Storm surge simulation in transformed coordinates

>. Avurip oo Cug,
TR 76-3
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Storm Surge Simulation in

Transformed Coordinates |

VOLUME I

Program Documentation

by
John J. Wanstrath

TECHNICAL REPORT NO. 76-3
NOVEMBER 1976

y
DOCUMENT |

Approved for public release;
distribution unlimited.
Prepared for

U.S. ARMY, CORPS OF ENGINEERS

COASTAL ENGINEERING
=) 6 RESEARCH CENTER

GB Kingman Building

) ASO Fort Belvoir, Va. 22060
ODES
Ys 16-3,

Reprint or republication of any of this material shall give appropriate
credit to the U.S. Army Coastal Engineering Research Center.

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Contents of this report are not to be used for advertising,
publication, or promotional purposes. Citation of trade names does not
constitute an official endorsement or approval of the use of such
commercial products.

The findings in this report are not to be construed as an official
Department of the Army position unless so designated by other
authorized documents.

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ION

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READ INSTRUCTIONS
T. REPORT NUMBER 2. GOVT ACCESSION NO|| 3. RECIPIENT'S CATALOG NUMBER
TR 76-3

4. TITLE (and Subtitle) 5. TYPE OF REPORT & PERIOD COVERED
STORM SURGE SIMULATION IN TRANSFORMED COORDINATES
VOLUME I. THEORY AND APPLICATION
VOLUME II. PROGRAM DOCUMENTATION

Technical Report

6. PERFORMING ORG. REPORT NUMBER

8. CONTRACT OR GRANT NUMBER(s)

7. AUTHOR(S)
John J. Wanstrath, Robert E. Whitaker,
Robert O. Reid, and Andrew C. Vastano

DACW72-73-C-0014

10. PROGRAM ELEMENT, PROJECT, TASK
AREA & WORK UNIT NUMBERS

9. PERFORMING ORGANIZATION NAME AND ADDRESS

Texas AGM Research Foundation
oe.) Box \H

College Station, Texas 77843
11. CONTROLLING OFFICE NAME AND ADDRESS
Department of the Army

Coastal Engineering Research Center

Kingman Building, Fort Belvoir, Virginia 22060
14. MONITORING AGENCY NAME & ADDRESS(if different from Controlling Office)

A31231

12. REPORT DATE
November 1976
13. NUMBER OF PAGES

WO ko LOS Woil, iti, LYo

15. SECURITY CLASS. (of this report)

UNCLASSIFIED

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16. DISTRIBUTION STATEMENT (of thie Report)

Approved for public release; distribution unlimited.

17. DISTRIBUTION STATEMENT (of the abstract entered in Block 20, if different from Report)

18. SUPPLEMENTARY NOTES

19. KEY WORDS (Continue on reverse side if necessary and identify by block number)

Computer program Hurricane Gracie

Coordinate transformation Numerical modeling

Hurricane Camille Orthogonal curvilinear coordinates
Hurricane Carla Storm surge

20. ABSTRACT (Continue on reverse side if necesaary and identify by block number)
A two-dimensional time-dependent numerical storm surge model using
orthogonal curvilinear coordinates is presented. The curvilinear coordinate
system is based on a conformal mapping of the interior region bounded by the
actual coast, the seaward boundary (taken as the 180-meter depth contour) and
two parallel lateral boundaries into a rectangle in the image plane. Three
regions of the Continental Shelf of the Gulf of Mexico and two regions of the
eastern seaboard of the United States are mapped.

(Continued

DD , FORM 1473 EDITION OF 1 NOV 65 1S OBSOLETE

JAN 73 UNCLASSIFIED

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Since the transformation is conformal, the associated modifications of the
vertically integrated equations of motion and mass continuity are minimized.
The coast, seaward boundary, and the lateral boundaries of the computing grid
are straight lines in the image plane thus facilitating the application of the
boundary conditions. The final coordinates allow for the greatest resolution
near the coast in a central area of principal storm surge development and
modification.

The model is employed in the simulation of the storm surge induced by
Hurricanes Carla (1961) and Camille (1969) which crossed the gulf coast of the
United States and Hurricane Gracie (1959) which crossed the east coast. Ana-
lytical interpretations of the wind and atmospheric pressure-forcing functions
are used in the computations.

2
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PREFACE

This report is published te vrovide coastal engineers with the results
of a study to develop an oper. .onal program for numerical simulation of
storm surges on a given segment of the Continental Shelf, using a curvi-
linear coordinate system. The report consists of two volumes. Volume I
discusses the theory and application of the transformation procedure for
generating the curvilinear shelf coordinate system for particular regions,
and the theory, numerical algorithm, and application of the storm surge
program for simulation of Hurricanes Carla (1961), Camille (1969), and
Gracie (1959). Volume II presents the program documentation and the coded
programs for carrying out the coordinate transformation (CONFORM), for
establishing the spatial lattice (GRID), and for carrying out the storm
surge calculations on the shelf (SSURGE). The work was carried out under
the wave mechanics program of the U.S. Army Coastal Engineering Research
Center (CERC).

This volume was prepared by John J. Wanstrath; Volume I was prepared
by John J. Wanstrath, Robert E. Whitaker, Robert 0. Reid, and Andrew C.
Vastano, Department of Oceanography, Texas A&M University, College Station,
Texas, under CERC Contract No. DACW72-73-C-0014. Most of the computational
work in the development and application was carried out at the National
Center for Atmospheric Research which is supported by the National Science
Foundation.

The authors express their appreciation to Thomas J. Reid for assist-
ance in program coding, and to Dr. D. Lee Harris, CERC, for very con-
structive comments on the draft of this report.

Dr. D. Lee Harris, Chief, Oceanography Branch, was the CERC technical
monitor of the report, under the general supervision of Mr. R.P. Savage,
Chief, Research Division.

Comments on this publication are invited.

Approved for publication in accordance with Public Law 166, 79th
Congress, approved 31 July 1945, as supplemented by Public Law 172, 88th
Congress, approved 7 November 1963.

Colonel, Corps of Engineers
Commander and Director

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CONTENTS

CONVERSION FACTORS, U.S. CUSTOMARY TO METRIC (ST)

I INTRODUCTION.

Il COMPUTER PROGRAM DOCUMENTATION FOR PROGRAM-CONFORM.

ONAN HRWNe-

Program Purpose
Program Description .

. Type of Computer.

Input Data.

Mathematical peoeeduees a Soy ye eean fami eaeions
Flow Chart.

Glossary and Description ioe Terms

Input and Output.

IIl COMPUTER PROGRAM DOCUMENTATION FOR PROGRAM-GRID .

ONIADMHHRWNe

. Program Purpose .

Program Description .
Type of Computer.
Input Data.

. Mathematical procedures rl Proce Lamitersions

Flow Chart. 6
Description of Teance
Input and Output.

IV COMPUTER PROGRAM DOCUMENTATION FOR PROGRAM-SSURGE .

APPENDIX

ONAN HBWNE

Program Purpose .
Program Description .

. Type of Computer.

Input Data.

. Mathematical Beawedhontes an ceoenam Lantlzerhions

Flow Chart. 5 6
Glossary and Doseeipelon “of Teens
Input and Output.

A FORTRAN LISTING OF PROGRAM CONFORM.

B FORTRAN LISTING OF PROGRAM GRID .

C FORTRAN LISTING OF PROGRAM SSURGE .

CONVERSION FACTORS, U.S. CUSTOMARY TO METRIC (SI)
UNITS OF MEASUREMENT

U.S. customary units of measurement used in this report can be converted to metric (SI)

units as follows:

Multiply by To obtain
inches 25.4 millimeters
2.54 centimeters
square inches 6.452 square centimeters
cubic inches 16.39 cubic centimeters
feet 30.48 centimeters
0.3048 meters
square feet 0.0929 square meters
cubic feet 0.0283 cubic meters
yards 0.9144 meters
square yards 0.836 square meters
cubic yards 0.7646 cubic meters
miles 1.6093 kilometers
square miles 259.0 hectares
acres 0.4047 hectares
foot-pounds 1.3558 newton meters
ounces 28.35 grams
pounds 453.6 grams
0.4536 kilograms
ton, long 1.0160 metric tons
ton, short 0.9072 metric tons
degrees (angle) 0.1745 radians
Fahrenheit degrees 5/9 Celsius degrees or Kelvins!

' To obtain Celsius (C) temperature readings from Fahrenheit (F) readings, use formula: C = (5/9) (F — 32).
To obtain Kelvin (K) readings, use forumla: K = (5/9) (F — 32) + 273.15.

STORM SURGE SIMULATION IN TRANSFORMED COORDINATES

Volume II. Program Documentation

by
John J. Wanstrath

I. INTRODUCTION

Presented in this volume are the program documentation and list-
ings of the coded programs for carrying out a simulation of a
hurricane-induced storm surge on the Continental Shelf in curvilinear
coordinates for a given reach of shelf. There are three separate
programs detailed here for this purpose.

The first program, CONFORM, is employed for evaluation of the
transformation coefficients which map the given reach of the Conti-
nental Shelf into a rectangle in the image plane, the shelf region
being delineated by a smoothed version of the coastline, the shelf
break (taken as the 180-meter depth contour in the examples), and
bounded laterally by two parallel straight lines. The theory and
several example applications of the transformation procedure are
given in Section II of Volume I. The example input for CONFORM con-
tained in the listings here are for the mapping of the region from
a section across Laguna Madre about 90 kilometers south of Brownsville
to Marsh Island. Particular care must be taken to follow the pro-
cedure exactly for the example if CONFORM is to be verified by obtain-
ing the given transformation coefficients. This procedure is given
explicitly in the CONFORM documentation.

The second program, GRID, develops the detailed computing grid
information, based on the mapping coefficients evaluated by CONFORM
plus coordinate stretching information supplied by the user (see
Section III of Volume I). Part of the output of GRID is a listing of
the grid positions which are required by the user in order to read
from appropriate charts the detailed bathymetry field, which is
necessary input for the final program SSURGE. The example data
supplied here for GRID are for the Laguna Madre to Marsh Island
region.

Program SSURGE (Shelf Surge) carries out the numerical integra-
tion of the storm surge equations in the transformed coordinate system
supplied by CONFORM and GRID, using a parametric representation of a
hurricane wind field and pressure field. The theory is given in Sec-
tion III of Volume I. The particular example data given here are for
Hurricane Carla and the Laguna Madre-Marsh Island grid system.

The appendixes to this volume contain detailed FORTRAN listings
of the three programs in this application. The data to be supplied
by the user for other applications are discussed in the documentation
of each of these programs.

II. COMPUTER PROGRAM DOCUMENTATION FOR PROGRAM-CONFORM

1. Program Purpose.

The purpose of the program is to determine the transformation
coefficients which will conformally map the interior region bounded
by the actual coastline, a seaward boundary curve, and two parallel
lateral boundaries into a rectangle in the image plane.

2. Program Description.

The program is written in FORTRAN IV language. This program and
the program GRID provide all the necessary computing grid data for
input to program SSURGE. The program GRID takes, as input, the
transformation coefficients and determines the computing grid infor-
mation (such as, scale factors, grid point locations, and, at each
grid point, the orientation of the &-axis to the x-axis).

The program CONFORM is composed of:

MAIN Defines constants. Reads and writes the
coordinates delineating the given coastline
and seaward boundary curve. Calls Subroutine
COEFFS.

SUBROUTINE COEFFS Determines the transformation coefficients.
At the completion of each iteration, the
coefficients, the variance between the
transform-generated curves and that Speci-
fied, and other pertinent information are
written. COEFFS interfaces all other
program subroutines and functions.

FUNCTION XTRAN Is the transformation function x(&,n)
FUNCTION YTRAN ' Is the transformation function y(é,n)
SUBROUTINE SLFAC Determines the scale factor and derivatives,

ox/9—& , dy/d& , for a given value of €& and
m6

SUBROUTINE CURLYB Determines the necessary parameters to fit a
spline under tension through the given coast-
line coordinates. The spline is fitted with
Y2 as a function of X2

FUNCTION CURVYB Interpolates the given coastline, returning
a value of y at a specified value for x.

FUNCTION CURDYB Differentiates the given coastline, returning
a value of dy/dx at a specified value for
xe

SUBROUTINE CUR2YB Determines the necessary parameters to fit a
spline under tension with X2 as a function
of coastline arclength.

FUNCTION CUR4YB Interpolates the given coastline returning a
value of x at a specified value for arclength.

SUBROUTINE CURSYB Determines the necessary parameters to fit a
spline under tension with Y2 as a function of
coastline arclength.

FUNCTION CURSYB Interpolates the given coastline returning a
value of y at a specified value for
arclength.

There are identical subroutines and functions as delineated above
for the seaward boundary curve specified by coordinates X2P and
Y2P . These subroutines and functions are recognized by the same
names as their counterparts with a terminal letter A. For example,
SUBROUTINE CUR1YA determines the necessary parameters to fit a spline
under tension through the given seaward boundary curve.

3. Type of Computer,

The program CONFORM can be run on any computer with minimum core
requirements of approximately 24K (based on the present sample program).
However, significantly more computer memory would be required if one
desires a large number of coefficients and/or numerous integration
points, The program has been executed successfully on IBM 360, CDC/
6600 and 7600, and GE/635, The present sample program requires no
auxiliary storage devices, peripherial devices, or magnetic tape input
or output. No site-orientated computer plot. routines are involved in
the program, Approximately 20 minutes of machine time on a CDD/7600
is required for the sample program (total number of coefficients,

2 x NMAX = 220; number of integration points, 0 < —€ <r , = 110; and
number of iterations = 80).

4, Input Data,

Input data, other than constants defined in MAIN, are read in
MAIN on IBM cards prepared according to the following list:

Gl) Gases i
IWANT, MQ, MOP, NMAX, Continuation flag, number of shore-
JMAS1, IL, VARWT line and seaward boundary coordinates

number of mapping coefficients, maximum number of
iterations, number of integration points, and the
convergence criterion in format 615, F5.3.

(2) Card Group 2

NZ NZ The x,y coordinates (units in x,y space) of
the given coastline in the region 0 <x<A
are read with one pair per card in format 3X,
B76 Sho 764 (laine i'50)) .

(3) Card Group 3

X2P,Y2P The x,y coordinates (units in x,y space) of
the given seaward boundary curve in the region
0 <x < are read with one pair per card in
format F7.2, 3X, F7.2 (limit 150).

Optional Card 4, Card Group 3, and Card 6

If IWANT = 1, indicating the program is being re-
started, the following cards must be supplied:

(4) Card 4

B, BZRO whe values of s)8ayand s Bossi units of length of
x,y space from the last iteration of the prev-
ious run in format 2E14.7.

(S) Card Group 5

COB, COC The NMAX cards containing the dimensionless
Fourier-type transformation coefficients from the
last iteration of the previous run in sequential
order with one pair per card (format 2E14.7). If
more coefficients are desired in the present run
than the previous one, blank cards should be
supplied for the difference.

(6) Card 6

WSX, WSY The value of the weighting factors for the sea-

WCX , WCY ward boundary curve (x and y component) and
coastline (x and y component) from the last
iteration of the previous run (4E14.7).

5. Mathematical Procedures and Program Limitations.

The conformal mapping relations, solutions for the transformation
coefficients, and the iterative procedure for determining the coeffi-
cients are presented in Volume I of this report. The mapping equa-
tions are sufficiently general to treat the situation where either or

both given curves are multivalued in y for a specified x value.
The only program limitation relates to the computer memory capacity.

No program error messages or consistency checks are incorporated in
this routine.

Ge RlownGhantr

MAIN - CONFORM

Read constants; MQ, MQP, NMAX,
IL, JMAX1, VARWT, IWANT

Read and write coordinates delineating
the coastline and seaward boundary curve;
X2 WA goal XA 5 AAP

a

inaktevauleezen Ay bee ROMs m NS X
WCX, WCY , COB(N) and COC(N)

Read transformation
coefficients and
weighting factors

CALL SUBROUTINE COEFFS

SUBROUTINE COEFFS

Compute coastline arclength

Arclength computed directly
from given coordinates,
O25 2)

Yes Consequently, assumed that
used has sufficient data
points to accurately define
the curve.

Stored in ARCYB

COASTLINE HAS
FOLDBACK,
TYB=1

No

_ CALL CUR1YB
Arclength computed from a spline
fitted through Y2 as a function
of X2 with 9 additional points
between the given coordinates to
further resolve the curve.
Stored in ARCYB.

CALL CUR2YB
Fit a spline through X2 as a function of ARCYB

CALL CUR3YB
Fit a spline through Y2 as a function of ARCBY

Compute seaward boundary arclength; the same

procedure as for the coastline is followed.

Arclength computed
directly from X2P,
WAP.

Stored in ARCYA

CALL CURLYA
Arclength computed from a
spline. Stored in ARCYA

CALL CUR2YA

CALL CUR3YA

Write coastline and seaward boundary
coordinates and arclengths.

Compute CS and SN _ where

CS = cos (nké) ,
: for n=1,2 --+ NMAX
SN = sin (nk ‘ Zl
ee ema ES(iet) Gite)
where
dai sAi oo Wb/A.)\ MWh ChyGine
or
i=1,2 °° IL/2+1, IL odd.

Determine the x and y coordinates of the given
coastline and seaward boundary curve from the arc-
length of the transform-generated curves.

ATXB , ATBY and ATXA , ATYA

Compute B and _ BZRO

Compute COB and COC
for n=1,2 -- NMAX

Write transformation coeffi-
cients, iteration number,
NMAX and IL.

Compute the error function and variance between the trans-
form-generated curves and that specified. Write the values.
The x and y coordinates of the given curves are deter-
mined from the arclength of the new transform-generated
curves for the next iteration. Write the values.

COMPUTED VARIANCE

ICOUNT = ICOUNT + 1

7. Glossary and Description of Terms.

Arrays

X2 Dimensioned MQ. Contains values of the x coordinate in
units of x,y space of the given coastline (max, 150).

WZ Dimensioned MQ. Contains values of the y coordinate in
units of x,y space of the given coastline (max. 150).

X2P Dimensioned MQP. Contains values of the x coordinate in
units of x,y space of the given seaward boundary curve
(max, 150).

WiZB. Dimensioned MQP. Contains values of the y coordinate in
units of x,y space of the given seaward boundary curve
(max. 150).

COB Dimensioned NMAX. Contains values of the dimensionless
transformation coefficient BY (max. 200).

COC Dimensioned NMAX. Contains values of the dimensionless
transformation coefficient ce (max. 200).

DUMB Dimensioned NMAX. Contains the iterative value for BO
(max, 200).

DUMC Dimensioned NMAX. Contains the iterative value for Co
(max. 200).

ARCYB Dimensioned MQ. Contains values of the arclength for the
given coastline coordinates X2 , Y2 (max.150).

ARCYA Dimensioned MQP. Contains values of the arclength for the
given seaward boundary coordinates X2P , Y2P (max.150).

SN Dimensioned NMAX x IL1 where IL1 is IL/2 for even IL or
IL1 is IL/2 + 1 for odd IL. Contains values of sin (nké).

CS Dimensioned NMAX x IL1. Contains values of cos (nké).

ATXB Dimensioned IL. Contains values of the x coordinate of

the given coastline as determined from the arclength of
the transform-generated curve (max. 400).

ATYB Dimensioned IL. Contains values of the y coordinate of
the given coastline as determined from the arclength of the
transform-generated curve (max. 400).

ATXA

ATYA

Dimensioned IL. Contains values of the x coordinate of the
given seaward boundary curve as determined from the arclength
of the transform-generated curve (max, 400).

Dimensioned IL. Contains values of the y coordinate of
the given seaward boundary curve as determined from the
arclength of the transform-generated curve (max.400).

Dimensioned IL. Contains values of & (max.400).

? Dimensioned IL. Temporary storage.

Constants

MQ

MQP

NMAX

IL

IWANT

B

BZRO

WSX, WSY

WCX, WCY

Number of coordinates delineating the given coastline
(max.150).

Number of cgordinates delineating the given seaward boundary
curve (max.150).

Number of transformation coefficients, BO or Cc. (max. 200).
Number of equally spaced integration points for 0 < € < xX
with IL > NMAX (max. 400).

If IWANT = 1, the program is to be re-started requiring
input from the previous run. If IWANT # 1, it is the
initial mapping of the region.

Desired variance (in units of x,y space squared) between
the transform-generated curves and that specified. .

h = X2(MQ) = X2P(MQP)
k = a/d

-B

B

BS

The x and y component of the weighting factor for the
seaward boundary curve.

The x and y component of the weighting factor for the
coastline.

8. Input and Output.

The input data required by CONFORM to determine the transforma-
tion coefficients which conformally map the Laguna Madre, Mexico, to
Marsh Island, Louisiana, region into a rectangle are presented here
as an example, The first card image gives the continuation code,
number of coastline and seaward boundary coordinates, number of coef-
ficients desired, maximum number of iterations, number of integration
points, and the convergence criterion, The next 47 paired numbers
are the coastline coordinates and the last 40 card images give the
seaward boundary coordinates,

Program CONFORM provides detailed and voluminous output concern-
ing primarily the rate of convergence. These output statements are
not necessary for program completion and can be easily deleted with
little alteration to the sequence of instructions. Optional output
statements are indicated in the program listing (App. A) by an arrow
(<) on the right-hand side of the page.

The results from CONFORM required by Program GRID are 8, B_ ,
and the coefficients, B, and C, . These are given in the next
section as input to program GRID.

Reference to Table 4 in Volume I of this report shows the values
for the number of coefficients, maximum number of iterations, and
number of integration points are only indicative of the final steps
of this particular application. Explicitly, in order to obtain the
given coefficients to conformally map the Laguna Madre to Marsh Island
region into a rectangle in the image plane, the following steps must
be followed:

(1) Set NMAX to 40 and IL to 80 for the first 20 iterations,
(2) For the next 10 iterations, NMAX is 60 and IL is 120,

(3) Set NMAX to 80 and IL to 160 for iterations 31 through
50.

(4) Take NMAX as 90 and IL as 180 for the next 10
iterations,

(5) For iterations 61 through 70, NMAX is 100 and IL is 200.
(6) Over the next 10 iterations, NMAX is 110 and IL is 220.
(7) Starting with the 8lst iteration, WCX amd WSX are set

to 0 and NMAX is 110 and IL is 220 through the 100th
interation.

(8) From iteration 101 through 110, NMAX is 130 and IL is
260,

(9) Over the last 10 iterations, NMAX is 150 and IL is 300.

Note that steps (7) through (9) utilize the alternate solution
to the mapping equations (9) and (10) in Section II of Volume I of
this report, The instructions which must be altered or removed are
indicated in the program listing by parenthesis with the proper
instruction enclosed.

The following input is required by CONFORM to conformally map the Laguna Madre, Mexico
to Marsh Island, Louisiana region into a rectangle in the image plane. Note the fourth and
fifth parameters on the first card image are only indicative of the final results. See
Input and Output text for explanation of procedure used to obtain the mapping coefficients
given as input to Program GRID.

Cc

Cc INPUT OATA

Cc
1 47 40 150 1500¢001 360200 110.00

(2 c

Cc Cc

Cc COASTLINE COORDINATES Cc SEAWARC ECUNDARY COORDINATES

Cc Cc

Cc Cc
000e00 C35e00 000-200 0CzZe00
005e60 040200 007¢50 OCE&e0C
010.50 044.80 016200 010200
018.200 C5100 021-250 014-50
028200 CSSe00 028200 0190200
040.200 O0€Ee80 036200 022200
050e50 C730e00 045-200 028-00
056-50 C7Ee50 054.00 022200
063-200 C8z2e00 060-¢00 036.00
070200 CS3e00 066200 040200
074.200 102.00 077.200 046200
076250 1C€e50 085e00 0£1200
079200 116200 089-00 055e00
082250 127200 095.200 0€4-00
9088.00 127.280 104.006 076200
095200 14€.00 110e00 084-00
106.200 154-00 11200 C8900
121200 1€2200 113-50 0S&.00
136-200 167200 114.50 1C1e00
153200 168.30 117.250 106.00
158e00 16€200 121-00 1120e00
161-50 167230 127.00 116200
166.200 166280 139-00 121-00
171200 169 00 153.00 122.50
183.200 168.00 165600 122200
197.200 16€.00 179200 119200
214.00 1€£206 193000 11560C0
230-00 16€-00 213-00 104-00
242.200 167290 229200 0922006
250200 1€€.270 244-200 080e00
253240 167220 257200 071200
257200 169050 267200 O0€€e00
265-00 169200 276200 062200
280200 166200 289200 O£€e00
291-00 1€2200 298200 049-200
296200 159e30 307200 042.200
298250 156290 318200 03£-00
303200 159-00 333200 027200
314200 193-00 347200 020¢00
323200 148.00 360200 014200
328200 142200
331-00 1372e20
335¢50 129220
342250 121200
349250 11€e00
352250 113200

III. COMPUTER PROGRAM DOCUMENTATION FOR PROGRAM-GRID
1. Program Purpose.

The purpose of this program is to determine the grid point array
in the stretched curvilinear shelf coordinate system and appropriate
scale factors needed for program SSURGE. The detailed grid is
needed in order for the user to read off depths from an appropriate
bathymetric chart of the shelf region at grid locations.

2. Program Description.

The program is written in FORTRAN IV language. This program
interfaces between Programs CONFORM and SSURGE. It is assumed that
the conformal mapping of the storm surge region has been completed
to the user's satisfaction. The program GRID takes, in part, as
input, the transformation coefficients and determines computing grid
information of scale factors, grid point locations, and, at each
grid point, the orientation of the €&-axis to the x-axis

The program GRID is composed of

MAIN Defines constants. Reads transformation coeffi-
cients outputed from CONFORM. Reads water
depths along a line near center of grid from
the seaward boundary to the coast. Computes grid
point locations, scale factors (uv, v , and FP),
and, at each grid point, cos 6 and sine 6
Writes computing grid information.

SUBROUTINE XUT Writes information transferred into XUT.

SUBROUTINE SHCOR Determines and writes grid point coordinates in
X,y space and the distance in nautical miles
between points.

SUBROUTINE TRAN Computes the x and y coordinates of the trans-
form-generated coastline and seaward boundary
curve.

SUBROUTINE TRAN1 Computes x(&,n) and y(é,n)

SUBROUTINE TRAN2 Computes 9x/d— , dy/d— and 6 = tan-1 (22/28)

SUBROUTINE CURV9 Contains the expansion curve Y = Z + B(x")
, where A,B, and C are constants. The term
Y is either Sp (units, nautical miles) or T
(units, minutes). The term X is cither S*
(units, nautical miles) or T* (mits, minutes).
This subroutine computes Y and dY/dX given
the coefficients and X .

20

SUBROUTINE CURV1 - Determines the necessary parameters to
compute an interpolatory spline under
tension through a sequence of functional
values contained in arrays X2 and Y2.

FUNCTION CURV2 - Interpolates the given curve, Y2 asa
function of X2 , returning a value for
Vi Salven gear

SUBROUTINE CURV3 - Determines the necessary parameters to
compute an interpolatory spline under
tension through a sequence of functional
values contained in arrays X2P and Y2P.

FUNCTION CURV4 - Interpolates the given curve, Y2P asa
function of X2P , returning a value for
Va SAN CT Xane

Type of Computer.

The program GRID may be run on any computer with minimum
core requirements of approximately 26K words (based on the
present sample program appropriate to the Hurricane Carla
surge simulation grid). GRID requires no auxiliary storage
devices, peripheral devices, or magnetic tape input or output.
No site-oriented computer plot routines are involved in the
program. Approximately 25 minutes of machine time on a
GE/635 is required for the sample program to determine the
computing grid information. This time is based on the
following pertinent program parameters:

a) 150 transformation coefficients, BO or C, >

b) 121 evenly spaced values of & for determining the
transform-generated coastline arclength as a function
OH EG 3

c) 51 evenly spaced values of n for determining the arc-
length along a particular isoline of &€ as a function
Oi 8

d) the computing grid of 47 € (or S*) lines and 15 n (or
T*) lines;

e) for determining the scale factor F , the area in x,y
space of each quadrangle is approximated by using 4
evenly spaced increments between €&€ isolines and 2
evenly spaced increments between mn isolines.

2\

For production runs, smaller sampling intervals might be required
in) D, © eiatel Csjxscaailily, © -

4. Input Data.

Input data, other than constants defined in MAIN, are read in
MAIN and are on IBM cards prepared according to the following list:

(1) Card 1

NMAX, NUMXI, Number of mapping coefficients, number of &

NUMETA, DELSS, lines + 2, number of n lines, AS* in nautical

SOPRA PDE ils miles; first value of S* in nautical miles,

ND, NS AT* in minutes, number of depths, number of
points in format 314, 3F5.1, 214.

(2) Card 2

BETA BZRO The value of 8 and By from the conformal
mapping solution in format 2E14.7.

(3) Card Group 3

COB, COC - The NMAX cards containing the Fourier-type
transformation coefficients, BL and C ,
in format 2E14.7. ”

(4) Card Group 4

SY Temporary storage for the ND values of the water
depth (fathoms) along a line from the seaward
boundary to the coast. This information is
needed to evaluate the traveltime coordinate T.

5. Mathematical Procedures and Program Limitations.

4

Information concerning the expanding grid procedure and the
relations transforming &,n to S*,T* space is presented in
Volume I of this report. The user is required to know the coeffi-
cients of the expansion function

S, = A + B(S*)”

for each region of the curve where S. is arclength (nautical miles)
along the transform-generated coastline. For the sample program,
there are five regions of the expansion curve. Selecting AS* =
6 nautical miles, the number of AS* intervals of each region and
the value of dSp/3S* at the end points of each region, we can
determine the coefficients of each region from three simultaneous
equations derived from the constraints:

22

For region I, 176 nautical miles < S* < 236 nautical miles
(10 intervals of AS* )

OSD 2 6205 Ae 8

do*

dSp
35*

S
p

Los) che | Sy

S ale Se
Pp

For region IT, 236 nautical miles <
(CMaintervallswore AS)

OSD 3 '1..8 at S*

dS*

eS aye Se

S aoe

For region III, 260 nautical miles <
(7 intervals of AS* ) ;

3S*

dSp
3S*

S
wy

Nos ge Ss

1.0 ae §=6O

302 nautical miles at S*

= 176 nautical miles

236 nautical miles

236 nautical miles

S* < 260 nautical miles

236 nautical miles

260 nautical miles

260 nautical miles

S* < 302 nautical miles

= 260 nautical miles

= 302 nautical miles

= 302 nautical miles

For region IV, 302 nautical miles < S* < 356 nautical miles
(9 intervals of AS* )

SP 1.0 at S*
SSP 1.0 aye }8=SR
S = 302 nautical miles at S*

23

= 302 nautical miles

= 356 nautical miles

= 302 nautical miles

For region V, 356 nautical miles < S* < 452 nautical miles
(16 intervals of AS* )

SSP 1,0 at S* = 356 nautical miles

asp

aoe = Zoll at S* = 452 nautical miles

S

p 356 nautical miles at S* = 356 nautical miles

A similar procedure is followed for the expansion function
T=A+t BT*)°

where T is the long wave traveltime (minutes) along the particular
isoline of &. The program assumes that there are, at most, two
regions of the expansion curve with the second region being AT = AT*.
The expansion coefficients are determined by the program. If the
user desires only one region (i.e., T = T* and AT* = total long wave
traveltime-number of n grid intervals), the program computes AT*.

24

6. Flow Chart.

MAIN - GRID

Define constants
NMAX, NUMXI, NUMETA, DELSS, DELTT, SSTRT
ND, NS, XLAMDA, SLMAP, and XIDUM

Read conformal mapping transformation
coefficients, BETA, BZRO, COB(N), COC(N)

Determine and write transform-generated coastline

and seaward boundary curve coordinates for N4 evenly

spaced ivalues ods: a (Om< ew) ae Store in ATXB ,
ATYB, and ATXA, ATYA.

Determine and write coastline arclength
‘Sp (nmi) as a function of &.

Store Sp in X2
Seo = a WA,

CALL CURV1
Fits interpolatory spline through Y2
as a function of X2.

Defines number of expansion regions (IRG)

for Sp(S*) , coefficients A, B, and C and

number of AS* grid intervals of each
region (IIC).

25

Determine appropriate values of & for
evenly spaced values of S* .

For i = 1,2 -- NUMXI

*
S5 = SSSTRT + (i-1)*DELSS

CALL CURV9

Computes Sp for a given S* .

FUNCTION CURV2
Returns a value for & given Sp.
Arrays X2 and Y2 cannot be al-

, tered from the cast call of CURV1.

Store THO YC1l

E

Determine and write coordinates and arclength
Sn (nmi) along a particular isoline of €
(=XIDUM) for NS evenly spaced values of n ,

=) <S i S Bo
S 15 Ao
n
tim WZ.

Store

Store n

CALL CURV 1

26

Read water depth (fathoms) at ND
evenly spaced positions from the sea-
ward boundary curve to the coastline.

Determine and write the long wave traveltime
T (min) and distance S, (nmi) along
XIDUM for the ND positions.

i ne 2P
S fm | VWZIP 5
n

Store

Store

CALL CURV3
Fits interpolatory spline through
Y2P as a function of X2P.

Define number of expansion regions (RG) for T(T*) ,

coefficients A, B, and C, and number of AT* grid

intervals of each region (IIC). If IIC(1) = NUMETA-1,
compute AT*

27

Determine appropriate values of jn
for evenly spaced values of T*.

For j - 1,2 .. NUMETA

*
T, = (j-1) * DELTT

CALL CURV9

Computes T for a given T*

Function CURV4

Returns a value for S, given

T. Arrays X2P and Y2P can-

not be altered from the last
call of CURV3.

Function CURV2

Returns a value for nn _ given

Sn AGkays exe andy 2yican

not be altered from the last
call of CURVI1.

Store n in ETA.

The values of & and n are now known.
Determine the computing grid data.

28

CALL SHRCOR
Determines and writes the grid point coordinates
in x,y space and the distance (nmi) between

grid points.

Store x coordinate in X and
Store y coordinate in Y for
i = 2,3 .. NUMXI-1
j = 1,2 .. NUMETA.

Determine and write scale factor u
X,y space-nmi).

Store u in SX for i=1,2 .. NUMXI-2.

(units

Determine and write scale factor vw (units of
X,y Space-time minutes).
Store v in SY for j=1,2 .. NUMETA.

Determine and write dimensionless scale factor F.
Compute area in x,y plane of each quadrangle
subdivided into IQUAD intervals between E

lines and JQUAD intervals between n lines.

Store area in X for i=1,2 .. NUMXI-1°
1,2 .. NUMETA.
=1,2 .. NUMXI-2.
1,2 .. NUMETA.

Store F anh SC Geo he

29

Determine and write cos 6 and sin 6 where
@ is the orientation of the & axis to
the x axis.

Store 6 in Z for i=1,2 .- NUMXI
j=1,2 .. NUMETA.

Store cos 6 in X and
Si -@ min SC stone ils .. NUMXI-2
is

1=
j= NUMETA.

30

7. Description of Terms.

Arrays

All arrays except COB, COC, XI and ETA are reused throughout the

program.

The user is cautioned to consult each major program divi-

sion for assessment of array contents.

XO Ge Dimensioned NUMXI x NUMETA.
X2,Y2
X2P,Y2P

ATXA,ATYA

ATXB,ATYB Dimensioned the larger of N4, NS, ND, or IQUAD.

Ne 135 (G ee

TIC Dimensioned IRG (the number of Sp(S*) expansion
regions).

SX Dimensioned NUMXI-2.

SY Dimensioned NUMETA.

COB Dimensioned NMAX. Contains the conformal mapping
transformation coefficients, Be

Coc Dimensioned NMAX. Contains the conformal mapping
transformation coefficients, ore

XI Dimensioned NUMXI. Contains the values of & for
determining the computing grid data.

ETA Dimensioned NUMETA. Contains the values of n for
determining the computing grid data.

Constants

NMAX Number of conformal mapping transformation coefficients,
Bo @e G@ .
n n

NUMXI Number of computing grid € lines. With respect to the
computing grid in Program SSURGE, there is an extra &
line at each lateral end. This requirement results from
the averaging procedure used in determining the grid
data.

NUMETA Number of computing grid n lines. This is the same

number as in Program SSURGE.

3!

DELSS

DELTT

SSSTRT

ND

NS

IRG

The value in nautical miles of AS* . This corresponds
to DXI in Program SSURGE.

Since in SSURGE the product, wAS* , is always computed in
the surge equations, we determine DXI in meters such that
u values are dimensionless, i.e.,

DXI = 1852 x AS * i cami | SLMAP —_2™ yy unit
X,y unit
The value in minutes at AT* . This corresponds to DETA

in Program SSURGE.

Since in SSURGE, the product, vAT* , is always computed in
the surge equations, we determine DETA in meters such
that v values are dimensionless, i.e.,

DETA = 1852 —4
nm

x AT* min. x SLMAP amt

x,y unit

The first value of S* in nautical miles.

The number of water depths (fathoms) inputed from the sea-
ward boundary to the coast for determining distance as a
function of long wave traveltime.

The number of points along XIDUM for determining n as
a function of arclength Sh 3

Horizontal extent of the mapped region in units of x,y
space.

The particular value of €&€ used in determining n(S_)

The chart scale relating distance in nautical miles to
distance in x,y units (i.e., nmi is equivalent to 5l
units of length in x,y space).

Acceleration due to gravity (feet-s~*).

The number of points used in determining &€ as a function
of arclength Sp.

The number of expansion regions of SP(S*) or t(T*)

32

IQUAD The area in x,y space of each grid quadrangle is sub-
and divided into IQUAD intervals between €& lines and
JQUAD JQUAD intervals between n_ lines.

8. Input and Output.

The first card input to Program GRID gives the number of conformal
transformation coefficients NMAX , the number of & lines NUMXI,
the number of n Jlines NUMETA, AS* in nautical miles DELSS, the
first value of S* in nautical miles SSSTRT, AT* in minutes DELTT,
the number of water depths ND , and the number of points US _ used to
establish n = n(S_)

The second card gives 8 and B, and the next 150 cards give the
mapping coefficients B, and C, . These 151 cards are the punched
output from Program CONFORM.

The remaining 31 cards are the depths picked off a bathymetric
chart of the northwestern Gulf of Mexico. These depths are on a
constant €&-line selected by the user.

Expansion coefficients, provided by the user, appear as statements
within the program after format 135.

The reader must refer to Section III of Volume I of this report
for an explanation of the parameters associated with the stretched
shelf coordinate system for the Hurricane Carla surge simulation.

Output from GRID consists of the transform-generated coastline and
seaward boundary coordinates, the transform-generated arclengths along
the coastline and seaward boundary, and for each of the five sections,
dSp/aS* , S* , Sp , and €&€ are listed. Additionally, the
transform-generated arclengths at values of n for evenly spaced
increments of T* and at n values for constant increments of T
along the chosen —-line, the n values along the specified €-line
such that AT* is constant, and the traveltime and depths along the
constant é-line are printed.

The following output is required by Program SSURGE for simulating
the Hurricane Carla surge. This includes the x,y coordinates of the
grid intersections, the scale factors u and vw related to the trans-
HORNACNOM Of & to Se ginvcl’ wy) wo We | megaeeemielhy, ie Seale iie-
tor F , and the sines and cosines of theta giving the orientation of
the €-axis to the x-axis. The program listing indicates when these
may be punched or written on tape or disk for convenient input to
SSURGE .

33

(s

Cc

Cc INPUT OATA
150 47 15 €eC17600 beO

Cc

2ellO9S529EE01 $e5078648EE01
— 403493873EE01-608176823EE01
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10242251 3EE0 1-101728220E600
— 50 06365C2EEC0-10447071 8E-02
106711 B64EE0 C= 106921 €334FE00
— 305976 884EE 00-6073 30E€28E-01
1041928E€0EE00 103667185EE00
-101794944EE00-200S13£91E-01
4e1190876E-01 1033577S3E-01
—307759354E-01 48326 202E-03
7eS716016E-02 £03212199E-02
-1200800307E-01 106050344E-01
—102900039E-C2=1 oS5SEB75E-01
—5eSE35S7EE-C2 164022560E-01
- 103795311 E-02-€ 03872108E-02
-305084819F-02 926181509E-02
606049823E-03-4 01274 305E-02
—1LeS301L171E-C2 201339S555E-02
Ge 3927 E01E-0 3-20 286E277E-02
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-102749315E-CZ €032075S5E-03
400667667E-03 104115754E-03
—106674068E-03 40€354670E-04
3099555 78E-0 4-206 3€ SEB TE-04
-1288E€218SE-C3 5e682€825E-04
900772255E-04 70€038S04E-04
10559158 3E-0 3-4 09427928E=04
Gel903310E-C4 30343SCS4E-04
=101438246F-0 3-1 60257530E-04
30e7952217E-04 167815754E-04
-607125224E-04 9.487E230E-05
4037 12E674E-04-507567649E-06
-Se8352281E-C4 10470€E47E-04
108351595E-04-3021 298S4E=-05
—205129E44E-04 300412098E=-04
= 100435642E= 05-2 04449992E-04
100222874E-05 20632669 26E-04
—5e8439003E-0€-10570C8E2E-04
—102788127E-CE 105418€1 6E-04
= 206134 267E-0 £=9 03255390E=05
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20 6851102E-C £-206723854E-06
~ 40311551 9E-0£-2003€ 34£2E-06
206351927E-O5 1014225£3E-05
— 203699234E-05 263080753E-06
10060475 2E=-C £-2081665S6E-06
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7e96S51199E-06-703427355E-07
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205921 445E=-06€-208028600E-06
~ 30163786SE-CE £020276€6E-06
800697 328E-07-3020493£3E-06
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31

Si

200821 353E-CE€-3010024£2E-07
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20 7314736E-C7? 1200232€4E-07
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109894453E-CS $e3920S30E-10
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6e7366526E-11 364544254F-10
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34

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636
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460
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32e
300

IV. COMPUTER PROGRAM DOCUMENTATION FOR PROGRAM-SSURGE. .
1. Program Purpose.

The purpose of the program is to numerically simulate the storm
surge in orthogonal curvilinear coordinates with the vertically
integrated form of the quasi-linear long-wave equations.

2. Program Description.

The program is written in FORTRAN IV language. It is assumed
that the conformal mapping of the region under investigation has
been completed. The transformation coefficients for three regions
of the continental shelf of the Gulf of Mexico and two regions of
the eastern seaboard are provided in Appendixes A and E in Volume I
of this report.

An interfacing program is required which inputs the coefficients
and generates a curvilinear computing grid to the user's satisfac-
tion. The output from the program (and, in part, also the input to
SSURGE) must be the scale factors, grid point locations, and, at each
grid point, the orientation of the €-axis to the x-axis.

The Program SSURGE is composed of:

MAIN Defines constants and interfaces the
subroutines.

SUBROUTINE ZERO Initializes all arrays to zero.
SUBROUTINE FIELD Reads data and writes the water depth field

relative to mean sea level, the wind
field parameters and the storm positions.

SUBROUTINE WINDF Calculates the wind and atmospheric pressure
fields.

SUBROUTINE ELEV Computes the water tevel anomaly, H .
SUBROUTINE FLUX Computes transports, Qox and Qn ¢
SUBROUTINE DRAW1 Outputs H and vertically averaged water

velocities, Qou/D and Qx/D ,

at hourly time intervals and saves the water
level anomaly at prescribed grid locations
for output at program completion.

35

SUBROUTINE METER Calculates and saves the vertically averaged
water velocities at prescribed grid locations
for output at program completion.

SUBROUTINE HUV Outputs the saved information of the simu-
lated hydrographs, simulated current meters,
and observed water levels at program completion.

Type of Computer.

The program SSURGE may be run on any computer with minimum
core requirements of approximately 30K words of memory (based on
the present sample program appropriate to the Hurricane Carla
surge simulation). The program has been executed successfully on
the IBM 360, CDC/6600 and 7600, and GE/635. The present sample
program requires no auxiliary storage devices, peripheral devices
or magnetic tape input or output. No site-dependent computer
plot routines are involved in the program. Approximately 14.4
minutes of machine time on a GE/635 is required for the sample
program to complete 66 hours of surge simulation.

Input Data.

Input data, other than constants defined in MAIN, are read
in SUBROUTINE FIELD. These data are on cards prepared according
to the following list.

(1) Card 1

NT1 - Number of cards (max.50) containing on each TIM, ROT,
RAD, VRMAX and PZRO (format I5).

(2) Card Group 2

NT1 cards with each card containing values of

a) TIM The time in hours at which the hurricane wind
and atmospheric pressure parameters are
recorded (format F10.1).

b) ROT The angle in degrees between the direction the
storm is moving and the region of maximum winds
(format F10.1).

c) RAD The distance in nautical miles from the storm

center to the region of maximum winds
(format F10.1).

36

(3)

(4)

(5)

(6)

(7)

d) VRMAX Maximum observed windspeed in knots (format

F10.1).

e) PZRO Atmospheric pressure in millibars of the storm

center (format F10.1).

Card 3

NT2 - Number of cards (max.150) containing on each

TIMPOS, XPOS and YPOS (format I5).

Card Group 4

NT2 cards with each card containing values of

a) TIMPOS The time in hours at which the hurricane

position is recorded (format F10.

A)

b) XPOS The x-coordinate in units of x,y space of
the hurricane center (format F10.1).

c) YPOS The y-coordinate in units of x,y space of
the hurricane center (format F10.1).

Card Group 5

GRID2 The fluid depth in fathoms relative to
level along each column, i=1,2..IM, is
a nested do-loop for j=1,2..JM (format
The program will zero those values for
prior to computations. The depth data

mean sea
read with
11F7.2).
even itj
are posi-

tive numbers which the program converts to nega-
tive values (in meters) to be consistent with

the coordinate system.

Card Group 6

S The values of the dimensionless scale factor relating

the (x,y) plane to the (&€,n) plane are read along each
column, i=1,2..IM, with a nested do-loop for j=1,2..JM

(format 5E14.7).

Card Group 7

DSDXI The values of the dimensionless scale factor, u,

transforming &€ to S* are read with
per card for i=1,2..IM (format E14.7).

SiG

one value

(8) Card Group 8

DTDET Values of the dimensionless scale factor wv
transforming n to T* are read with one
value per card for j-1,2..JM (format 2X,E14.7).

(9) Card Group 9

HOBS1 The values of the observed hourly water level

in feet at grid location (IH1, JH1) are read
with 19 values per card in format F4.1.

(10) Card Groups 10 through 14

HOBXk The values of the observed hourly water level
in feet at grid location (IHk, JHk) are read
sequentially with the same format as above.

(11) Card Group 15

YOSo 2 LAL The paired x,y coordinates (units in x,y
space) of the computational grid points are
read along each column, 1-1,2..IM, with a
nested do-loop for j=1,2..JM (format 10F7.2).

(12) Card Group 16

COSG, SING The paired values of the cosine 6 and sine 6
where 6 is the angle between the € and x
axis are read along each column, i=1,2..IM, with
a nested do-loop for j=1,2..JM (format 10F8.5).

A computer printout of the sample program and data cards are given
later in this section.

5. Mathematical Procedures and Program Limitations.

The storm surge equations, solutions, and algorithms are described
in the text of Volume I of this report.

Basically, the model utilizes a centered difference, leapfrog
analog of the vertically integrated, quasi-linear form
of the long-wave equations. The algorithm treats the time
dependency explicitly and employs a computing lattice in which
the transports, Qox and Qa >» are computed at the same location

but are staggered in time and space with respect to the water

38

level anomaly. The program assumes that seabed scouring does
not occur. No program error messages or consistency checks are

incorporated in this version.

Flow Chart.
MAIN - SSURGE

Define constants

CALL ZERO
Initializes all arrays to zero

CALL FIELD
Reads data and writes depth field,
wind field parameters and storm positions

CALL HUV
Output saved information of
simulated hydrographs, simulated
current meters and observed
water levels.

39

CALL WINDF
computes wind and atmospheric pressure
fields WIND(i,j) , P(i,j)

CALL ELEV
computes water level anomaly GRID1 (i,j)

CALL DRAW1 _
outputs H, Qcox%/D and Qrx/D at hourly time
intervals and saves information for output
at program completion

CALL FLUX
computes transports, GRIDI. (i,j) and GRID2(i,j)

40

Glossary and Description of Terms.

Arrays

GRID1 Dimensioned IM x JM. Contains H values in meters

at oad a) and Qo in meters squared per second
at even itj

GRID2 Dimensioned IM x JM. Contains fluid depth values,
Dy » in meters at odd itj and Qps in meters squared
per second at even itj

S Dimensioned IM x JM. Contains dimensionless scale
factor, F, determined from the conformal mapping of
(x,y) space to (&,n) space.

DSDXI Dimensioned IM. Contains the dimensionless scale
factor, u, transforming §& to S*

DTDET Dimensioned JM. Contains dimensionless scale
factor, v , transforming n to T*

XX Dimensioned IM x JM. Contains the x-coordinate in
units of (x,y) space of the grid point locations.

YY Dimensioned IM x JM. Contains the y-coordinate in
units of (x,y) space of the grid point locations.

COSG Dimensioned IM x JM. Contains cosine values of 0
relating the orientation of the &-axis to the
X-axis at the computational grid points.

SING Dimensioned IM x JM. Contains sine value of 6
relating the orientation of the &-axis to the
X-axis at the computational grid points.

WIND Dimensioned IM + 1 x JM. Contains values of the
windspeed in meters per second. At a transport
computational grid point (i,j), the S*-component |
is stored in WIND at (i,j) and the T*-component
is stored at (itl,j).

P Dimensioned IM x JM. Contains values of Hg (the
hydrostatic elevation in meters of the sea surface
corresponding to the atmospheric pressure anomaly)
and stored at odd itj.

HOBSk Each array is dimensioned 67. Contains the observed

en.2. 6 hourly water levels in meters at grid locations
a (IHk, JHk).

4I

ROT

TIMPOS

XPOS

YPOS

Constants

IM

JM

NMAX

Each array is dimensioned 300. Contains values
of the computed water level H in meters at
grid locations (IHk, JHk).

Each array is dimensioned 300. Contains values of
the vertically averaged water velocity, Qc»/D , in
meters per second at grid location (ITk, JTk).

Each array is dimensioned 300. Contains values _
of the vertically averaged water velocity, Qyx/D ,
in meters per second at grid location (ITk, JTk).

Dimensioned 50. Contains the time in hours at
which the wind and atmospheric pressure field
parameters (ROT, RAD, VRMAX and PZRO) are
recorded.

Dimensioned 50. Contains the angle in degrees
between the direction the storm is moving and
the region of maximum winds.

Dimensioned 50. Contains the distance in nautical
miles from the storm center to the region of

- maximum winds.

Dimensioned 50. Contains the maximum observed
windspeed in knots.

Dimensioned 50. Contains the atmospheric
pressure in millibars of the storm center.

Dimensioned 150. Contains the time in hours at
which the storm position is recorded.

Dimensioned 150. Contains the x-coordinate in
units of (x,y) space of the storm center.

Dimensioned 150. Contains the y-coordinate in
units of (x,y) space of the storm center.

Number of grid points in the S* direction.
Number of grid points in the T* direction.

Maximum number of time steps.

42

INC

DXI

DETA

DELT

F
THk , JHk
keig25 56

ITk, JTk
k=1,2..6

CORIO
PHI

PINF

Comments

Number of time steps between saving of surge
results in HYDk, UCMk and VCMk. INC must be
an even integer number.

Grid increment in meters in the S* direction.

Grid increment in units of meters in the T* direc-
tion. The units of DTDET-DETA (that is, v-AT*) are
in meters.

Time increment in seconds. DELT must be less
than that required for numerical stability and,
also, be an even integer multiple of 60 minutes.

Acceleration due to gravity (=9.8 meters per
second squared).

Dimensionless seabed drag coefficient (=0.0025).

Grid point location for saving the computed water
level. The sum of the indexes must be odd. Index
THk cannot exceed IM. Index JHk cannot exceed JM.

Grid point location for saving the vertically
averaged water velocities. The sum of the
indexes must be even. Index ITk cannot exceed
IM. Index JTk cannot exceed JM-1.

Coriolis parameter (=6.70875 x 10 © second 1)
for latitude: 27s 256252! Ne

Wind ingress angle in units of degrees.

Far field atmospheric pressure in millibars
(=1016 millibars).

1. A symmetric analytical hurricane wind field representation
as given by C. Jelesnianski (1965, A numerical calculation
of storm tides induced by a tropical storm impinging on a
continental shelf, Mon. Wea. Rev., 94, 379-394) is
employed in the surge model).

2. The wind stress coefficients, K, and Ky , are 1.1 x 107°
and 2.5 x 107-© and are defined in SUBROUTINE FLUX.

3. Constants YRANGE, THIT, XHIT and YHIT are not used in
this program version.

43

8. Input and Output.

Data statements in Program SSURGE establish the number of computa-
tional points, number of time steps, output interval, values of the
grid and time steps, acceleration due to gravity, bottom stress coeffi-
cient, and locations of the simulated hydrographs and current meters,
and the Coriolis parameter corresponding to latitude 27° 23.232' N.

Card input provides the number (NT1) of hurricane description sets,
followed by NT1 cards giving the time (in hours after start of com-
putations) and the three required storm parameters. These are suc-
ceeded by one card giving the number (NT2) of hurricane positions to
be used, followed by NT2 cards providing the time and storm center
positions in the original arbitrary Cartesian grid (see Volume I of
this report). NT1l need not be equal to NT2, nor must the observed
storm parameters and positions coincide in time.

For the Carla surge computations, the hurricane characteristics are
stipulated at 6-hour intervals for the first 18 hours and at 3-hour
intervals for the remainder of the 66-hour prototype time simulation.
Note the radius to maximum winds are in nauttcal miles, the maximum
winds are in knots, and the central pressures are in mtllibars. Due
to the erratic movement of Hurricane Carla, the coordinates of the
center of the storm are specified at hourly intervals, except for a
single 6-hour interval spanning the end of the simulation. These
coordinates are specified in x,y space (100 units = 219 kilometers).

The depth field, in fathoms, is introduced followed by the array of
scale factors, S . The array of scale factors, DSDXI (an alias for
u ) is read along the specified row and the scale factor array DTDET
(an alias for wv) is specified along the chosen column.

The following six arrays are the observed hourly water levels (in
feet) from Padre Island (HOBS1), Aransas Pass (HOBS2), Port O'Connor
(HOBS 3), Pleasure Pier, Galveston (HOBS4), Mud Bayou (HOBSS), and
Sabine Pass (HOBS6). These data are not necessary for any phase of the
surge calculations and may be omitted.

The last two arrays input to SSURGE give the coordinates (XX,YY)
of the grid points in units of x,y space, and the sines and cosines
of theta at the computational points.

All input not in the MKS system of units is converted internally
to the MKS system.

The storm parameters, hurricane center coordinates, and depth field
are printed out immediately following input. At hourly time intervals,
the water level anomalies and the depth averaged velocity component
fields are printed. The six simulated hydrographs and current meters
are printed out with their positions and the corresponding observed
water levels at the completion of the surge simulation.

44

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67

APPENDIX A

FORTRAN Listing of Program CONFORM

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84

GNW/32av =GB00WX

VNIW/VIYNV =VGOWX

1X1530*aWNS =GN1V

IxXWadkVWNS =VNI1V

(TIIA*S9O = CT)AxS°O = GNIS =GWNS

CT1)MkS*O = €T)N*S*U = YWNS =VANS

( aAouay ) YVeS/EXIAD =XIM
( 8AOWdY ) 4VGS/VXS3 =XSM
uVa3/9A43 =ADM

YVGS/VA43 =ASM

O° v7 4335 =4VER

GA 3354GX55S+VASI40 X49 =43
ADM*xLA*GABWNS =GAS4S
XIJMeLd*DXSWNS =EX33
ASM* 14% VABWNS =VWASS
XSMeLIeVXAWIS =VWX 43
GAWNS*S*U = FAAWNS =BASWNS
OXWNS*S°O = GX3SWIS =8XaWNS
¥AWNS*S°O = VARNIS =VABWNS
VXWAS&eS°O = WX3WNS =v X3WNS
H)v4a9S =(L1)A

vovaoS =C(11)N

c*x( CCT Me (T)GALVISESY )*xS°O
exe( (CCTPVLECTIZ+( 136K 4VISBV 2*S°9
cxx( CET )SH=CT)VALYV) SEV bkS°O
CxeC COTY (CL IZ4¢CT DVXLVISGV eS °0

GAWNS + DAIZWNS =GASWAS
BXWNS + GX3N1VS =X SINS
VAWNS + VARANNS =VASWNS

VXWAS + VXSWNS =VX3WNS

ceux (CC LI) M@CLIDGALVISaV) =GAWIS

Sex CCC LI) LC 21 ¥Z4(0 11 DAXLV)S3SV) =aXWNS
Cex (CC LIDS@(LIFVALVISEV) =VAWNS

2xxC (CLL) 4=€ 41) 24011 )VXLVISEVY) =VXWAS
dov39S + JAWNS =BWNS

dic

85

TWX /VAUVA =0 ANWA
TVIX/SVXYVA =VXUVA

JANI LNOD

GGONX*ENNS =8NTV

: VOINX*VANS =VN1V

CO °2/ IX TS0x( CITA + CLIDA } + BWAS =AwWAS
O°c/I XTS0%( €11IN + (4109 ) + VWOS =VWWAS
T+il =I

eZ2 Gi OD ( TI 23° LI )SI

cxx (CLT )Me( LI) GALVY SGV + GANVA =GAUVA
Cex EC LIDL=C LI YGXLVY SGV + 3XYVA =GXHVA
CexX(CLIIS=(LIDVALV JSG + VAYWA =VAUVA
c*xx¢€ CLI Do@C LI) VKLVY }SEY + VXHWA =VXUVA

( 41 *© VWWOIS * GNIV )JAASYUND =CLI)GALY

€ Ml * WWOIS * GNIV JdAHYND =C4L1)EXLV

( XI © TVWOIS * YWNIV )WASUND =( LI PVALYV

( 41 © IWNOIS © UNIV FVAHHND =(LI)VXLV
SNW =CLIDA

VNIv =( LI )x

ei

Vl*t=il e242 Od

0°0 =BAXNVA

O°0 =GXYVA

O° OG =VANVA

0°0 =VXAVA

0° Oo =9N1V

o° =VN1V

0°Q =Gwns

0° 0 =vwns

eid

86

TW*t=I Se2e OG

(i NV b= 0UVHE6
‘ X bo NSATS@AHZ XZ © CWLA® IK AHS © Xb ENSAI OmKAL* KZ 8 (VES SIX) XHOS X86
e IH 1° Xp *NVYL=OYUVHE6
‘ Xv © NSAT9@AHZ © XZ OC WLS SEX PAHS OXY ONSALO@ X42 SKS * (VW LSS 1X) XHO* XE* 76
C B=Via © VONVWIe=IxXe=e0 * SGYO0D SNIVLSVODHSE® XEZO

6y=VisS © VOWVTe=IxX==0 * SGYOO3 AYGE GUVMVASHZH*xST® 7/7) iN WHOS
(Tez*9}31L1UM

( 2° pla*xOdct sz * vIa*X tL* 46
C 2° vla*x*2*yvla*xc6
*seyla* A ONISN G3SI4I3DSdS LV4t WOAXS AYGD AYvNnvaS NYOSASNVYLHIG* X76
6 2° y71a* x0Z6
*2°y1ls* X ONISN G3ISIDSdS LVHEL WOUS ANGE GHUVMVSS WUCGSSNVYLHIS *xo*76
© 2° vt axed “oH 1a*x 26
© 2°7135* A ONISN GaISTIDSdDS LVHL WOUS SNIFiSVOD WHOASNVHLHE HS KLS 46
C ; 2°vta*xoce
6 2°pl3* xX INISN GSILSIIAdS LvVAL NDYHS AJNIDWLSVOD NYISSNVYALHB YS XLS 776
$ LONS YSHL*°X9ZT* SONVIAVAAE*xX99 *y77* He BH =6
JA 1VYSNS9 *WHISSNVOS x OL HIONSTDIYUV ANGE Gav4¥vsaS SO GIL V4HSS *xX2e*/76
G b?sd* = Gaivea6

N39 4NYOASNVYHLx OL HLONSTONV ANITWLSVOD SO OITLVYHISG*XSseE di Vwuos
Sd * HVA *VASS*S VET AS VAYWA *y¥Xd5° VX YWAG

C GA 45* GUVAS GANA *Gxda *OxXuVA *VIOWX*gdIWX CU0S8d* 9) SLIM

( GAYVAtBAXYTA )*SG°C =Guyva

€ VAYUVATVXUVA )*S°C =VYdA

O° ¥/s/( GAHXVAFEXYVAFVAXTARTX YUVA ) =UVA

TVIXSEANVA =BANVA
TIX/EXYVA =EXUVA

ls¢

osc

87

XVWN*S HX *ONZaE ° 3205 * 805d 471 8X04 /NOWWOD

(CO¢)dD0D * (90¢)}805 NOISN3WIG
( vis © IX INVHLA NOTLONAS

GN3

NUNLaa

IX + NVYULX =NVULX

JONI INOS

CIXANXINIS & ( HNS*(N)} 609+ HS Dx (N} DOD JENVULX =NVULX
(VANX)HNIS =HNS

(¥XNX})HSOD =HSD

1X WX #NX =1X INK

VAXaNX =VINX

N =NX

XWWN*E=N OF OG

IX*4X = IXHK

VL3* HX =V HX

0°O =NVuyLx

XVWN* HX *OuNZ3 * 20> * 309 /¥1TYO4/NOWWOD
(002 )3909 * (002¢);80) NOI SN3wWIG

( WLS * IX JNVYLX NOTILONAS

GNA

NYN134

GS Oi 39 ( IXWAF %3T® LNOOSI }SI

SZ€ OL OD ( LMHVA °37° HVA )SI

1 + LNNODI =iLNNODI

]°h sexy SZ tL a*XEf?) PL SEKS SZPLATKE SS L4 PKC SEI xv6

IHL *X 7S Se Sade XH* J° L4* XE* SP LAE KGS SLAP KES SPL ASX SEIS XY JLVWYOS

CIDASEL GALES CI IMS (LT) aAXiv*® (1) 1°16

CI)K* CHEV VALVE CTIS SCL VKLVE CIT IYST (982 °9 PALI UM

ot

Sle

98¢

S8¢

88

VLSGA + VLAGAG =VLAGAYG

vi30x + ¥iadxXG =Vv1isdxd
( I X4N xX) SOD x ¢ HS D*(N)} 309+ HNS* (N )GGD)* ¥X*NX =VLIAGA
(I XHNX INI Sx ¢ HNS*(N)}D094 HSD*(N)GOD) *4AX*NK =V1L SOX

(VOANX}HNIS =HNS

(VANX DHSOD =HSD

I XAK%*NX = IX HANK

VAX*NX =V HN X

N =NX

XVANS T=N OT OG

0° 0 =v LS0AG

6°00 =vVLsaxXGa

IX*xWd xX = 1X

Vilidk HX =V HK

XVWNSHX*SONZE ° 305 * 899 7¥ 1Y0S3/ NOWWOD
(062)303 * (00¢)35) NOISNAWIG

( 3S © 3VS0S © vLSGAG * VLEGXG * WLS © IX JOVSIS ANI LMOYSENS
ONS

Naniay

Ouzea + Vid + NVYLA =NVULA

SNNTLNOD

( TX ANX 2S ODx ( ANS* (NJ DOD+ HSD*CNJAODIPtNVAYLA=NVYLA

(VOANX}HNIS =HNS
(VANX)HSOD =HSD
I XHX*NX =I XUN X
VAXeENX =VOUINK

N =Nx

XVWNST=N OT OF
IX HX =1XK XK
VLAxAX =VHAK

O°O =NVHLA

ot

89

SVIOdS*¥NIOVIG = (1)S
(S1T3G=SHN ISJ*ENIHNIS = OVIOdS
€TddIS=IXQ)*NIOVIG = (1)V
toviIGs°t = NIOVIG
(SHNIS=(SdX3/°T4+SGX5}*S°*S TAG) XNIHNIS = I9VIG
CSHNIS*TX7350)/°t = NIHNIS

; (SdX3/° l™=Sdx3)*S° = SHNIS
€S73G)dX3 = SdX3Z

IXTS0*xdVNDIS = 5730

CCT ISKX=(NIS XIAO TEN DRC VWDIS)S3S¥ = dVWOIS
Nd IS = NdditS

T4715 = IddItsS

GS OL OF (°O*ilYWOIS) SI

IX THG/ CCL) CA@ECZ FZA), = IXG

(t}exe(ec)ex = TxX130

T+N = TdN

T=N = TWN

OW =N

ex 30 NOTIONAS V SV CA @=4NI 16S Sills

OW © 2A 2X /SBA/NOWWOD

S /SNHOF/SNOWWOS

V /YNHOF/NOWWOD

€80¥)S * (O6ST)¥ NIISN3WIQG

(OST IZA * (0S1T)2X NOI SNIWIG

€ VWOTIS * NIIS © Id1S JEATHND SNILNOYSEIS
GN3a

Nanlas

( 3S JLayOS =DVSI3S

cee (( VLSITAIT)S3IV) + Sx* (CV LIGXDG)SEV) =4S
O°T + WLSGAG =VLAGAG

SON IL NOD

Ot

90

(CN )}2K=(NICXK = NNWG

(c@N) cxeCINN)ICX = TWN13d

(TAN) CX#(N) CX = NWBC

CEVCAXED + (ZIZAXED + (TIZAKTD = TddIs
EXTVAG/CIXTVSAG/ TX Wad = £9

ex WaG/TIX 139d 721X150 co

TX V304 21X 1307 ( IX 130+21X 150 }= = 13

€ 1) oxX=(t)ex = CIXIG

(c)exel(E)ex = 2x1ad

a ai) GS SWEIPIN Shi!

Nadnlaa
(T4+4VElT}V x dVaGI DS ea( aval)? = (47d1)V

I=tdN = AVEI
UG eS i ap OG

COCTWNIVY *9VIGdS=2 XG=NddIS)*xNIOvId = (N)¥
CCTWN)S *OVIGdS=TOVIG)/°t = NIOVIG

eSVIG = t9VId

ecxG = 1XG

O¥VIGdS*¥NIOSOVIG = ¢€1)5S
(€S ISG2*®SHNIS}*NIANIS = DSvVIGaS
(CT#1)V xOVIGGS=IXG=2exQ)xNIOVIC = (I}V
€(T=1)S x*OVLAGdS=dcOVIG+TOVIG)/°t = NIOVIG

(SHNIS=( (SdxX 3/°T+Sdx3)*S* )kS THO) *XNIANIS = COVIG

(SHNIS *¢€X7350)/7°T = NIHNIS
(Sd xX3/° t@SdX4)*SG* = SHNIS
(S150 }dX3 = SdXx3

eK 1W3d*dVWOTS = S$ 13G
2X tas ET }CA=(T+I) CA) =CXG
CT)oxke(T+1)2x = 2x 130
TNNe? = I 2 OG
(Sp a) CEPOSSNY si

ina)

9I

(tel) 2xe(1l)ex = S1lad
de(l)cx = 27159

CTel)yoxKemi = 11350 ¥
T OL OS
c= Wk
7 (st wo — Cex Vale? kh CRMC NM Cao Clot pas Nel 3
N = I

3NNILNOD eA
€*ed*c ( amC lI )ex )aAl
Votre lt @ OG %
GS UR CUES wWsyo Li yy she
S/( TEN) X(VANDISJSHV = dVWOIS
(T)ECX=(N) 2X = S
OW =N
*sySIWVv3 JQ5s71V5 38 L1SNW BATHVD SNILNOYBNS 2)
*x LV A SO SATIVA SHL SI GAAYND NI GS3NGNLSY SBNIWA SHE OD
*x ets SNIWA NSAIO v iv BNITLSWOD SHL SALVWdsSLNI NOILONAS SIHL 39
DOW © ZA ex SAASNOWWOD
VY 47 NHOF/SNOWWOD
(O35 1)¥ NOISN4AWIG
(0ST)ZA © €0S61} 2X NIISNSAWIA
-( LT © VWWOTS © 4 }EHAAGND NOTLINAS
ONS
NYNL3y
(2)v
Or = SCN a9
i G1 O9
(N}CAKXTD + CEWN)IZA*CD + (CHENIZARED = Ndd IS
TWN THG/NNTSG/N135U = £D
TWNW3SI/N 13290 /NN13502 co
NTAG/ NN 43Q0¢ (N 1S CG* NN 1350 ) | Se)

®
Q
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92

(Tel }exmi = TED ¥

2 GWE Ci vars Cae Owe WN Oey (Eo

t OL OD
e¢ = TI
1)¢x )SI
N = I

SINILNOD €
efe%e ( tei }CXK ) SAI

NSTI

= I ¢ oat

@ = Tl (#03 °41) SI

S/T ™=N)& (vWD 1S) Sa¥
CL )Zx=¢

*e3ltdVv3 Gsrivd 3G t5NW JZATHND
°xX iv XxG/AG 3D SNIVA 3AL SI FGAGHND NI G3aNdNisa 3VIVA

SNIVA NSAIDS ¥ LV SNI VWiSvOD S3Ht SFLIVILNSa55519 NO
OW £ 2A *2xX 48

VY /SY¥NHO

(CST)V N

COST)ICA * (ds 1)ex N
© woul Ss Sk aA AD

SWau/( ¢ WOx*( (Tel )y =
THG*( C1} V=C1) CA) )+SHNIS/(CGHNISx( Te] )V +TIGHNIS«(1)V >}
(SdxX3/7° TeSax3)%*S°
Sdxk3*1TSdXx
CS dX 37° T=SdX 5) x5 °
(2 130% dvwIIS} Gx
€b5 dX 37° T=1SdIxX35) xs °
(TWO *d VANDI S) 35X35

= dVWOIS
N}eX = S
OW =N
SNITinoy 8ns
SHL *X wYOa
TAiSNANA STHL
A/NOWWOD
FSNOWWOD
GCISN3WIG
QISNIWIG
NOTLONAS
GN3
Nanisa
fos UU
(Ter) cA} 6
BAAYND
SHNIS
3 = SdXx3s
= ¢GHNIS
4d = SdX4
= IGHNIS
= SdXx3

Ul

2)
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93

COTIGADYUV=(NIBADIV) SE TREN) &(VWIIS)SAvV = avVWOIS 1
Nd TS = Ndd1sS
Td IS = tddIs
G OL OD (*UP%LTIPVNOIS) GBI

TxWsag/((l)exe(d)cxX} = xd
(LYSADYV=(2)GADNV = 1xX735a
T+N = TdN

T=N = TWN

OW =N

gADav JO NOT LINAS ¥ SV 2X ==4NITWdS SILI4 2}

Ow * 2A *2X S3A/NOWWOD

SAIYVY SBAVZNOWWOD

S /SSNHOS /NOWWOD

2 SINHOC/NOWWOD

(800)S * (0S1T}3 NOISN3AWIG

COSTISA * €O0ST}¢e€x NOISNISWIG

(OST }HADaV NOISNAWIG

€ VWOIS © NdIS © TAS )EACHID SNITLNOUENs
QN3

Nanlas

I = Tl!
STA0/(C(T#1I}¥¢2€ Tel) 2A. @( C1 T

YV= (TCA) Y+SHNISZ €CCHSODK (CTI }V =tGHSOd*( I) } =SGaAayNd
SVNOISZ (Sd 34 Pt @SdK3)*S* = SHNIS

Sd X4d3* TSdx3a3 =, SdXF
(SdX3/°t+SdxX3a)kS° = @GHSOD
(2TAG*KXdVWITIS )GdX3 = SdKAZ
CETSdX3S7° 1+1Sdx3)*S° = IGHSOD
(TTAG*d VWOES )dX 5 = I1Sdx3

€ tI) cx=(1}ex = S13a

L=(1I)¢cxX A

94

Cy, (OL OS} (SP WSO IN xi)

Nani aay
CT+HAVEI FD *¥€ HVIIDS =( +Va1})9 = (avd 1)9
IelTdN = HAVEGI
Veg s i 3) GG
€C TAN)JD *5 VII dS=ed xG=NddIS)*VIOvIG = (N)D
CCTWNIS *OVIGdS=19OVIG)/°T = NIOVIG
cOVIG = t9VIO
e¢xd = xd

OVIGdS*NIOVIG = €1)S
(STSG@SHNIS)¥*¥NIHNIS = OVIGdS
C(T@LJD *¥OVIGdS#I XG=2xG)*NIDVIG = (1)9
€CtT=1)S xOVIGdS=cOVIG+EOVIG)/°T = NIDVIG

(SHNIS=((SdX3/° T+SdX3)*¥S°)*S 7139) HNIANIS = ¢cOVIAG

(SHNIS¥2xX739)/7°T = NIHNIS
(S dX a7 °T=SdXx3)*5° = SHNIS
(sms a)isixXSae = Sd x

2X dk dGVWOIS = S$ 14d
ex1aag/((T)exe(t+1)2xX) =exa

C(1yaADNTe( THI) SADSY = €xXT13a0
TWNS2Z = I ¢€ OG
(SOY DS USIMSIONY Eh

OVIGdS*NIJVIIJ = ¢€T)S

(S TSG@SHNIS)*NIHNIS = OSVIGdS

(tTdd IS* TxXd)*xvViOvId = (1)9
TOVIG/°T = NIOVIG
(SHNT S=@€SdX4/7° 14+5dK5) *G°*S THQ) ¥NIHNIS = TOVIG
( SHNIS*TXI59)7°T = NIHNIS
(SGX 37 PT =SdX5)*S° = SHNIS

(S130}dxXS = SdX32
tx Wd* dVWOIIS = S130

95

S/ETEN XC VNDIS)S

@V = dV¥wWwOIS

(CTIBADYV=(NFSADYV = S

DW =N

*34u0338 G3TIW) 3E LSNAW SACHND SNILNOXYSNS *HIONSTDISV AVIIDILYVd S1iI iV

X 3O SNTIVA SHE SI SAVYND NI GSNANLIY AN WA 54h

eGs SNIWA NSAIO v iv SANTIVLSVOD AHL SALV Wad SsiNI
OW © CA *exXx

HAD YV

°445N35 12049 SHI
NOTLONAS STHL
SBASNOWNWOD
7341 SNOWWOD

D> SINHOF /NOWWOD

(USt)

COST 2A * (0ST)2xX
COST) GAD

( £1 * VWOIS © Lb JEAVE

(N}CX*ID + CIWNICX*#ZD * CEN) CXREED

TWN T3G/ NN 13

> NIISNAWIQA
NOISNJAwWI G
WV NOIISNSWIG
NM NOT LINAS
GN3
NYALsay
(2)9
(t)3 9
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Ndd 1S
G/N1S30 = £95

iy)

°¢
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oo

INN TAD/ANIAG/NNTIS0@ = ¢d

NTSU/NNI30/4(N7139 9+
(EmN) GADYUVe (NEGA
(Z2=@N) 3 AdD40= CLWN}GAD

NN1320) = 15
DuV = NN1AG
4¥ = TWN 130

CT WNIEGADSVE(N) GADYY = N 130

CE) 2X*ED + (E}CX*CD + (TEX
Z2XVWIG/2TXVI7

*1TD = ItddtS
Ixdad= = £9)

ex IG/TX 1aG/2tTXISsd = 22
TXVSG/2UX1VSG/( TX 1WaG+d2TxX13G)= = 19

CT YGADUVE(EVIAD

uv = 2txX1sad

(2 GADNVNCEGADYNV = CX 14d

2)
2
2)

96

GAIJ4HV AG NOITLOINOA WV SY ZA =23NI1dS StI4 2)
OW © CA *2XK SBASNOWWOD
BAIS SAAYCNOWWOD
S /SNHOF/SNOWWOD
G SGNHOF/SNOWWOD
€eCv)S * (0St)G NITISNAWIG
(GST CA * (€usTtJ2x NOISNSIWIA
(0ST)SADSV NOISNAWIG
( WWOIS © NdIS * Td IS )BAEYND ANI LNOSENS
QN3
Nanisa
L = I!
ST3BU4 ( Z15GKC(T=1)3 = (T#1 2x) 6
TWSGx¢¢1 )O—=(1)ExX))+SHNIS/S(CGHNES*( Ter} S +TGHNIS*(1I)3 ) =HAVAND

(SdK3/° l=SaGxX3)*S*° = SHNIS
SqGX3* 1SdKa = SdXA
€Sax37° TeS4X%3)*S° = COHNIS
(2 13G*x dVWOIS)dxX3 = SdX3
(1S dk 3/7 °° l= 1SdxkX35)*5° = ITGHNIS
(TIBI xdVNDIS)SXS3 = TSdXA
(T=I )eADaVeRCIdSADSE = STAC
Le(I)eaAduv = 2130
(tT@=1)8A05Vv—2i = 1150 BF

T OL OD
GS (hil

t @©L O09) [l ySkoUNW, Saat IS dO he ers ae Cee Tea ADS Vile) Pir €
N = I

SANTILNOSD ra
e*e%d? ¢ te(l)aAdUG ) AT
NOUN Ss th eo OG ¥
SS Un GUM CN MW) =H

Si

OVIGdS*XNIOVIG = ¢€1)S
(S TSQ"SHNIS)ENIANIS = SVIGdS
€CT=I)0 *9VIGdS=I XG=2x G)*X¥NIOVIG = (1)d
€¢TsI)S *OVIGAS=cOVIGtTOvVIG)/°T = NIOVIG
(SHNIS=( (SdxX S/°T4+S dX 3)*S° DXSTAG)*¥NIANIS = eOVIG

(SHNIS *cXT350)4°t = NIHNIS
(Sd XS3/° T=S4K 3) *G*° = SHNIS
(S139 }dx3 = Sdxa

eX 130*agWWOIS = $730
ZX 130/70 (1) EA=( 141) ZA} =2xG
CI YBADNV](14+1)GA0N7 = 2x 7aG
Invéz = I 2 oa
irae) irADPEIOIN wee
OVIGdS*NIOVIG = ¢1)S

(STAG#SHNIS)*NIHNIS = OVIGGS
(TddIS@#—EXG)*NIDSV¥VIQ = (13)d

TIVIIQ/°E = NIOVIG

CSHNIES@(SdX37° 145 dX3)*G° *S TAG) *¥NIHVIS = I9OVIG

CSHNISXTX1393G}7°T = NIHNIS

(S dX Sf °T=SaGxX5)*&S° = SHNIS

(S7T232G}dxX4a S dX 4
TX I30*dVNDIS = S130
CCT) BA DYVE(NISADUV IZ (TEN) X(VWOITS ISEAV = dVWOIS

Nd 1S -= NddJs

KdmS= —satdamS

SS OL OD Ms aan VND ES®)) shir
TEXTa0/¢€¢€T)CcA=€2) CA) = IXO
(TISADaV=( 2) aGAXYHF = TX159
T+N = IdN

T=N = TAN

OW =N

98

(630)G NOISNAWIG

COST) 2A © €O0ST)2X NIISNIAWIA
(OST)HADYW NOTISNAWIG

( ti © WNOTS © L )BASYND NOTLONAS

GN4

NYnNLas

"OQ = (230d

°o = (1340

§ Ody ws

(N) ZA*TD + (CITWNIZAXZD + (2@N)ZAKED = NddIlS

TWN TSG/NNTSG4N730 = £9
TWNTSG/N 139 /NN 199028 = ¢€d
NVSG04NN1WS3 0/4 (NTISCG+HNNISG) = 89
(2eN) FAINV=(N) GADISV = NN 130

(Z=NJGADYVECTWN}AZADSZ = TWN1A0
( TWN) GA Dave (N)BADYV = NIBd
(EVZAKED + (2S) ZAKZD + CTI CAKID = IddIS

2x 130/72 UX G/T XK 1aG=@ = ED
eXTaG/ IX 13d/2TXtTaG = 2d

Ux dwg /2tx19dg/( tx73a0+ 2txtad)= = 5
(LT YSGADHNVeE(e DGADYW = 2TX 13d

(Z)GADNV=(E£)GADYUV = 2x735d
9 OL OD (2°03°N) SI
NYNLay
(T+x4vd1)G x(4V8I)S e=(HVEGI)G = CxVEI)G
leldv = 4VEI
NCB = i ) OG
(CIWN)G *9V10dS=Z2xXdeNdd1S)*NIOVIG = (N)G

CCTWNIS eOVIAdS#=1)VII)/7°T = NIOVIG
eOVIG = LOVIG
éxg = Ixa

N

Ss)

Si30/( cox (Te=I 39

+ TT3G%(€1)G=@(1)2A) )+SHNISZ (CGHNIS*#(T@1)G +TGHNIS*( 1)
(Sd a7 °L=Sdx 3)

Sd xXax*xI

(S dX 3/7 °T=SdX 4) &

(21V30*dVNDIS

: €TSdxXS/° T=1Sdx3 )*

(TVAG¥dVWOIS )

(T=1)aAd3 V2 (138

te(I)o

ONS
Nyni3ays
I= Tt!

= (T=I)}cA)6
G ) =S8ASHAND
*S° = SHNIS
Sdx3 = SdX3
3° = 2GHNIS
YdxX3 = SdX3a
S* = IGHNIS
dX3 = 1Sdx3a
Addy = S13G
Adtt = 2€13G

(Tel )GADNV=L = T1130 F

T OL OO
ce = ti

bo OL GD € CTI)SGADHV %37° L °HO® L &35t® CTI yGADNV JAI

E>aeg ( hot iD

N¢

e¢ = il ¢

S/C THN) * (VWOTS )S
(TIEAD av =¢

N= I
SANILNOD 2
GADNV )4I
MS 2 8 OG 8
1°05¢41) 41
dv = dvWOIS
N)3A.uW = S
OW =N

°3d0530 GATWSD 38 LSAW SAEXYND SNILNONHSNS *HLONAWDYY YVIADILadWd Sil iv

A JO SN IVA SHEL SI BASUND NI GSANYUNLAY AN WA JHL

uO4 ANWA NSBAIO Vv itv SANIMWMSVOD SHL S3LV TOdy¥siNI
DW * cA * ex

GAD
Ga /Qa

°415N3 D9V SHL
NOIFLONAA STHL
ZEASNOWWOD
43 Ad /NOWWO)D

NHOS SNOWWOD

2)
2)
>)

100

(S139 )dx3 Sdx4
eX TAGxXdGVWOILS $130
CX HIAZ( (EF dCASCT+I)d cA) =eXxO
CI )dgx(T+I)czx = 2X 130
TAN®2Z = I ¢ OG
FDI DS (SOMME sy
OSVIGdS*NISOVIG = €1T)S

iH)

€STSAG*™=SHNIS)*XNIHNIS = OVIddS
€LddtS=1XG)*NIDVIG = (T)a

TOVIGdZ*L = NIOVIG

CSHNI S=€ Sd XS/7° T#SdKA) *G® *S TSO) *XNIHVIS = TOVEIG

(€SHNIS*UX739)7°T = NIHNIS

(Sax 37° leSdX4a)¥*5 ° = SHNIS

(€SW390}d xa = SdX3

TX 130k GVWOTS = S73d

(( T)dexe(N dex) (TAN) & (VND ITS SEV = dVWOIS T

Nd tS = NddIS

TdtS = tddls

Sade (a) VEO CHU OVMMONS ys ai

Tx 90 7(( T) de Ae CCA CA) xd
(T }dex=(2azxX = 1x 1aG

T+N = TAN
TeN = TWN
dOW =N

42x 40 NOTL9NAS ¢¥ SV déABPBSNIIdS Sita 3
dOW © ddA * dex STASNOWNOD
S /SSNHOF/NOWWOD
8 /3INHIPF JNOWWOD
COST )dZA © (OS5ST}dexX NOISN3AWIG
(807)S © (CST)S NOISN3AWIG
( VWOIS © Nd IS * Id IS )VATYND 3JZNIiNnousns

101

(1)8e
61 O9
CNY GCA*TD +°CTWNIGZAKZD + (CN) AGCAKED Ndd Ws
TWN 1S3C/NN T4G/N135C = ££)
TAN VTSAIG/NISU/NNISU= = 2d
NTSC0C/NN 14 0/(N 139+NN1530) = TD
€2=N)dex@(N)GZX = NN1SG
CcmN) dcoX= (TAN ACK = TWN1350
(TWN) dgx=(N)dex = NTSC
(EVA CZA*XED 44°C) dCA%ZD +01 )dZAXTD = TddIs
CXTWG/ZUXTWI/IXISI= = £5
2X WAGs 1x Wad/2 TX 14G Ae)
TX13S0/c2UKVSGU/¢ 1X13aGt!cEXTs0)=" = 19
(Ti déxe(f£)déX = 21x 130
(c}dexe(e)dzZxX = €X 130
2 Si IS GPP) sis
NdM134

CT+HVGI DG x€aHVdT)5 ®(4VdI)G = €AVYI)Y

TelavN = AVE!I
N‘¢e = I ¥ OG

(CC TWN) G *IVIGdS=CKGSNddIs)*xVIOVIG = (N)E
CETWNIS *¥OVEGdS=(d719)7°t = NIOVIDG

20

-

i]

cOVIG = 1T9VIGQ 2

2G = UG
OVIGGS*NIDVII = ¢1)S
(ST3SG=SHNIS )KNIHNIS = SVIGdS
(C1=1)G *)VIIGS=IxXd=7Zxd) xNIDVII = (1)9
((l=1)s xOVIGdS@Z9VIG+lIVIIJ)7°1 NIOVIG

€SHNIS=((SdK3/7° 14S5SGKX353)*S°)*S 14G) ¥NIHNIS = €OVIG-

(SHNIS*¢ X150)7°T NIHNIS
(S dX 3/°T=Sdxa)*S° = SHNIS

9

&

102

7 ON BSG Ch peer Cele

€ SdX3/° T=SdXd}*S° = SHNIS

€(SdxX3/7° tTeS4X3)*G° =

SdX3*ISdXS3 = SdX32
2GHNIS

(2 130% DVWOTIS)dX3 = SdX3

CIS dX3/°T=1SdX5)*5° = IGHNIS
(1190 *40UNDIS)AXS = TSAX3
(l]=1)déxe(I)d2gx = S73aG
etn (le) icicxe = cals
(T=I)dzx=i = I173G ¥
i ty ©
Ca nlel
i *yoe ea (t=l) dex )4I
Niel
JIN ILNOD 2
2*2*2 ( te( I) dex )4I
NOUkh = hk @ OM jf

[hig (Uw O@S!O Mis) Shu

S/(T#N)* (VWOIS)SHVY =%dVWOIS

€ClydeCXx=tNJdex = §$
: GOW =N

*Hd3SITVS GSAIIV)D 36 LSNMwW VATUND SANILNOYENS

MN AY S&S SQ sini siRuy si
*x wa anavn NSATS EV AUG QuVa Vas

VAAUND NI
SHL SGLV WOdGuSsiNI
elas ©

dOW *¢

COST) dZA *

(24425

VWOTS

é

G3NaNL3SH ANIWA SHI
NOILONOS STHL
acx /SVAZNOWWOD
& SAINHOF/SNOWWOD
(051)a@ NOISNAWIG
(0Sl)4acéx% NITSNAWIG
4 JVAAGND NOI LONAS
GNA
Nanisga
so) = (cis

2
>
2

103

+

y OL OO ¢

( T)} dcx

(SdX 37° I+SdX3)%5° = ¢CQGHSOD
(2 1W30*d0NITS)dX3 = SdX3
(TS dX 34 PT +ESdx3}eS° = TGHSOD

CTIS0*d VNOIS)dXS = ISdKS
(TI )dexe(I}dex = S130
dh = (I}dex = 2730
(tel )acexXsei = Tad 4
tT OL O93
e¢ =TI
CANTO si Osa IE OeyG Guo upeays sii
N = I

SNNILNOS ra
d=(I}d¢ex }SI
NS = lca OG

GS Ut Cb CWE UY sul
S/ET#N)XCVNDIS) SAY = dY¥WOIS
(ET ddex=(N)dex = S

dOW =N

SUBS (

*ysi Wva JarW> 38 ISNN VATHND SNILNOYNENS

*X LV XG/AG 30 ANTVA SHE SI
ANIA NSAID V IV

1130*¢(1}G=(1}dZA) }4SHNISZ (CCGHNIS*ET@1 2G *+TGHNISk*(1I)3 }

¢

VAGYND NI GQS3NYUNLSe ANIWA SHL °X AOA

AXYGG GHYVMVSS BHL SSIVIINSYS4S510 NOILONNSA STHL

dzX /¥A/NOWWOD
@ /GNHOfF/NOWWOD
. (0S1)3 NIISNAWIG
(0ST)dz2x NOISN3WIO

dOW * ddA *

COST }dcZA *

41 © GNDIS © £ YVAGHND NOTLONIS
GN 3

Nanta3y

Tara e
ST30/€ 2WI*( CTel d= (le ld cA)S

=V AAYND

3
3
3

104

TOV¥VIG/°T = NIOVIG
(SHNIS=(SdX S/ ° 34S dX53)*G°*S TISG)*NIHNIS = ITOVIG
(SHNIS*TX13G)/7° 1 = NIHNIS
(SdX37° T=SaGx3)*%S*° = SHNIS
(S130}dx3 = SdX3
TXT3G*dVNDIS = S173ad
COT IVA DYUV=(NIVADEV ISOC TEN) KC VWDIS)S37 = dVWOIS T
Nd tS = NddIS
T3735 = IddIS
G OL 0D (°O0°LTSYNODIS}! STI
IX 130/ (( 1) deéxe(é}dzZx)}) = IXG
CT)VADSVe(2Z)VADSW = IXT3G
L+N TdN
T=N TWN
dJOW =
VADYW SO NOITLONN4A V SV d2éX==3NI1dS SiI4 9D
dOWw © ddA © dex SVASNOWWOD
VADIUV 47 Ad /NOWWOD
S “SSNHOfS/NOWWOD
3 /3NHIF/NOWWOD
(O51 )dZ2A © (051) d2x NOI SNAWIG
€807)S * €05T)3 NOISN3EWIG
( OGT)VADS¥V NOIISNSWIG
( VWWOIS © NdWS * TdTIS )WAcYND SNIiNaysns

ONS

Nanise

I = Tl!
STSO/((CT#L Ga Tel kdcA de ( (1 T

YG=e(1)dZA) )+SHNIS/( ZOASOD*( T=1)a stTIHSOI*€I)a ) =VWAGYND
AVWOISZ (SdX a4 PT =SdGX3)kS*° = SHNIS

SdX3*ISdGX3 = SdX3a

105

(2 ydzZxK xe D +¢(2)dzgX*2ZD +01 )dCKRED = TddIs
2X1V390 72 1X73 0/1xX13G= [25
2X WG/ IX 130 7Z1x13G eS)
UX TV4G/ ZIXT3EGZ ( 1X TAG+7Z1X 7350 J= 5D)
CT} ADS Ve (E)VADNY = ZIXIZG
€2)VADdVeE(E)VAIUY = |@xTad
9 OL 3D (Z*DS°*N) JI
Naniay
(T4+HVGI)3S x(4VEI)S =(HVGI)3 = €xAVdI)S
I=I1dN = HVGI
VC e hf wv OG
(CTWN)S *£9VIGdS=2XG=Ndd1S)*xNI9vIG = (N)J3
CC INNIS *xOVIGdS=I9VIG)7°l = NIOVIG
ZDv1d = t9VIG
AG = Ty6G)
OvIddS*¥NIOVIG = €1)S
(S732 =SHNIS})*NIANIS = S¥IGdS
(€T=1)3 *9VIGdS=1XG=2xG)*¥NIOVIG = (193
C€T=1)S *OVIGAS=ZOvVIG+I9VIG)/°T = NIOVIOG
(SHNIS &¢ (SGX 3/° 14S dK 5}kS° )kS13G)*NIHNIS = ZOVIG
(SHNIS *2X730)4°T = NIHNIS
(SdX3/°T=S3X5}%S° = SHNIS
($139 } xa SdXxa
ex WaG*xdVWOIS = S730
2x WI/( (1 aAexe(t+I)agx) =2xa
CT )VADSV=(T+1)VAR4s = Z2xX13G
cwMoe =& k SG AG
€ OL 3D (€2°DS°N)} GI
OVIGdSXNIOVIG = (T)S
(STAG=*SHNIS)*NIHNIS = OSVIGdS
(Tl ddIS=IXG)*NIDVIG = ¢1)3

106

lay (by (© (Sesh Oe qo J Ope GL OBO 6 oat Norell —)) shi
Nie—sel
SANTIiLi NOD c
OBC (CHO) yoy yeh
WOT IC s I 4 aya |
Ars (i VWs} CIE)» 21
S/( TEN} *X(VWITS) SHV = dVWOIS
(TIVADEVFCN) VADYVY = S
dOW =N
°340S35H GSTIVOI SE LSIW VACHND SNELANOYSNS "YLONS WAG SVINITI¥VG Sil IV
K 43D SNIVA SHL SI VAbUND NI UGSANOUNLAd AN WA SHt *ALONA WV SHL
YOS ANIWA NSAID gv iv ANY3qd GavMV3S BH SALVIIDHRINI NOILIONAS STHi-
dOW © ddA * dex SVASNOWWOS
VADaGV /VAV/ZNOWWOD
a SANHIF/NOWWOD
(GsT)3 NOISNIAWIO
(OStIdcdA © (CS T)dz2K NOISNSIWIG
( 0ST} VADISI NIISNGWIG
( EY © VWSTS © 1 )WAaMd! "N@IeEONAS
GNA
Ndnisga
Se = (e)) S)
Po = (1b le! 9g
bo Ob. gS)
(N}déx KID #0 TWN) G2KX*29 4+(22N)dcdxeeD = Ndd Is
INN TSACG/4NNTSAG/NISG = £9
IWNTAG/NVAI/NNISG@= = 23

NTSG/ NN 1302 (N TSG#NN150) >= T3
€ CEN) VADYAVEEN) VADYUV = NN1TSG
(Ze@N)VADUVNECTWN)VAIYF = TWN13G

CTWNI VA DGVeE(N)VADGV = N13BG

2)
2)
2)

107

S OL OF (*0°%171"VWOIS) SI
IX 130/ (( T)dZA=(2)dzZaA) = 1xQ
CU ADSv=(2) VAIN = ITX130
L+N = IdN
T=N = TWN
dOW =N
VADYUW 4O NOILINNS VW SV dzZA==4NI1dS Sild 9
dOW * d2A * 4exX /SVA/ZNOWNOD
VA av /¢ Av /NOWWOD
S /SNAOF/NOWWOD
4 /4NHIP/NOWNOD
(80v)S * (051)4 NOISN3WIG
(OST)dZA * (0531)d2X NOISN4WIG
( UST) VAI4¥¥ NIISNAWIG
( WWSIS * NdtS * Td1S )WAEaND ANI LNOASNS
ONS
NYNiay
I = VI
S130/¢ €130x( (=I a= (lel )dzx 6
+ [1S30x€¢1)d=(1)d2X) )+S4NIS/(Z0HNI S(T") 3 +TDHVIS*(I)3 ) =VAvEND

(SdX 37 Pb e@Sax53)kS° = SHNIS

SadxXS3*xISdXS = SdX3

(SdX3/7° T=SdX3)*S* = COGHNIS

(€2130*xGVWOIS )dX3- = SAKS

CISdXS/° T#1SS XS} *S° = TOHNIS

(1 73G*dVWDIS)dxX3 = TSdXA

CT#I) VA ONVE CL YVADUY = S713ad

de(IyVADHV = C130
(Tel)VADGVeEL = 1130 9

tT OL OD

e= TI

108

I=IdN = »>VGI
MOG e& i (KG)
(CINN) 4S *9VIGdS=@=2XG=Ndd1S)*NIOVIG = (N)4
NIOVIG

(CTANDS xOVIIDS@IDIVIII/S ET =

eJVII = 19VIAd
exG = xa

OSV¥ILGdS*xNIDSVIG = (1I}S
(S TSG=2SHNIS JXNIHNIS = OVIGGS
CC TelL}S *3VIO0dS=IxXG=cxd)*xNIOVIG = (1I)}4

(€teI dS eOVIIDGdS#=Z29VIIF+tTSOVIG)/° = NI

OVIG

(SHNIS =€¢S dx 34 °14+SdX5)}*S°)*STAG)*NIHNIS = cCOVIG

€SHNIS*2X130)/°T = NIHNIS
(SdX3/7° T™SdX5)*S*° = SHNIS
(S730)dx3 = SdX4
eX WGxdVWOIS = S130
2X WHG4( C1 DdCASBCT+1}0G2A) =2XO
CI}VADYVECT+I1) VADNVY = 2X 140
Inv‘é¢e = I ¢ OG
£€ OL OD (2°03°N) AI
OVIGdS*NIOVIG = (T}S
(STSG="SHNIS)*NEANIS = SVIGdIS
(TddIS=IXGJ*NIOVIG = (TIS
IOVIGs*T = NIOVIG
(SHNIS@=(SdX3/°T4+SdxK5)*S°*S BOGV*NEIHNIS = IDVIG
(SHNIS5*41X130)/7°T = NIHNIS
(Sd 34 *Te=SdX 5)*S5° = SHNIS
(€S7t3G}4dxX3 = SdxX3a
TX 13G*edVWOTIS = S390
COTPVADYVE EN) VADSVIZ EC TEND XC VWOIS)SAV = dGVWOIS
Nd 1S = NddIS

td1IS = [dad1S

&
2

109

°34y0438 GatIIvD 3G LSNW VAENND SNIANGwINS *HLONA WY HV INI TLYVd Sil 1Vv

A SO SNTIWA SHI SI
yuOsd ANTWA N3BAIO Vv iv

WASYUND NI GSNYNLSY AN WA SHL °419N3 19e8V AHL
ANG@ GHvMVSS SHi SHLVIWIdGYSINI NOTLOINAS STHIL

dOW © dcA * dex SVA/NOWWOD

VWADUW SVAVSNOWRWOD

4d SANHIFSNONNODD

(GST)43 NOISN3WII9

(OST¥dZA © (05S T)d2x NIJISNAWIG

€ 0ST) VADHYY NIISNAWIG

é

( LI * WWOTIS i )VASHYND NOI LONNS

ONS
NYnlay
°o0 = (2)4
Co eS Gil ep
tT OL 2D

(N}AGCAxXTD F¢TWNIGZA%ZD +(C=N)dcARED = NaddTIS

TWN TS0/NNTSG4N13G = £9
LTWNTVSO/N139/NN 1802 = 23d
NTSG7NN1W3G/ (NTSC#+NNT3G) = U3
(2@]N) ¥ ADHVE(NIVADYY = NN IAG

(Z=N)VADUVECLWN} VAIN = TWN 180
( TWN) VA DHVE(N)VADYUW = NISZG
(EVGZAKED +€ 2) dZAK2D +¢T)ACAxXTD = Idd TS

ZX 1W90/Z1xX WAG/IX Wd] = £9

Exdadg/ IXVaG/eTXI13G = cd

Ux Wd /ZEXTVSO/( ExXTaG+2TxtdI)= = 19
(Tt }vA DYVe(e VADUV = 2IX 130

€2)VADIUVeE(E)VADSVY = eX 130
9 OL OD ¢€2°D3°N) SI S
Nanisd

(t+ nVGEI da eC AVAIDS e(AVSI}4S = CAVET}S WZ

2
2)
2)

110

Nanisd
°¢e435 =HNIS
(AY GXd=(X} dXxS=S
Xe=A
(XHNIS NVOTLONAS
QN3
NYUNL AY
I= IU!
ST3G4€ 2 T50*€ (TI bd= (T=1) d2A)6
+ T13G0*(€1T)4=C1)deA) }+SANIS/(€(COANIS*( 1l#1} 5 +TQHNIS*(I}3 } =VAGUNAD

€SdXd/7°T=SAXK3}*¥S® = SHNIS
(SdK a4 PE @=SdxXa)kS° = SHNIS
SdXsaxTSdx3 = Sdk3Z
€SdX3Az7® T=SdxX3)*S*° = ZGHNIS
(2 IAGe dWWOIS)GdXS3 = SAX
€TSdX353/7°T=1SdX3)*S* = TGHNIS
(C(t T3G*dV¥WOI1S)axX3 = ITSdkKS
C T=] ) VA D6V= C(I }WVASDdV = S130
4e°(I)VADY4V = 21hG
(tl }vVADGV=i = 1 HA
iS (aut {aj}
e= tI
{Je @S) (iA ars) - CalajO, Je Seo f CSS CSpot Yi
N = I

SNNITLNOD

£*2*2Z ( L=(T}VADHV )AI
NeW SB if & (BG)

@ = (hl CUOweyO Sys 2)y
S/CT=N)x(VWOTIS)SEGY = GdVWOIS
CTIVADHTS(NIVADYY = S

GOW =N

QN3

Nani3s

°2/S =HSOD

(A) GXS340K% ¥dXS =S
Xm= A

€X)HSOD NOI LINAS
GN3

I2

APPENDIX B

FORTRAN Listing of Program GRID

3

°09E =VGWV 1x
®US/ °09=d VWIS

(vie *L°Sse *S*vle)ivVWwaod
SN *GN*L4950%LH1SS*SS 190% VLIWAN * IXWAN * XVWNO TSS) OVSy

SiNIGd 30 YSEWAN *SHId3aSCG 40 Y3SEGWAN *S3ENNIW NI aviS 1 Virsa
JO SNIVA *WN NI avWLS S SO ANIA LSHYIS *WN NI YvtS S VWiT3G0 40 3NIWA
*S3NI7 VL SO YSBEWNN S$2+SSNIT FIX 30 YAEWNN *S355309 30 H3SEGANNN QV3a

XVWN° SX *OdZESD0D°EG03 /SV1YXYOS/7 NOWWID

LEWZ8* L6d7E8°VLSExAxX /OS/NOWWOD

AS®XS SAXS/NOWWOD

A*X®WL3° IX SAKLAIX CNOWWOD

GOW SdcAtdcx 4VA4 NOWWODD

OWS CAS SCX SAAS NOWWOD

(SIDJTIEESID*(S)8* (S)¥ NOISNSWIG

(OOTIAS © (O00T}xXS NIISNSWIG

(SES¢LMyZ © (SE®ZHIA © (ST*20)xX © (STIVID © (2e)IX NOISN3SWIG
(OOOT}IEALVY © (OOTIEXLV * (OO0O0T)IVALVY © (O00T)YXLVY NOISNSwWI1d
(OOOCTIGZA © (O00TIdexX NIISNSWIAa

€OOOECA * (OGOT)2X NIISNSWIG

(G0¢}I209 * (002)80D NIISNAWIG

VTYVD SNVWOIIYYNH eOsa GIYo

®°Sa55b9

SHL SO LNdGING HLIM ( WHOSNOD WVdOOXd } ALATVIWOD SI NOIOSY 3HL 30

ONIdGdVW WWHYOSNOD BHI LVHL G3SWNSSV SE 41 °3949NSS WYYDOdd OL
LNdGNI 803 ViVG GIYS SNELNDWOD SHI SSNIWHS1350 *adl¥S wyveS0ud

T

YU OVY YU

VUVUVUUVYVOYN

114

(T=9N)/VOGNV IX =1X 13G
tet =9N
T=EN

AYGG VSS UNV SNITLSVOD GSLVYHAN]AD WYHOSSNVWYL ANIWYSL9G

*3wil WAVYL
3AVM ONODW SO SATLIN3SS3SY¥d35u SI INV 7h DL TWIWWAWd SI SIxXV ¥1

°419N3 19u¢

3NIVLSVOD SO SATANSSSYday SI GNV IX OL VAWVWUVd SI SIXV *S

°(*x1L*xS) NSLSAS SLYNIGUOOD
STSHS GSHOLAULS v 3O SWUSL NI (VWLS*IxX) SALVYNIGYOOD SNITWSYOHS ANI WYS13d

VLSG*AK =VLIENK
Vina = OxZe =LEWZ5
Viadad + O0YZH =1LEdZA
(2° vtadc) ivweos
SONNEI LNOS
€N}DO03*(NIFOD (22°S) QVA4
XVWNST=N 92 00
O¥ZE*viad (24¢°S) AVY

c=IXWON =CIXWNN
T@=IXWAN =TT XWON
Tev LAWN TL AWIN
eck =9
T= VWOIS

VGNV IX/Id = AX
e6STtyl®& = Id
o°EE7O0°92Z%O0°09E =WNAGIX

N

Ze
9¢

ike)

( AWSGeATSG + X 13G*exX130 )LYOSkdvyWIS =(I1)¢2x

¢
(

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(S149) dxKa = Sdx3a
2X Wdk dVWOITS = S130
EXTSI/ EEL Vd cAe(T+I)dcA) =2xa
(1l)ddX=( TtI)ddgéx = cx t3aG
TWNS2 = I ¢ OG
Ss OL DS (S°OSION) an
IVITGS*NIDVIG = (1)S

€ST3SG"SHNIS)J*NIHNIS = OVIGdS
€TddIS=tTxXd)*NIOVIG = E€ET)A

tTOvIGs*t = NIOVII

(SHNIS=( SdX3/° T#SdX3)*SG° *S 1930) ¥NIANIS = 19VI90

(SHNIS*IX730)/°t = NIHNIS

(SdxX 3/7 °TesdkK3)*S° = SHNIS
(S139) 4X3 = SdX3

UX WaG*kGdVWOIS = $130

COR) AEXPENDdCXIPSC TEN) *(UWOIS¥Sdv¥ = dVWWOIS
N37S = NdadTS
Td1IS = tddlsS

GS Ji OD €°O°11*°VAOIS) SI

TX T3040 (Td) dzgAe(2)5dcA) = IXI

(T)dexe(2 dadzgx = Ix 15d

ttn = TAIN

T=N = TWN

dOW =N

146

(O03)A NI ISNAWIG
(O00T)dZA © (0001) d2X NOISNAWIID

CJHS COs: OC Ij iNelaey Navan) ve!

ONS

NyYAtay

*d = (€2)A

°G = CI}A
by Ble

EN} GEAKID +€C TWN) G2ZAKZD #+0( CHEN) ACAXED = Ndd IS
TWN T3G/NN 1390/N150 = E€)
TAN TAG/NISG/NN1TSEG= = ¢d
NIZI/NNTIBO/SONTIItENNGISI) = 19
(ZN deg X=(N)G2é@xX = NN WI
(2=N) dceXe(twN)dex = TwWN1W¢G

C TWN) doxe® (NJ 4eX = N39

(EVACAKED +°¢2)d2A*2ZD + (T)deAexTD = TddIs
exVaAI/?2 TX 13G/TX tWaAG= = ED

ex Wad/txwd/2tx1sd co

IX T3G/2TXK 1404 C1X 13G+21K 130 }= = 19
(l)dcxe€&)dex = ctx tad

(2 )dexe(edadzgxXx = ¢e€x139d

9 Of - aD (2°dseN) aI
NYnisy
(T4HHVELE)DA *xCAVEI IS =(HVEI)A = ( AVAGI)A

l=IdN = YVEI
oe Sr oO TG

COINNIA *9VIGGS@eXG=Ndd1S}*NIOVIG = (N)A
(CTAN)S xIVIGDS@19VIFJ4°L = NIOVIG

Zdv129 = LdVvIG

zxGQ = Ixa

mM

147

S/(T#eN)&(¢WDIS) S37 = dVvNDIS
(l )agx=(N}dzx = 5
dOW =N

*SN1VA GSLVIOGHSLNI SHL SNIVINOD (GHOOD=A=)} HASAND

eeiNdinO NO

®* pASND NI GSxe¥S3LIWNO Jav 3S34L ONV EAHND 3D LNG1ND SHL NOUNS GA4SLTYNNA
GSLLNAGNI 3A QINOHS WWOIS GNV A %dedA*dZXxX (dOW)N SHYSLAWVYVS SHI

*( déAtdZ2x ) SSGON 341 Lv 3A8TD AAL 3D BATIVATAHRO
GNOD335 B4t GL AVNOITLYOdOdd SANWA 4HLIM AtS40 NV_LSI A
*SENIOd GILVIDdYeSLNI IWNISTSO
SHi 30 SSLVNIGYHO INV S¥SSID30 34Ll ININIVINID SAvV3sS¥ SHV GecA* dex
°BAMND 344 JNIWH3Z1LSG
OL GSLVWWdNS4LNI S3535M HOTHM SANIDd 433 ‘YSGNAN SHi SI GOW =N
SAS) SHL GQLNO Gaddvw 3G O01 SANIWA 1V5N V SNIViNOD (359905=xK} +t

selNdiNI NO

VUVYVVUYGUYVYVOEYVYGBYNYOYYG

FOO IO OO CGO OO OGIO OG IOI OGIO ICO RK kK mack (#2
- 3
*Sy¥sLSWV ad AYUVSSSDIN NIvViusdD SNEWHS159 O1 HSI TH75 BRAID 3d OD
QIABDHS ENAND ANILADYSNS BHL *NOFISNSL YSONN SNI TdS V ONISN 9D
INIOd NSAIS V LV SASND V S3LVWdasiNI NOI LONAS SIHL D
2)
FORO CO III GIGI RO IGOR I OGIO OO I IOI RRR AOI IC RGK gk a akc a ak aki ak 2k
: 3

DON* dZA*dZX /SVAS NOWWOD

A /9YNHOPSNOWWID

148

+

O° 2/ANIS

(NVGAYDXS = (NVOX}AXS
nv 0X

(NVGX)HNIS NO

0°2/HSOD

(NVGA)IXS + (VOX }dx3
o NVOXe=
(NVGx }HSOD NO

SIad/¢ 2IWI«Kl Cll pA] CT
TVSG*k CCI )A=C€1)dZA) DFSHNISS OC 2GHNISx( T=] )A +TIHNISXCIDA 9)
(Sd a4 PT eSaGX3)«S° =
Sdk3*TSAXS

(SdX 34° T]=SdX5d)}«S° =

(2 V30xdVWOIS )dX3
€TSdXS/° T=TSAXxX3)«S® =
(T1V30*dVWSIS)dxX3 =
(T=I)d2éxe(I1)d2x

do = (I) dex

(tT=1)dexer

T

yy se DS NE Caloak CBO Lf Seo Cuca pel

SANI

B 2092) 10 bei id
N° TI =

e = TI (1°03°

agN3
Nanas
=HNI5S
=HNIS
=WVTA
IL ONS
GN3
NY M434
=HSOD
=HSOD
=WVGA
ILONNS
GN 3
NY nisy
I =TIl
I )d2A )6
PAUND
SHNIS
= SdX3
COHNIS
= SdX3
TGHNIS
TSdX3
= $7130
= 2739
= 17139
OL 939
c= Tl
2x )al
N = I
1NOD
ex jal
I 2 0a
4I)} GI

149

APPENDIX C

FORTRAN Listing of Program SSURGE

150

ft
SIAC

7S200° *8°6
44 *AVAD
Avis XGlee

: THa®
$ i THA ¢

*3e “sl
“OLT “SHE
ADs Cae

L7530* 01N03 = ED
L7450%*0 °c=c)9

€° SvGE*xAWYUD) LHOS So)
(4 13G*xAWVU9D) LYOS=59
GS0eaG4804°9 =OTHO05S
c=nwl=cn We

Tew =nwe

fen =EWAI

c@=WI = CWHI

Tew l=wwt

“86 §)T *2iee eaiSilegeoe
‘OHI *GiIf *Sil ‘SHE *SHI viva
(St 2b “ot Oiesictesson

SVIFSVELIS DHE Se PHISE LF SEL I SEHF Sc HI vivd

Cts OO} 15

SGT Sol Sv Set Si eer

AZLFSCLIS2HE S2HI ST ar et al *tHFStTHE viva

*O°0e8T S2°lveT Se °Lecet
‘iad *y 13g ®I xd viva
7O StcEet *SI *Se
ZINISXVNN SAE SWI Viva
ANId
£ ciNn* Tin
TI3¢ QTD
LIHX * KISS SINVYUA STYUNHS NOWWOD

QL SOLIS GHES OHISSLFESLIS SHES SHI/9 ANE/ NOWWOD

INI SPL SVL S vor ® pA S ELF S ELIE CHF SEMI SP CLFSCLISCHFt

*2ZHIS TLE ST LIS THES THIS xX VAN

*AW AD 4c 4% Wd /NOWWOD

CWNT SENNT® CNNISLOYNS OS EDSSISOINID SUCH IE/S NOWWOD

N *WWE So WW

“wr SWI 6 3a*t W90S W130 SI xd sv LaF IDSI K19%99* 2O/eo 1G /NOWWOD

S39aNnss

S|

Of

GN3
dOLs

ANH WWW>

“SANTI LNOD

xn14 Wvd

SNNILNOD

volt OL O95

IMvVaG TIVO

As13 11)

4QNIM WW)

OL DO(N°OS*SKES/N) ) AI
XVWN® [=N OO1 OG
G731I4 WWW>

Guaz Wo9

(GVeS3SG*I Hd) NIS=TID
(GV89S0xI Hd )SOD=O0TD
* OB T/t6SIyl®*e=Gv4o3z0d
*OTGTt =4aNId

*O¢=I1Had

* ZT =3ONVYA
SE°6H=LIHL
°89 1=LIHA
°O9T=LIHX
We=AI
WI=XI

VLS0/ 4 TAG*eAVYO=HVIL3AI9
1xXO/i 730xAVYS=IXID
ee) = 104

EO*ED = VO

OGt

Ce

152

°O =(F*S TJONIS
°o@ =(F*T)9S95
°G =C(FFST)AA
eu =€FY1)Kx
°UG =(F STFA
°G =(fF*I)xX

20) = CGS lia
°G=(FSI)S
*U=(F*S1)2QlHo
°c=(F ST) TO1YS
WEStT=F GUT Od
°G =(1)1IxdSG
WI*t=I 3CI Od

(ST®SHIAS (STS GH)X SVELXS/ NOWWOD

COSTISOGAS (OSTISOdX*(OSTISOdWIL*S (65) 08ZaG T

. (OS)XVWHAS. C€OS)IVAs (0S) Lox * (OS JAIL Zann NOWWOO
(ST*Sy )ONIS * (ST *Sbh }9S05% (SI *St AAS (STSSbIxXxK ZSNVYLS NOWWOD
CSTE OV)IUNIM®S (ST* SY )d ® SARA S35A35% © GN/WOODLS/SNOWNO OD

(Z9)9T

SHSIHS (2Z93SSEOH® (29 HS EOH® (29) ES 8GGH9 (29 )CSGOH £( 29) ISHOH/2 Hd JNOWNDD
: (COE VIWDA® (U0e )SWIA® CODE FI

GWIN* (COE ISWIN® ( COE FODAH S$ ( COE )SAGAHS (SGT) 13G10*(S%) I XGSG/SA TE /NOWWOD
ONIS PLES oL I*o He * pHi tear Sea PSE He PCH Sel seal S2Hert

*CHIS TIS Til THE SETHI S KVAWN *AVUS ZEA TES NOWWOD
N ° WWE S WWI “wl “WIS 361 130% Vi3d0 SI XGSVLISTOSI X1I9*99* 29/2 ATA /NOWWOD
(OOE) YNON® (OGE JENDA*S (OCE DEWIANS( COE) TIWDAS(O0E DS

WIM *( COE JDEWIN* ( CCE DZWINS ( COE) TWIN® ( OOE} HGAH* (008) EGAH* (COEF SGAH!S (OT
CE IGAHS (ST*&Sb)S *“(ST°Sb I2GIT YD * EST SS) LAILHYD/T ANE /NOWWOD
CyusZ SNILNOUANs

153

29°T=mH GUS OG
SNNILNOS

*¢ =(f) L3GLa
we*t=f O0Ov OJ
SANTILNOSD

ee Oo = (FSTI)GNEM
we*t=fF oof OG
XVWIST=I OGt OG
TtwIl = XVWI
SNNILNOD
*G=(A)OIWIA

© O=( 4) GNWIA
°O=(HAIYNDA
C= CNJEWDIA

© G=( A) CWIA
SO=(A)JIWIA

° v= (9%) GWON

° G=( 5%) SWON
°G= €H)vWON

° G= CA} EWIN

° O=( H}ewoN
°O= (4) TWO

G* G=¢ 4) 9GAH
G° 0=(H}ISAGAH
C°O0=€H)}7GAH
ue G=€) EGAH
S°0=(H)¢ GAH
6° 0= (4) TGAH

XJGN*T =" OGe OG)

LtONIZX VAN =X3GN
SNNILNID

COT

COe

O0€

vGt

154

(ST*SPIQINIS*ESIT*®S¥)9OSID* (SIS SAAS CST*&SY)XX ZSNWYHLZ NOWWOD
(Z29)9T

SIOH *( 29) SSGOH £ (29) VSBOH® (29) CSADHS (29) dS BOHS (29) 1S GOH/42ZATd/ NOWWOD
( OGE)GWINS COOESNIA® (OOE)T

ONIN (COE ISGNINS (OGE )FGAH® (O00e JSGGAHS (ST )L3010*(S4)1 XISG/SA1B /NOWWID
N *wwf *wal SWE NTS ASL TSOS VLSOSIXGs VAS TOS IXT9D°D0* CO/2AIE/NGWWOD
€ OOE) PWOAN SC COE JEWDAS( OOE) CWOIAS EF COE) INDAS COCE) 12

WINS (COLD EN INS (COLE JVZNIN® (OOE) TWINS COOE JX7GAHS (OGE JEGAH *( G0E)ZGAH SCOT
Ge YTGAH*®(ST*SH)S ©(SI°S%) CGIYD® (ST* Sv} IGIYD/ TA Ta/NOWWOD
GwIida JNILNOYANS

GN3

Nant ay

SANILNOD

°Q =(H}SOdA

°GO =(%}SOdxX

°G =€H) SOdWI L

Ost*T=% O02 OG

SaNNILNOD

°o =() 082d

°G =( HX VWYA

°G = €)QGV4

°G =(4) 194

°O =(M)WIL

-US* T=% 009 OG

AJMNI LNOD

°G =(>4)9SS8GH

°u =(%)}SS804

°G =( 4) ¥%¥S80H

°O =(>%)£SS8OH

°C =(%) 2SSO0H

°o =(>%)TS8E0H

cod

009

155

ANNILNOD COV

(ce°24stt)LivWwdoa SG

(We? tare (ret eGlyuo) (S*TAN)Gvay

wI*l=1 00% O29

Ss fan

(1° O0L4c£0 *Sa* Xt) LVWuOS

(ZANST=I1°( I )SOdA*( I )SOGX*C1 )SOGWI L)( 8159) 3LIMM

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€(£1T*9 PSLIUM
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(7* (Gw) CSLOND) (ANN) (930) (SUH) HOSd
*xTSs* SS3Yd NAD SGNIM XVW SNTOGWS NOI LV 10a JwWIk HTtSTt

®XTS//7* SHV SHSLAWVYUVd WHOLS SHLHe *XST® THT )LVWHOS
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(T° OTtsae) iVwHOoS

(2LIN* T=1% CL PSOdGAS (CI YSOdK * ( L}SOdWIL} CET SS JGVAd

ein (€TT*S)GVS4

(t°otTsaS JALVWHOS

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( SI )ivwyos

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1

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dt

GT

st

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156

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NI*T=1 tC OG
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BCE °O* (11)9S58OH=(11 ¥9SEG0H
870° Ox (11) SSADH=(11}SS80H
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BvOecOx (II) ESGOH=(11)2£S 80H
87 0£° Ox (11) cSHO4=(11) ¢SE0H
B7OE°Oe (II) ISG0H=(11 }1S80H
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29*T=II GOf OQ
€T°v a6 )LVWYUOSs
(29° T=1E*€1E1) 9SG9H) (6° TAN) GVad
(29 *T=1I *€11)SSd0H) (6 SE ON) GV3y
(29° T=HTTS (II) 9S80H) 05° TAN) GV3AY
(29°T=I1* (TI) €S89H) (6% INN) QVSH
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ANNI LNOD
(2°vta*X¢e )Lvwuod
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we*et=F COs OG
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158

( LSLOS (LOSA=dOLA JFIXSO/ (539 WNHLIGN, P¥OD HALF *T)YTGIYD = (FSET) TOL1SD
(TAF ST IS*k(TWret)ecGlad = LOA

(TdF PT)S* (IGF *1)2CGTHS dOLA

(FS TWIT Se (fF STWE) TITad 430n

(FE TAGE ISKIF ST GI DIEGIYD = LIYN

Ixd*x(I1}1XUSdG=IxSd

(FSIIS/CF S1)9S/i7450 =659

iT]

T=I = TWI

1+I = IdlI

e*SwWI*SI=!I Oe€Tt OG

Tef = Iwe

T+f = tof

CWWI = SWI (£° 03° SI )41
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