*SUMF1/(FNTS*XNM1*XNP1)-6,0*SUMF/(XNM
*1*FNTS))
B=SUMF/FNTS-XM*XNPl*DT/2.0
111
FMEAN=SUMF/FNTS
WRITE (6,9) FMEAN,XM,B
9 FORMATOX, 'MEAN-S F 10 .5, 3X, • SLOPE = • , FIO .5, 3X, • I NTERC
*EPT =• .F10.5,//)
DO 103 1 = 1, NTS
XI = I
103 FX(I)=FX(I)-(B+XM*XI*DT)
C SUBROUTINE FOR CALCULATING VARIANCE, STO DEV, SKEWNESS
C *, KURTOSIS
U2==0.0
L 3=0.0
L4=0.0
SUMU2=0.0
SUMU 3=0.0
SUMU4=0.0
00 151 1=1, NTS
U2=FX(1)*FX(I)
U3=U2*FX(I)
U4=U3*FX(I)
SUMU2=SUML2+U2
SUMU3=SUMU3+U3
SUMU4=SUMU4+U4
151 CONTINUE
FNTS=NTS
L2=SUMU2/FNTS
URMS=SQRT(U2)
t3=SUMU3/(FNTS*U2*URMS)
L4=SUMU4/(FNTS*U2*U2)
RETURN
END
112
SUBROUTINE SPEC ( Fl, M, INV, S, IFERR)
C
C SUBROUTINE TO CALCULATE THE POWER SPECTRUM OF A SIGNAL
C *USING RHfiBt*,
C
C
C MEANING OF THE PARAMETERS
C
C
C FKINPUT) IS THE DETRENDED ARRAY OF Y VALUES. A MODIFI-
C ED ARRAY APPEARS AT THE OUTPUT
C M( INPUT) IS THE INTEGER POWER OF 2 OF THE EXPRESSION
C 2*2**M
C INV(ARRAY) ,S(ARRAY) ,AND I FERR (OUTPUTS ) ARE OUTPUTS OF
C SUBROUTINE RHARM
C
C
DIMENSION INV(515),S(515) ,Fl(515)
CALL RHAPM(F1,M,INV,S,IFERR)
NP=2*«M-1
NF=2**M+1
NM=2*2«*M
NL=NM+1
F1(1)=F1(1)*F1(1)
CO 500 I=1»NP
J= 2*1+1
L=I + 1
XR=F1{ J)*F1(J)
XI=F1(J+1)*F1(J+1)
F1(LJ=XR+XI
500 CONTINUE
F1(NF)=F1(NL)**2
RETURN
END
113
SUBROUTINE READOT ( MUMYEA, X, Y, YNB, XNB, NP )
C
C
C THIS SUBRCLTINE READS AND PRINTS THE DATA
C
C
C MEANING CF THE PARAMETERS
C
C
C KUMYEA(INPUT) IS THE NUMBER OF YEARS OF DATA TO READ
C X AND Y( INPUT) ARE DUMMY ARRAYS FOR SCALING PURPOSES
C IF SOME PLOT IS INTENDED TC BE DONE WITG TflE OUTPUT
C ARRAYS
C YNB AND XNE(OUTPUT) ARE THE ARRAYS CONTAINING THE DATA
C VALUES AND THE GENERATED ABSCI SSAS, RESPECTIVELY
C NP(OUTPUT) IS THE NUMBER OF DATA POINTS
C
C
DIMENSION X(4),Y(4)
DIMENSION YNB (1250) , XNB (1250), I YEAR (105)
100 FORMAT(A4,7X,I4,12F5.2)
200 F0RMAT(1X,A8,2X,I4,I2F5.2J
210 FORMAT COM
ISTART=1
I£ND=12
DO 10 I1=1,NUMYEA
READ (5, 100) NAME, I YEAR (ID ,(YNB( INB) , I NB= I START, I END)
WRITE(6,200)NAME,IYEAR(I1) ,(YNB( INB) , INB=ISTART, I END)
ISTART=ISTART+12
IEND=IEND*12
10 CONTINUE
NP=:IEN0-12
NUMM0N=NUMYEA*12
114
DO 20 I2=1,NUMM0N
XNB(I2)=FL0AT(I2I
20 CONTINUE
XNB(NP-»-l) = X(3)
XNB(NP+2J=X(4)
YNB(NP+1)-Y(3)
YNB{NP+2)=Y(4)
WRITE(6«210)
RETURN
END
115
SUBROUTINE CIFFER ( I DELAY , I COM , YNB» YAS, X , Y , DYNEAS, OXNBA
C
C
C THIS SUBROUTINE PERFORMS DIFFERENCES BETWEEN TWO DATA
C SETS, PRINTING THE RESULTS
C
c
C MEANING OF THE PARAMETERS
C
c
C IDELAYdNPUT) IS THE NUMBER OF MONTHS OF DELAY BETWEEN
C THE FIRST AND SECOND TIME SERIES ( POSI T IVE IF THE SEC-
C CIMD TIME SERIES BEGINS EARLIER THAN THE FIRST ONE)
C ICOMdNPUTJ IS THE NUMBER OF COMMON MONTHS TO BE
C SUBTRACTED
C X AND Y( INPUT) ARE DUMMY ARRAYS FOR SCALING PURPOSES
C IF SOME PLOT IS INTENDED TO BE DONE WITH THE OUTPUT
C ARRAYS
C YNB AND Y/S( INPUT) ARE THE ARRAYS CONTAINING THE DATA
C POINTS TC BE SUBTRACTED
C OYNBAS AND DXNBAS ( OUTPUT) ARE THE ARRAYS CONTAINING
C THE RESULTS FROM THE DIFFERENCE COMPUTATION AND THE
C COMMON ABSCISSAS, RESPECTIVELY
C ND(OUTPUT) IS THE NUMBER OF DATA POINTS CONTAINED IN
C THE OUTPUT ARRAYS DYNBAS AND DXNBAS
C NCTE-THIS PROGRAM ASSUMES THAT THE MISSING VALUES OF
C THE CATA ARE REPRESENTED BY 99.9 OR 00.0. IF
C SOME MISSING VALUES OCCUR, ND WILL BE LESS THAN
C ICCf'
C
C
CIMENSION YNB(I250), YAS(1250)
DIMENSION DXNBAS{999) ,DYNBAS(999)
116
DIMENSION X(4),Y(4)
210 FORMAT CO')
220 F0RMAT(1X,2F6.2, 110)
NN-1
lNIC=(IABS(IDELAY)-IDELAY)/2^-l
A6C=FL0AT(INIC)
IST0P=INIC-1+IC0M
DO 20 JN80=INIC,ISTOP
JNBA=JNBO+IDELAY
IF{YNB( JNB0).GT.90.0.0R.YNB(JNB0) .LT.O.DGO TO 2000
IF(YAS(JNBA).GT.90.0.0R.YAS(JNBA) .LT.O.DGO TO 2000
DYNBAS{NN)=YNB(JNBO)-YAS(JNBA)
DXNBAS(NN)=ABC
WRITE(6,220)DXNBAS(NN),DYNBAS(NN) ,NN
NN=NN+1
2000 ABC=ABC+1.C
20 CONTINUE
ND=NN-1
DXNBAS(NC+1)=X(3)
DXNBAS(ND+2)=X(4J
CYNBAS(NC+1)=Y(3)
DYNBAS(ND+2)=Y(4)
WRITE(6f210l
RETURN
END
117
BIBLIOGRAPHY
Balazs, E.I., and B.C. Douglas, "Geodetic Levelling and the
Sea Level Slope Along the California Coast," National
Oceanic Survey, NOAA (unpublished).
Bloomf ield. P., "Trend Estimation with Autocorrelated Errors,
with Physical Application," lectures delivered at the
Naval Postgraduate School, Monterey, California, July
1980.
Bretschneider , D.E., Sea Level Variations at Monterey, Cal-
ifornia, M.S. Thesis, Navy Postgraduate School, Monterey,
California, March 1980.
Gutenberg, B. , "Changes in Sea Level, Postglacial Uplift, and
Mobility of the Earth's Interior," Bulletin of Geological
Society of America, Vol. 52, p. 721-772, 1 May 1941.
Hicks, S.D., and W. Shofnos, "The Determination of Land
Emergence from Sea Level Observations in Southeast Alaska,"
Journal of Geophysical Research, Vol. 70, No. 14, P. 3315-
3320, 15 July 1965a.
Hicks, S.D., and W. Shofnos, "Yearly Sea Level Variations for
the United States," Journal of the Hydraulics Division -
Proceedings of the American Society of Civil Engineers,
HYS, p. 23-32, September 1965b.
Hicks, S.D., "On the Classifications and Trends of Long Period
Sea Level Series," Shore and Beach, April 1972a.
Hicks, S.D., "Vertical Crustal Movements from Sea Level
Measurements Along the East Coast of the United States,"
Journal of Geophysical Research, Vol. 77, No. 30, p.
5930-5934, 20 October 1972b.
Lisitzin, E. , Sea Level Changes, Elsevier Scientific Publish-
ing Company, 1974.
Merry, C.L., "Processing of Tidal Records at Hount Bay
Harbour," International Hydrographic Review, Monaco,
LVII (1), p. 149-154, January 1980.
Roden, G.I., "Sea Level Variations at Panama," Journal of
Geophysical Research, Vol. 68, No. 20, p. 5701-5710,
15 October 1963.
118
Roden, G.I., "Low Frequency Sea Level Oscillations Along the
Pacific Coast of North America," Journal of Geophysical
Research, Vol. 71, No, 20, p. 4755-4776, 15 October 1966
Smith, R.A. , and R.J. Leffler, "Water Level Variations Along
California Coast," Journal of the Waterway Port Coastal
and Oceanic Division, p. 335-348, August 1980.
Smith, R.A. , "Golden Gate Tidal Measurements: 1854-1978,"
Journal of the Waterway Port Coastal and Ocean Division,
p. 407-410, August 1980.
Whitcomb, J.H. , "Regional Crustal Distortion Events in
Southern California; A Confirmation of Jerky Plate
Motion?, EOS, Vol. 61, No. 17, p. 368, 22 April 1980.
Wilson, M.E., "Tectonic Tilt Rates Derived from Lake-Level
Measurements, Salton Sea, California," Science, Vol.
207, p. 183-186, 11 January 1980.
Wyss, M., "The Appearance Rate of Premonitory Uplift,"
Bulletin of the Seismological Society of America , Vo 1 .
67, No. 4, p. 1091-1098, August 1977.
119
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189906
Abreu
Determination of land
elevation changes using
tidal data.
Thesis
AI725
cj
189906
Abreu
Determination of land
elevation changes using
tidal data*
thesA1725
Determination of land elevation changes
illliiililliililllll
3 2768 001 90890 8
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