>. Avurip oo Cug, TR 76-3 ‘ : é Vol. TE 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. Limited free distribution within the United States of single copies of this publication has been made by this Center. Additional copies are available from: National Technical Information Service ATTN: Operations Division 5285 Port Royal Road Springfield, Virginia 22151 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. MBL/WHOI ION 0 0301 005900595 7? UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) 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 15a, DECL ASSIFICATION/ DOWNGRADING SCHEDULE 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 ————————— EEE ee ee SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE(When Data Entered) 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 UNCLA B SECURITY CLASSIFICATION OF THIS PAGE(When Data Entered) 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 a Ni MM i i iat mn Ly Hi = 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 < —€ 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 , =) » 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. 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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 " 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) 5) 2) 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 OL OS Ndd 1S G/N1S30 = £95 iy) °¢ 2 @) 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 . 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SNILNOYANS SHL °NODISNAL YSGNA SNIIdS Vv ONISN ANIQd NSAID V-tLV SAUGND V S3FLV1WOds3SLNI NOTLINOA STIHL BEAR I ROK RR RO aK OR ke RFE 2k 2k ae ae akc a aC 2k ak ak ak a CE 2 2K CK 2 aK aK OIC OK akc ok 2k akc 2 2c aie ak ic yY STNAOF/SNOWWOD DN & cA * 2K SEACNOWWOD (CO8)4 NIISNAWIG CO03T IZA © €0990T)2K NOISNAWITD VYUVUYYMVUYNVYVYVUVIUIUVYVUYVOYVYYVVYUVVIVOUYY 142 SNNILNOD ro e* dtc ( le(I)ex Jar NUH aS Ut @ ad TWASXSSALNI YOS HDYVAS Geo un ChOwsyo lit) shi OSNINNID 33S WONSd LH7LS SFSTIMHSH19 SGSLVNINYSL AISIVOIASYd SYSHM HOaVSS LYVIsS T°AN° Lt Al S/C TPN) k¥C7WITS)S3SF = aVWOTS (T)}eKe(N)CX = S VNOIS AZIIWAYXONAG OW =N “(NFCA =CAUMD* (NISX "19° LF BOs CT PZA SeNAdND “(T) ex vane 2 soa SS CINSMS Nik °SNTWWA GSlLv1IOdy¥SiNI BSHL SNIVINOD (GadO0Id"=A=) 2ANND eelfNdiNnd NO GSYS1WNA Suv 3SSH1 GNV TAYND 30 LNdind 3AL NDYHA AZHALTIVNA GSLINdNI 38 GINDHS WWOIS GNV *4 S2A 2X *(OW)} N SHSILSWVAVd SHI *T=y 4LlIM Sidt4S L ININIVINOOD CC T442X* (42K) AVAYSLNI FHL GOS HOUVAS SH4l 1 ST II SI "WS SHi 30 XSGNT 34a D4 LT ONTLLSS JNV ONISVAYONI SSNIVWA SHL ONTYSGNO AG JQANIVD SI AINITIII 3433 *SINIOd 40 tS) BPUUYUYUYHMYUYVYUYYYUYSOY 143 ( COOT)dZA * (OO0OCT} deX NII SNAWIG ( VWWOIS © NdIS * Id1S J=EAYNND ANILNOUSNS ana NYAiay R 2 2y STB0/( 2737 eC (Lely © (lel) ZAd.6 + 1750% €(1)d=€1) ZA) )FSHNISZ(ZOHNIS *(L@L aS +TGHNIS#¢(1)Y ) = ZAYND (SdxX3/°1L=SdxX3)eS* = SHNIS SdXSe1SdXa = Sox] (SdX3 7% TeS4X3)«S* = ZGHNIS (2 130* dvWDI1S)dX3 = SdXx3 (USdX3/ *T=15dK3)*5° = TOHNIS (11ad*xdvW91S)dx3 = 1SdXx3 (t=I)ex=(1)2x = $73a Le(I)2x = 2730 (teI)ZxXelL = 1730 ¥ NOTLVVId3SiNI W43455334 GNV gd 135 { OL 8 QA a ii ( LOSYUOONT SVM sellee ANGNI ) TI 45S34 QNV HOYUVSS LYvisas b OL OD ( (12x *an® 1 esos 1 *aa° (ter)]ex yar WAUSINI £ OUGW >] 493498999 SYNSNI JL AXDISHD JD N = I 3 144 *Q3SyYSLWNNA 3e8V VWNDIS *NdIS *IdIStadcAS dcx * (dOW )N ( ddA*dex ) SAGIN NSAATIO SH1L LV SAND AHL 340 SATLVAIYS9 GNODSS 341 JDL WNIT LHId3N4 SAINIWA SNIVLNOD A 2exiNdiIO NO °¢ S9VuOLS HOLVYDS } N LSVWVAT LV HLONAT SO AVUYW NV SI Ss ( 39VYHOLS LNINVANYSd) N LSV735T LV HLON3ST 39 Avsuv Nv ST A SSNWA 3414 30 S3LVNITHO N 3D AVN NV SI dea S3SNTWA AVNOILOINAS SHL 3O 3VSSIDGV ONISVAYONI N SO Avddv NV SI dex (2 °39* ) GSLVTOdDYSINI Jd OL SANTIVA JO HSDNNN JAHL SI dOW =N e=a=iNdNI NO 2k ok ke 2K i A KF ORR kK kk aio ki kk ei i kr a etki ok i ke ee ie fC ee eo Re ae ek ok *VASYND NITILINIA AHL W1V5 OL AYWSSSIDAN SI LI SAYND AHL NO SINIOd 30 NOT ivViNdwod WOiSY YO3 °Da4LTIWO 4D G3ISTID39dS 3G AVN 3AYND SHL 3O SONS OML SH1L LV¥ S3d071S 344 °SSNTWA WNOILINAS 3D SINANDAS V HONZISL NOISNSL YSGNN ANI WdS AXOLV 1O0ddy3a3lNI NV 3LNdWOD OL ANYVSS3DIN SH3LINVY7d J4L SINIVGYSL3ATI ANILNOYSENS STHL FOO ROO CII IO GIO GOO IOI OOOO III FOR IOI I KORO a doi a JDOWS dc ASdGZX SAAS NOWWIDD A ZVNHOF SNOWWOD S S2ENHIF/NOWWID (008)S § (008)A NOI SNAWII YVYVYVYVYETYODYUYBVYVYVYVYYVYVYVVIVYVYVVUEYYYO 145 OVIGQdS*xNIDVIG = ¢€1)S (STAD@SHNIS)*NIHNIS = OSVIGdS CETHT IA *£OVIGdS@IxXd=Z2xXG) eNISVIT = CIDA (( TeI }¥s *OVIddS=c9OVIGt+IOVIG)4Z°T = NIOVIG (SHNES@¢ (SdX37° 1+SAX3)}%S°)*ST30) *aNIHNIS = e29OVIAd CSHNIS*2xX13G)4°T = NIHNIS (SdX ae US dX 35}%S5° = SHNIS (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! 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TaqQUAaD yoreesoy BuTrAvsouT3uq Teqseop *S*n — 3aodex TeoTuYyDe]) “TTT : “A Z "9/6, ‘1eqUaD YoARPeSey BuTiseuTsuq [eISeoD "stn : °eA faATOATeg 340g — *[*Te Je] **szeyNeITYM “7 Jreqoy ‘yJezZSUeH "e uyor Aq / sajzeuTpiood pewszoJZsuer uT uoTIeTNUTS |81ns wW107¢ ‘ce uyor ‘yjzerqsuemM 4318cn* £9 G9 Ou 1918SN° €0ZOL *(WLOO-0-€L-ZZMOVA 2981IU0D *AteqUeD YOIRaSey ZuTIse8UTSUg qTeaseop *s*n : sataeg “III ‘“soujne qurolf ‘*q yasqoy ‘1ayeITYM “Il 4 “OTATL *1 °6S6, Setoeazy ouedTAInNy °*G “196, ‘SeTAeD eUrOTAANY °F "6961 SeTTTwe9 oueotainy *¢ ‘sweasoid zeqgnduog *Z ‘sadins wi0ojS “| *sourpoTiiny peqoetTes Aq paonput 98ins w1i0qs BuTzepNwts ut pesn St Tepoy] *soqeuTpA00D APaUTTTAAND TeuoZoy_1O BuTsn Tepow adans wiojs [TeoTIeuNU JUapuedep-suUlT, TeUOTSUelITp-omM] eB sassnosTp j10dey *satydeasorTqrq Sepnpouy *uoTJeqUeUNIOp WeaZ0I1g °Z'A = *uoTReDTTdde pue AzoayT *1°A : SINALNOO (¥L00-0-€/-ZZMOVa $ 19389 yoieesey ButrseuTSsuq Teqyseog *S*m — 3OeAUOD) (E-9/ “OU * 19azUaD yorleesey ButaveutTsuq Te3seoD *s*n — JAO0der TeoTuYyDeL) “TTT : *A Z "9/6| ‘2eqUaeD YOTeeSsy BuTIeeuTsuq TeqSeOD “stn 2 ‘ea fatoaTeg 310g — *[*Te Je] °**seHeITYUM *| Jteqoy ‘y7erWZSUeM *c uyor Aq / sazeuTp1009 pellaoysuet}z UT uoTIeTNUTS e3ains wio0ojs *c uyor *yze1QZSUeM iis) eee Gee i. ¢ i hy i i on tr .\ ‘F i i q Me cen ; « wa As | OD tre ee ah a we Mars Pre ie ta1gcn* Le9 SCWYL Poh 4319¢n" €07OL “(VL00-0-E€2-ZZMOVA JOPAUOD *Te}UeD YOTessSoy Sutiveutsuq Teqyseop *s'n : settes *[II ‘Aoyane jurol **q Jaaeqoy SzayeITYM ‘Il “e8TIFL ‘I «4°66 SeFOeAN oUPOTAINH *G “1961 “ETIBD eueOTAINH *y “6961 ‘OTTTwe9 oueoTainy *¢ ‘sueagoid azaqndwop *Z “sedans w104g “| *soueoTiiny peqoetes Aq paonput e8ains w1ojs dgupJeypnuts ut pesn ST Tapoy *SeqeuTpAOOD ARSUTTTAIND TeuoZoyIIO Butsn Tepow e3ans w1IojJS TeOTIoWNU JUspuedep-oWT] [TeUOTSUSUTP-0M}] eB Sassnostp j10dey *satydeaZ0TTqrq sapntouy *uoT}JeqUSUN.Op weAs0Ig *Z°A - ‘*uoTReDTTdde pue Atosayy *|°A : SILNULNOD (¥L00-0-€2-ZZMOVd * 19789 yoreesey BuriseuTsug Teqyseoy *s*m — 20e1RU0D) (€-9/ “ou : 1aqUEeD yorpesey BuTAeeuTsuq Teqseog *S*n — 4JxzOdea TeoTUYyDeL) “TTT = *A Z ‘Q/6| ‘19]UeD YOTeesey BuTIssuTsugq [Te seo) “stn : cea fatoaTeg qzog — *[*Te Jo) **eHeITYM °| Jteqoy ‘y,eaqsuem *p uyor Aq / seqeUTpOOD pewTossuer UT UOT}eETNUTS asins w10}S "rc uyor ‘yjzeaqsueM a31g¢6n° L709 G9) OU) 2318¢N° €07OL *(WL00-O-€L-ZLMOVA JORAIUOD *Ja}UeD YOTPESey ZuTrseUTZUy TeqseoD °S'n : setzes “III ‘sz0y;ne qutof ‘**q 3azeqoy ‘zexeITYM ‘Il ‘eTIFL ‘I 4°66, ‘etoeap aueotTaanyH *G *1 961 “BTID eueOTAANH *y “6961 ‘@TTTwe9 ourotaang *¢ ‘swea80aid zaqndwopg *Z ‘*se8ins wi03g *| *seueoTiany peqoetes Aq peonpuy a3ins wiojs BuTReTNUTS UT pesn St [epoW *saqeuTpiood AesUT[TAAND TeuoZ0YyI10 B3utTsn Tepow esans w1i0}S TeOTiownu jJuepuedep-owT} [TeUOTSUeUTTp-OmM] eB SessnostTp j1odey *satydeaZotTqrq sapntouy *uoTqzequounsop wei80ig *z°A = *uotjeoTTdde pue Atosyy *|°A : SINALNOO (¥L00-0-EL-ZZMOVG + 19837UeD yoiresey B8uTzeeuTZug TeqseoD *S*n — 39B13U0D) (E-9/ “ou $ AaqUaD yoreesoy SuTleeuT3uq TeqseoD *S*n — 3aoder TeoTUYydDeL) “TTF : °A Z “9/6| ‘19}UeD YDIeeseYy BuTseeuTZuq TeIseoD ‘"s°n : °eA faATOATOg Jaog — *[‘Te Jo] **TeHeITYM °Y Jseqoy ‘yIesjsueM *p uyor Aq / sezeUTpiIooOD pewaojsueiz UT UOTIeTNWTS adins w107g *r uyor *yjzerqzsuemM 4318S" £9 Srey, ~Ow A918SN" £0201 “(WLOO-O-€2-ZLMOVA 9PARUOD *1Aa}UeD YOTessey BuTirs9UTsuq qTeaseop *S'n : setzeg “JIT ‘aoyane jutof **q Jaeqoy *‘r99eITUM ‘Il ‘OTITL *1 ‘°6S6, ‘etoezy9 aueotTaany *G *196| ‘eTIeD euedTAany °y “6961 SOTTEweg ourotiany *¢ ‘sweagoid zaqndwop *Z ‘*sadans wi0qg *| *souroTiiny peqovetes Aq paonpuyt a8ins wi0qs BuTzepNuts ut posn ST Tapoy] *SazeuTpIOOD APaUTTTAAND [TeuosoyjAO Butsn Tepow adins wiojs TeOTAownu JUuepuedep-owT] [TeuOTSUeUTp-0mM} eB sassnostTp j10day *satTydeasorTTqtrq SapnyTouy ‘uoTqequounos0p wea8org *Z°A - ‘uoT}eOTTdde pue Artoayy *1°A : SLNALNOO (YL00-0-€L-ZLMOVG * 19} UeD yoteesey BurrseuTZug TeqIseoD “*S*n — FOeAIUOD) (E=-9/ “OU $ 12aqzUaD yoreesey BuTaseuTZuq Teyseog *S°*N — JAOder TeoTuYyDeL) “TTF ? “A Z "Q/6| ‘1eqUe9g YOIReSey BuTivseuTsuq [Te Seo) ‘eA SatToATOg q40g — *[°Te Je] °° *seyWeITYM “| Jteqoy ‘yIerZSUeM *e uyor Aq / sazeUuTp1I00D pawxOFsueirZ UT UOTIETNUTS |B34ns W105 *c uyor *yze1qSsueM TA1gsn* 19 EOL “Ou 4318Sn" €02OL “(VLOO-O-EL-ZLMOVG 29P1IUOD *Je}UeD YDIeESey SuTiseUuTsUg qTeaseop *s*n : setteg “III ‘1zoy;ne jutof ‘*q jrzaqoy ‘19y9eI TUM ‘II “STIFL ‘I °666, SeToeaAD SueOTAINH *G *1 961 “eTAPD eUPOTAANH “y “696, ‘eTTTwe9 oueotTainy *g¢ ‘sweadsozd aeqnduopj *Z ‘sedans w107S *| *soueoTaany peqoetes Aq peonput aedans wi0ojs BurjepnutTs ut pesn ST [apo *sozeUTpA0OD AeaUT[TAAND TeuoSoyjio ZuTsn Tapow a3ins wi0js TedTiewnu juepuedep-owT} TeUOTSUsUWTp-om] eB sassnosTp j10dey *sotydeaZotTqrq sepnjTouy *uoTqequeunosop weaZ01g *Z'aA = *uoT}JeOTTdde pue Artoayy *|°A : SINALNOO (pL00-0-€2-ZLMOVa *$ 293UeD yoieesey BSuTarseuTSug TeqseoD *S*N — 30R1RU0D) (E-9/ *oU * 1aqUaD yoreesey ZutTiseutT3uq TeqIseOD °S*n — JAOdeAr TeoTuYyIeL) *TTF : “A Z “9/6| ‘19qUeD YOTeeSey BuTrseuTsugq TeISeOD ‘ea fATOATEg 340g — *['Te Je]***rTeReITYUM *| Jreqoy SYyIezAWSUeM *p uyor Aq / saqeuTpioos pewizozsuez} ut uoTIeTNUTS a8ins W107 “rf uyor fyjzerzZSUueM NL eee ee —_—— i : i ee