CS5.W. NGV-4\ NOAA TR NOS 41 A UNITED STATES DEPARTMENT OF COMMERCE PUBLICATION #«°'c+. •%rts o' ' NOAA Technical Report NOS 41 U.S. DEPARTMENT OF COMMERCE National Oceanic and Atmospheric Administration National Ocean Survey A User's Guide to a Computer Program for Harmonic Analysis of Data at Tidal Frequencies R. E. DENNIS AND E. E. LONG ROCKVILLE, MO. July 1971 NOAA TECHNICAL REPORTS National Ocean Survey Series The National Ocean Survey (NOS) provides charts and related information for the safe navigation of marine and air commerce. The Survey also furnishes other earth science data — from geodetic, hydrographic, oceanographic, geomagnetic, seismologic, gravimetric, and astronomic surveys, observations, investigations, and measurements — to protect life and property and to meet the needs of engineering, scientific, defense, commercial, and industrial interests. Because many of these reports deal with new practices and techniques, the views expressed are those of the authors, and do not necessarily represent final Survey policy. NOS series of NOAA Technical Reports is a contin- uation of, and retains the consecutive numbering sequence of, the former series, ESSA Technical Report Coast and Geodetic Survey (C&GS), and the earlier series, C&GS Technical Bulletin. Reports in the series are available through the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402. Those publications marked' by an asterisk are out of print. COAST AND GEODETIC SURVEY TECHNICAL BULLETINS *C&GS 1. Aerotriangulation Adjustment of Instrument Data by Computational Methods. William D. Harris, January 1958 (Superseded by No. 23.) *C&GS 2. Tellurometer Traverse Surveys. Lt. Hal P. Demuth, March 1958. *C&GS 3. Recent Increases in Coastal Water Temperature and Sea Level — California to Alaska. H. B. Stewart, Jr., B. D. Zetler, and C. B. Taylor, May 1958. *C&GS 4. Radio Telemetry Applied to Survey Problems. Richard R. Ross, February 1959. *C&GS 5. Raydist on Georges Bank. Capt. Gilbert R. Fish, April 1959. C&GS 6. The Tsunami of March 9, 1957, as Recorded at Tide Stations. Garrett G. Salsman, July 1959. Price §0.25. *C&GS 7. Pantograph Adjustment. G. C. Tewinkel, July 1959. *C&GS 8. Mathematical Basis of Analytic Aerotriangulation. G. C. Tewinkel, August 1959 (Superseded by No. 21.) *C&GS 9. Gravity Measurement Operations in the Field. Lt. Comdr. Hal P. Demuth, September 1959. *C&GS 10. Vertical Adjustment of Instrument Aerotriangulation by Computational Methods. William B. Harris, September 1959 (Superseded by No. 23.) *C&GS 11. Use of Near-Earth Satellite Orbits for Geodetic Information. Paul D. Thomas, January 1960. *C&GS 12. Use of Artificial Satellites for Navigation and Oceanographic Surveys. Paul D. Thomas, July 1960. C&GS 13. A Singular Geodetic Survey. Lansing G. Simmons, September 1960. Price $0.15. *C&GS 14. Film Distortion Compensation for Photogrammetric Use. G. C. Tewinkel, September 1960. *C&GS 15. Transformation of Rectangular Space Coordinates. Erwin Schmid, December 1900. *C&GS 16. Erosion and Sedimentation — Eastern Chesapeake Bay at the Choptank River. G. F. Jordan, January 1961. *C&GS 17. On the Time Interval Between Two Consecutive Earthquakes. Tokuji Utsu, February 1961. Price 80.10. C&GS 18. Submarine Physiography of the U.S. Continental Margins. G. F. Jordan, March 1962. Price 80.20. *C&GS 19. Analytic Absolute Orientation in Photogrammetry. G. C. Tewinkel, March 1962. *C&GS 20. The Earth as Viewed from a Satellite. Erwin Schmid, April 1962. *C&GS 21. Analytic Aerotriangulation. W. D. Harris, G. C. Tewinkel, C. A. Whitten, July 1962 (Corrected July 1963) *C&GS 22. Tidal Current Surveys by Photogrammetric Methods. Morton Keller, October 1963. *C&GS 23. Aerotriangulation Strip Adjustment. M. Keller and G. C. Tewinkel, August 1964. *C&GS 24. Satellite Triangulation in the Coast and Geodetic Survey, February 1965. C&GS 25. Aerotriangulation: Image Coordinate Refinement. M. Keller and G. C. Tewinkel, March 1965. Price $0.15. C&GS 26. Instrumented Telemetering Deep Sea Buoys. H. W. Straub, J. M. Arthaber, A. L. Copeland, and D. T: Theodore, June 1965. Price $0.25. *C&GS 27. Survey of the Boundary Between Arizona and California. Lansing G. Simmons, August 1965. *C&GS 28. Marine Geology of the Northeastern Gulf of Maine. R. J. Malloy and R. N. Harbison, February 1966. C&GS 29. Three-Photo Aerotriangulation. M. Keller and G. C. Tewinkel, February 1966. Price $0.35. (Continued on inside back cover) ATMOSP/, r/ W£NT Of U.S. DEPARTMENT OF COMMERCE Maurice H. Stans, Secretary NATIONAL OCEANIC AND ATMOSPHERIC ADMINISTRATION Robert M. White, Administrator NATIONAL OCEAN SURVEY Don A. Jones, Director NOAA Technical Report NOS 41 A User's Guide to a Computer Program for Harmonic Analysis of Data at Tidal Frequencies R. E. DENNIS AND E. E. LONG a J - a (!) Ok OFFICE OF MARINE SURVEYS AND MAPS OCEANOGRAPHIC DIVISION ROCKVILLE, MD. JULY 1971 UDC 525.63:517.512:681.3.065.4 517 Mathematics .512 Harmonic analysis 525.6 Tides .63 Tidal variations 681.3 Computers .06 Computer programs .065.4 Instructions for stored programs For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402 — Price 40 cents II Contents Page ABSTRACT 1 BASIC HARMONIC EQUATION AND DETERMINATION OF TIDAL CONSTANTS FROM OBSERVATIONAL DATA 1 Astronomical Adjustments 3 Calculation of Orbital Functions 3 Computation of the Equilibrium Arguments 3 Computation of Node Factors 3 Inference of R', /, £' for Secondary Constituents 6 Elimination of Perturbations From Secondary Constituents 6 Harmonic Constants 6 USER'S GUIDE TO PROGRAM 7 Explanation of Control Cards 7 Master Job Control Card 7 Identification Cards 8 Program Control Cards 8 Astronomical Parameter Card 8 Format Specification Card 9 Correction Control Card 9 Explanation of Subroutines 9 Subroutine FORAN 9 Subroutine SASTR 9 Subroutine ASTRO 10 Subroutine ORBIT 10 Subroutine TERPO 10 Subroutine SORT 10 Subroutine EDAT 10 Subroutine XMEAN 10 Subroutine DAYXX 1 Subroutine TEMP 1 Subroutine FITAN 1 Subroutine ONEPI 1 Subroutine TWOPI 1 Subroutine AZIM 1 Subroutine DETAZ 1 Subroutine TILT 1 REFERENCES 1 PROGRAM FOR HARMONIC ANALYSIS AT TIDAL FREQUENCIES 13 III CONTENTS-Continued TABLES Page 1. Formulas for Computation of Orbital Elements 4 2. Formulas for Secondary Orbital Functions 5 3. Formulas for Calculation of Node Factor Reciprocals 5 4. Formulas for Inference of Secondary Constituents 6 FIGURES 1. Order of Control Cards 7 IV A User's Guide to a Computer Program for Harmonic Analysis of Data at Tidal Frequencies R. E. DENNIS AND E. E. LONG Oceanographic Division, Office of Marine Surveys and Maps ABSTRACT. This report describes a FORTRAN-IV program for the harmonic analysis of a series of 15 or 29 days of uniformly spaced tidal data (observations). The program is usable with minor modifications on any computer with a 140K mem- ory which accepts FORTRAN input. The mathematical basis and equations for the determina tion of tidal constants from observational data are given. This report should be used with Manual of Harmonic Analysis and Prediction of Tides, C&GS Special Publication No. 98, Revised (1940) Edition, published in 1941, for reference to for- mulas and tables. It was recognized in ancient time that tides follow a relatively regular cycle, recurring periodically. Since any periodic motion or oscillation can be re- solved into components consisting of simple har- monic motions, a method for reducing tidal motions to these components was inevitable. Lord Kelvin devised such a method about 1867. His method followed the suggestions and speculations made by Laplace, Young, and Airy during the earlier part of the 19th century, but full credit belongs to Kelvin for making such analysis practical. In the late 19th century and early 20th century, William Ferrel and Rollin A. Harris of the U.S. Coast and Geodetic Survey both made important contributions to har- monic analysis of tides and interpretation of the results. In 1885, Leland P. Shidy of the U.S. Coast and Geodetic Survey designed a set of stencils for the "Standard Method" of harmonic analysis of obser- vational data (Report of the U.S. Coast and Geo- detic Survey, 1893, Vol. 1, p. 108). A stencil with apertures was fitted over tabulations of observed data to obtain sums for computing the harmonic constants of the tidal constituents. Only after this — a process involving computation of values for about 20 coefficients for use in the tide equations — could the tide prediction begin. Operational har- monic analysis was continued by hand until the presently used computer system was implemented in 1965. The original version of this program for the standard harmonic analysis of tidal data was com- pleted in 1965. Since then, improvements in the method and refinements in programing have yielded a program that can analyze the observational data for a station in as little as 1.5 seconds on a CDC- 6600 computer. BASIC HARMONIC EQUATION AND DETERMINATION OF TIDAL CONSTANTS FROM OBSERVATIONAL DATA For analyzing equally spaced short-period data (15 days or 29 days), this program utilizes the standard Fourier analysis and traditional methods of the former Coast and Geodetic Survey described by Schureman (1941) with extracts from Doodson (1924) and Harris (1897) reports on tidal analysis. Harmonic analysis of tidal data entails three basic processes: 1. Initial separation of the tidal constituents from the data 2. Orientation of the constituent tides with the astronomical elements 3. Elimination from each constituent sought of effects of other tidal constituents The equation h = Ho +^2f n H n cos [a n t — (*„ — [V + u] n )] n (1) describes the height of tide at any time for which Ho — mean value of tide for observations 1 N ~ l Ho — -z} 2-i hi N i=0 t — time reckoned from some arbitrary point /„ — node factor of constituent H n — mean amplitude of the constituent a n — speed of constituent K n — epoch of the constituent at t = for period of observation (Fo+u)„ — value of argument of constituent at t = for period of observation n — any particular constituent being computed iV — total number of data values used in analysis The factor H n is the mean amplitude of constituent, n, for an entire nodal period, whereas the adjust- ment using the node factor /„, denoted ., (N — 1)5 where N8 = 2w provided that J^-n Jn **-n (2) is the amplitude pertaining to a particular time. The amplitudes R n are derived from observational data. Although the constant K n is used ultimately to characterize the constituent, the quantity k„ — {V -\-u) n may be designated $ n = Kn — (V -\-u) n (3) where f„ is derived from the data. £"„ represents the phase of the constituent at t = for a particular series of observations. To use the tidal equation in harmonic form we follow Schureman's (1941) discussion of the Fourier series technique. The curve described by h = H + Ci cos 0+ d cos 20 + . . . C k cos &0 + Si sin 0+ S 2 sin 20 + . . . S z sin Z0 (4) will pass through the N coordinates given by ordi- nates ho, hi, hi, hz, . . ., Ajv-i and abscissas 0, 5, 25, 2 y^ 1 C p = -r7 2-i hi cos ipd J\ ;=o and N— 1 s D = Tf /Li hi sin iph (5) (6) Since the limits of k and I are N for N even lim k = lim / - 1 AT-1 for TV odd, lim k = lim I = z N the limit of p is correspondingly always less than — . When represents the constituent day, Ci and Si corresponding to p = 1 represent the diurnal con- stituents, and Ci, Si corresponding to p = 2 repre- sent the semidiurnal constituents. For tidal work values of p greater than 8 are seldom used, so the limit of N/2 has no significance. Using the notation a = speed of constituent sought in degrees per solar hour 5 = length of sampling period in solar hours the coefficient equations may be written more ex- actly for real data: C * = !f§ hiCos[(i-l)[^a8p)\ (7) (8) From the Fourier coefficients in equations (7) and (8) desired harmonic constants are defined: r^tan-d) R'(A) = (S P 2 + Cp 2 ) 1 ' 2 (9) (10) The quantity $'{A) is the phase at t = 0, and R'{A) the amplitude for the constituent during the par- ticular series of observations. Both symbols are primed, designating that the effects of other con- stituents have not been removed. For each constit- uent the quantity p in the coefficient equations equals the coefficient of the hour angle of the mean sun, T, in the equilibrium argument of that con- stituent. Subroutine FORAN in the program does the entire Fourier summation for each desired constituent. R' and f' are determined from the original series for five major constituents M 2 , S 2 , N 2 , Oi, and Ki and the harmonics M 4 , M 6 , Ms, S 4 , and S 6 . In the case of a 15-day series of observations, the N 2 constit- uent is approximated from the constants for M 2 . Astronomical Adjustments Modifications of R' and f ' must be made to reduce these constants from the particular time of obser- vation to mean values for an entire nodal period. The epochs of each constituent (F + u) n must be computed to adjust f '„ to k n and node factors f n determined to resolve R' n to H' n for the entire nodal period. Calculation of Orbital Functions Prior to either adjustment, certain orbital elements and functions of these elements must be calculated. These are: Primary Orbital Elements s — mean longitude of the moon p — mean longitude of lunar perigee h — mean longitude of the sun pi — mean longitude of solar perigee N — longitude of ascending lunar node i — inclination of moon's orbit to plane of ecliptic co — obliquity of the ecliptic Secondary Orbital Functions I — obliquity of the moon's orbit v — longitude in the celestial equator (right as- cension) of the moon's intersection with the equator £ — longitude in the moon's orbit of the moon's intersection with the celestial equator v' — a function of the moon's orbit (this term appears in the argument of lunisolar con- stituent Ki) 1v" — another function of the moon's orbit (this term appears in the argument of lunisolar constituent K 2 ) P — mean longitude of lunar perigee reckoned from lunar intersection in the celestial equator Each of the orbital elements must be calculated for the beginning of the series before the epochs can be computed. The subroutines ASTRO and ORBIT compute these elements. Formulas for their determi- nation used in the program and found in table 1 are compiled from Newcomb (1912) and Doodson (1924). Detailed discussion of these elements can be found in these references and in other works on celestial mechanics. Computation of the secondary orvital functions are based on the assumption i and co are constant. This assumption may be reliable for as much as 4 centuries prior to or after the year 1900, as co is de- fined by the function co = 23.45229444° — 0.0130125°T — 0.000001638°r 2 and varies only 0.1 degree in 4 centuries. (T defined in table 1.) Formulas for the secondary orbital func- tions are given in table 2, based on Baird (1886) and Schureman (1924). Computation of the Equilibrium Arguments Equilibrium arguments ( V + u) are combinations of the lunar and solar elements in the tidal equations derived from lunar and solar motion and are de- noted by the symbol E(A). They serve to identify the constituent and determine its speed and period, fixing times of extrema for the corresponding tidal force. When referring the argument E to the time at the beginning of a particular series, the argument becomes Vq + u. The basic equations for computing the equilibrium arguments are taken from Schure- man (1924). Equilibrium arguments for (MK)j, (2MK) 3 , (MN) 4 , (MS) 4 , (2SM) 2 , Mf, MSf, Mm, Sa, and Ssa are omitted because they have negligible effect upon the maxima and minima for a short - period analysis. M3 and Si are also omitted because their effects cannot be determined from a short period analysis. Computation of Node Factors The ratio of the true obliquity of the moon's orbit for any particular value of J to the mean value of the obliquity over an entire nodal period is called the node factor. This factor applied to the mean constituent amplitude for a nodal period yields the amplitude for the constituent at that particular value of /. Conversely, the reciprocal of the node factor applied to the tidal coefficients determined from a particular series reduces them to a mean value for the constituent for all positions of the lunar node. Equations used for computation of the node factors were extracted from Harris (1897) and Schureman (1941) and are shown in table 3. Table 1. — Formulas for computation of orbital elements Formulas for adjustment of orbital elements to beginning of century: P h = 270.43742222° + 307.892°T + 0.002525°T 2 + 0.00000189°T 3 + s z = 334.32801944° + 109.0322055°T - 0.01034444°T 2 — 0.0000125°T 3 + p . = 279.69667778° + 0.768925°T + 0.0003025°T 2 + h z = 281.22083333° + 1.719175°T + 0.00045278°T 2 + 0.00000333°T 3 + p lo z N' = 259.18253333° - 134.1423972°T + 0.002 10556° T 2 + 0.00000222° + T 3 + NoZ Formulas for adjustment of orbital elements to beginning of observations: [For beginning of the series] s = s' + 129.38482032 [Y-C] + 13.176396768 [D s + X] + 0.549016532 [grbs] p = p > + 40.66246584 [Y-C] + 0.111404016 [D s + X] + 0.004641834 [grbs] h = h' - 0.238724988 [Y-C] + 0.9856473288 [D s + X] + 0.0410686387 [grbs] pi = pi + 0.1717836 [Y-C] + 0.000047064 [D s +X] + 0.000001961 [grbs] [For middle of series] p = p > + 40.66246584 [Y-C] + 0.111404016 [D m + X] + 0.004641834 [grms] N = N' + 19.328185764 [Y-C] - 0.0529539336 [D m +X] - 0.0022064139 [grms] s , Po, h , pu, N — Speeds of the elements (degrees/solar day) s', p' , h', pi, N' — Values of the elements at the beginning of any century s, p, h, pi, N — Values of the elements at any given time T — Nearest whole number of Julian centuries (36525 days,) from Greenwich Mean Noon, December 31, 1899, on the Gregorian Calendar to the beginning of the century for observations Y — The exact year in which observations are recorded C — The century during which observations are recorded z — A factor representing the time difference (in solar days) formed between the Gregorian calendar date and the time T as calculated in Julian centuries D a — The day of the year during which observations begin D m — The day of the year upon which the midpoint of the series falls X — Correction for leap years X = 0.25 [F— (c+1)] truncated to no decimals grbs — Greenwich hour at the beginning of the series grms — Greenwich hour at the middle of the series Table 2. — Formulas for secondary orbital functions I = cos- 1 [0.9136949 - 0.0356926 cos N]* v = sin- 1 [0.0897056 sin iV/sin I]* _j I" (0.206727 sin N) (1 - 0.0194926 cos N) >* * ~ tan Lo.9979852 + 0.206727 cos N — 0.0020148 cos 2NJ . T • I , 0.334766^1** " = tan L sin ' + V cos " + ~^2r) J i T ■ o ( o , 0.0726184 M** 2j>" = tan -1 sin 2^ -J- I cos 2^ + . ) P = [p (for middle of series) — £]* u of Z, 2 = 2 ({ - v) - R ± 180°* where R = tan" 1 [sin 2P / { (cos 2 0.5/ / 6 sin 2 0.57) — cos 2P} ] uofMi = U-v) +(? + 90°* where Q = tan" 1 [ { (5 cos I — 1) / (7 cos I + 1) } tan P] *Schureman (1924) **Baird (1886) Table 3. — Formulas for calculation of node factor reciprocals F(A)-/(A) = 1.000 /(A) = node factor of constituent A. F(M 2 ) = (0.91544) ^- (cos 4 0.5 I") F(S 2 ) = F(R 2 ) = F(T 2 ) = F(P0 = F(S 4 ) = F(S«) = 1.000 F(Oi) = (0.37988) -f- (sin 7 cos 2 0.5 I) F(Ki) = (0.8965 sin 2 2/ + 0.6001 sin 21 cos k + 0.1006)" 1/2 F(K 2 ) = (19.0444 sin 4 7 + 2.7702 sin 2 7 cos 2v + 0.0981)" 1/2 F(L 2 ) = F(M 2 ) X [1 - (12 sin 2 0.5 I cos 2P) + (cos 2 0.5 I) + (36 sin 4 0.5 I) -i- (cos 4 0.5 J)]~ 1/2 F(J0 = (0.72137) -f- sin 21 F(Mi) = F(Oi) X (2.310 + 1.435 cos 2P)~ 1/2 F(OO) = (0.016358) -=- (sin J sin 2 0.5 I) F(N 2 ) = F(2N) = F(X 2 ) - Ffx 2 ) = F(v 2 ) = F(M 2 ) F(Q0 = F(2Q) = F(P0 = F(O0 F(M 4 ) = F 2 (M 2 ) F(M«) = F 3 (M 2 ) F(Mg) = F 4 (M 2 ) Inference of R', k', f' for Secondary Constituents Following the discussion of Schureman, the re- lations existing between observational constants of similar constituents are assumed the same as those existing between the corresponding theoretical values. Since values for only 10 constituents are sought (M 2 , S 2 , N 2 , Oi, Ki, M 4 , M 6 , M 8 , S 4 , S 6 ), the other constituents, although minor in effect, must be approximate before the process of elimination can be applied to the major constituents. In the program this is done using the formulas given by Schureman (1941). Elimination of Perturbations From Secondary Constituents The disturbing effects upon S 2 by K 2 and T 2 and on Kx by Pi may be considerable in a short series owing to the small differences in frequencies of the respective constituents. To correct for this, account is taken of the phase displacement and augmenta- tion of amplitude in the major constituents S 2 and Ki by the respective minor constituent disturb- ances. This process is described by Schureman (1941) and performed by the computer using the imbedded data from tables 21-26 from the Manual of Harmonic Analysis and Prediction of Tides. The final process of elimination frees the constit- uents from the disturbing effects of the other con- stituents. The constituents sought are M 2 , N 2 , S 2 , Oi, Ki. Only constituents with the same coefficient for the factor T in their equilibrium argument are considered interfering, that is, effects of diurnal upon semidiurnal and vice versa are negligible. Complete discussion of constituent interference is found in Schureman (1941). The program com- pletes the elimination process, using tables prepared from equations 389, 390 from Schureman (1941, table 29). Harmonic Constants Inference of secondary constituents from primary constituents utilizes the ratios derived by Schure- man (1941) and shown in table 4. Table 4. — Formulas for inference of secondary constituents Diurnal constituents //(JO = 0.079 tf(Oi); *(Ji) = k(Kx) + 0.496 [k(Kx) - *(Oi)] #(Mi) = 0.071 jff(Oi); «(Mi) = k(Ki) - 0.500 [ K (Ki) - k(Oi)] #(00) = 0.043 i/(Oi); k(OO) = k(Ki) + 1.000 [*(Ki) - *(Oi)] f/(P x ) = 0.331 //(KO; k(Pi) = k(K x ) - 0.075 [k(Ki) - k(Oi)] #(Qi) = 0.194 #(00; k(Qi) = k(Ki) - 1.496 [«(Ki) - k(Oi)] H(2Q) = 0.026 i/(Oi); k(2Q) = k(K x ) - 1.992 [k(K0 - K (O x )] H( P1 ) - 0.038 //(Ox); «( Pl ) = K (K X ) - 1.429 [k(Ki) - k(Oi)] Semidiurnal constituents ff(K 2 ) = 0.272 H(S f ); k(K 2 ) = K (S 2 ) + 0.081 [/c(S 2 ) - /c(M 2 )] H(U) = 0.028 tf(M 2 ); «(L 2 ) = k(S 2 ) - 0.464 [k(S 2 ) - /c(M 2 )] = 0.143 H(Ni); = «(M 2 ) + 1.000 [k(M 2 ) - /c(N 2 ) ] i/(N 2 ) = 0.194 H(M 2 ); «(N 2 ) = «(S 2 ) - 1.536 [«(S 2 ) - X (M 2 )] H(2N) = 0.026 H(M 2 ); *(2N) = *c(S 2 ) - 2.072 [k(S 2 ) - /c(M 2 )] = 0.133 H(N 2 ); = k(M 2 ) - 2.000 [*(M 2 ) - k(N 2 ) ] i/(R 2 ) = 0.008 ff(S 2 ); /c(R 2 ) = k(S 2 ) + 0.040 [/c(S 2 ) - k(M 2 )} H(T 2 ) = 0.059 H(S 2 ); k(T 2 ) = k(S 2 ) - 0.040 [k(S 2 ) - «(M 2 )] J7(A 2 ) = 0.007 i/(M 2 ); k(X 2 ) = k(S 2 ) - 0.536 [k(S 2 ) - /c(M 2 )] H(n2) = 0.024 H(M 2 ); *0*,) = k(S«) - 2.000 [«(S 2 ) - k(M 2 )] J7(y 2 ) = 0.038 i/(M 2 ); k(v 2 ) = k(S 2 ) - 1.464 [k(S 2 ) - k(M 2 )] = 0.194 H(N 2 ); = k(M 2 ) - 0.866 [«(M 2 ) - K (N 2 ) ] USERS' GUIDE TO PROGRAM The present program is written in FORTRAN- IV, executable with minor adjustments on any compatible machine having a 140K memory and access to arcsine and arccosine systems functions. Computing time is approximately 1.5 seconds on the CDC -6600. Either a vector (polar form) or scalar variable may be analyzed. For vector series, the program allows either a major-minor axis analysis or a north- east component approach. No data series may ex- ceed 7,000 terms without redimensioning in the program, and no series of other than 15 or 29 days of uniformly spaced data can be analyzed. The program accepts input via magnetic tape or punched cards in any format with the restriction that, for vectors with magnitude and direction in the same record, the angles must precede the am- plitudes in the record. For vectors specified by one file of amplitudes and one file of directions, the amplitude file must be read first. Output comprises the mean amplitudes and phases of 26 tidal constituents. Explanation of Control Cards The order of the control cards (fig. 1) input to the machine is of utmost importance. The discussion of each card or set of cards is in the order in which they are read. The master job control card appears only once, whereas all other cards appear for each new data series. Master Job Control Card Immediately following the program deck must be a card containing the quantity NJ. This specifies the number of different jobs to be done. READ 221, NJ 221 FORMAT (15) , Apr 2431 m. ™L 45| LRD LX 39 -51 _64j 60X.F3.0.24X.F4.2) AQ 00001 60001 7257 m MAP GONL -13| CVAR U10X,F3.0> ~N) CORRECTION CARDS CORRECTION CONTROL CARD J FORMAT SPECIFICATION CARD 2 I ASTRONOMICAL PARAMETER CARD INC 61 100004325 NSPH TFAC 76 EAZI si I) (] PROGRAM CONTR h L las LftT. 410141N LONG. 723413U BEGIN 8'25/66-OOOOH END 9/22^66-2350H * 60U STATION NO. 92. LONG ISLAND SOUND ** 1966 *» 29 DAYS «""15 FT" «* Figure 1. — Order of control cards. If data are on cards, data deck(s) follow the format specification card and the correction cards are not applicable. Identification Cards These two cards follow the master job control card and are read by the statement. READ 808, IDENS 808 FORMAT (9A8/9A8) The purpose of these cards is to introduce any de- sired alphanumeric information to be used as iden- tifications. Contents of these two cards will be printed verbatim immediately preceding final results. Only columns 1-72 of each card will be printed. Program Control Card The program card specifies the operations to be performed on the data series, the mode of input for data series, and any adjustments which are to be made to the data. The variables are read by the statement READ 76, AZI, LSORT, JJ, N, MAP, K, CVAR, INC, NSPH, TFAC, EAZI 76 FORMAT (F5.0, 515, F5.0, 215, F10.8, F5.0) The variables listed have the following significance: AZI — Approximate mean flood direction (F5.0) LSORT — Specifies when a sort of the input data is required (15) LSORT = 0. No sort LSORT = 1. Sort to be performed JJ — Indicates the type of sort desired (15) (See subroutine SORT) N — The number of data points in the series (15) MAP — Controls input mechanism and format. MAP is determined partially by whether the data are vector or scalar. (15) MAP = 1. Scalar quantity, card input with format AQ. (See format specification card) MAP = 2. Magnetic tape input for vector quantity with format AQ. Angles must precede magnitudes in the record MAP = 3. Card input of vector data with magnitudes of format AQ and direc- tions of format RQ. MAP = 4. Tape input of vector data accord- ing to format AQ and tilt corrections from tape (if available) according to format BQ. Use MAP = 4 only for making corrections to data series. (See subroutine EDAT) K — Indicates whether analysis by components is desired (15) K = 1. Major axis analysis only (use K = 1 for scalar) K = 2. Minor axis analysis only K = 3. Major and minor axis analysis K = 4. North and east component analysis. In this case AZI = 0.00 CVAR — Compass variation of the vicinity. CVAR is negative when West and positive when EAST. (F5.0) INC — Indicate if tilt corrections are to be applied. (See subroutine TILT) (15) INC = 0. No tilt corrections INC = 1. Tilt corrections to be applied NSPH — Number of samples of data per mean solar hour (15) TFAC — Time correction factor for a slow or fast chronometer in meters. (F10.8) For no time correction TFAC = 1.000000. To determine TFAC use the expression —,„ . „ _ True time covered by observations Time reckoned by the meter clock CAUTION: If NSPH or TFAC is zero, the program will terminate immediately. EAZI — approximate ebb azimuth (F5.0) Astronomical Parameter Card The card specified in the statement READ 2038, XYER, MONTH, DAY, STT, TM, GONL 2038 FORMAT (F5.0, 15, F5.0, F5.2, 2F6.2) inputs information which is pertinent to the partic- ular times of the series of observations. XYER — Year in which the series is observed (F5.0) MONTH— Month during which series starts (15) DAY — Day on which series starts (F5.0) STT — Time in decimal hours at which series starts (F5.2) TM— Time meridian to which station is referred (F6.2) GONL— Longitude of the station (F6.2) CAUTION: The convention of west longitude being positive and east longitude being negative is fol. lowed in entering the parameters TM and GONL- Format Specification Card This card allows variability in the format of data on magnetic tape or cards. This card is read by the statement READ 2039, AQ, BQ 2039 FORMAT (4A8, 3A8) The maximum field length of AQ is 32 characters and of BQ is 24 characters. The format specification BQ must always begin in column 33 of this card. Parenthesis must be punched and counted as part of the field length. Limitations are according to the variable MAP: MAP = 1. Assign AQ a card format; do not assign BQ. MAP = 2. Assign AQ a tape format; do not assign BQ. Angle of vector must precede the magnitude in the record. MAP = 3. Assign AQ and BQ formats for card reading. AQ must be magnitudes of the vectors and BQ must be directions. MAP = 4. AQ and BQ must be as follows: AQ = (60X, F3.0, 24X, F4.2) BQ = (110X, F3.0) This is necessary for tilt corrections. Correction Control Card If data are on cards, data decks immediately fol- low the format specification card. For data read from magnetic tape, corrections can be made using the correction control card. READ 222, MX, LRD, LX 222 FORMAT (315) MX — Total number of records which will be read to attain desired series MX = N + LRD LRD — Number of records at beginning of tape to be skipped LX — Number of corrections to be read from cards and substituted in the series for those values found on the magnetic tape Correction cards follow the correction control card and contain the interval number of the point to be changed and the desired angle and magnitude of the vector at that point. The statement READ 555, (ICOR (LY), CD (LY), CV (LY), LY = 1, LX) 555 FORMAT (15, F10.0, F10.2) reads these parameters. Explanation of Subroutines The subroutines called in program CURAN can be grouped into three categories. The mathematical anal- ysis and astronomical adjustments are performed in four basic subroutines— FOR AN, ORBIT, ASTRO, SASTR. The subroutine DAYXX is a utility routine used for adjusting the time of beginning of series to astronomical reference frame (Greenwich Time Me- ridian). Operations peculiar to tidal currents are performed by subroutines AZIM, DETAZ, and TILT. Utility routines which are called or in some cases are optional include SORT, EDAT, MEAN, TERPO, TEMP, FITAN, ONEPI, and TWOPI. Subroutine FOR AN (SER, SP, RES, ZEP, N, M) This subroutine is designed to determine the Fourier coefficients of the data (SER) for the first approximations of phases (ZEP) and amplitudes (RES) of the various constituents. The angular dis- placement per sampling period (SP) of the required constituent is computed in the subroutine according to the expression: SP = (Speed of the constituent in degrees per mean solar hour) (Number of samples per hour X 180) The numbers N and M represent the number of data points and the number of harmonics sought, respec- tively. The subroutine uses equations (289) and (292) from Schureman to obtain the Fourier coefficients. Subroutine SASTR This subroutine calls as input the reciprocals (CF) of the node factors of the various constituents com- puted in the main program according to equations (73)-(80), (141)-(150), (207), (215), and (235). The input also comprises the node factor for K 2 (CKKF), the mean longitude of the sun (SL), the arguments for v', v" (VP, VPP), and the mean longitude of solar perigee (PS). Tables 21-26 from Schureman are em- bedded in the program. The following effects of secondary constituents are computed: ACS — Acceleration in S2 from K 2 and T2 APS — Augmentation of amplitude of S2 from K 2 and T2 ACK — Acceleration in Ki from Pi APK — Augmentation of amplitude of Ki from Pi This subroutine calls subroutine TERPO to inter- polate the appropriate tables. Subroutine ASTRO (XYER, DAYB, DAYM, GRBS, GRMS, JOBX) COMMON/COSTX/CXX(30), OEX(5) This subroutine uses input from subroutine DAYXX (DAYB, DAYM, GRBS, GRMS) with the input value XYER, the year in which observations begin, to compute the orbital elements s, />, h, p u N, I, v, E, /, and 2v " (table 1) relative to the period of observation. The subprogram calls subroutine ORBIT to compute all values for the beginning of the century, then adjusts each to the time of the observed series. The array CXX are the appropriate values of the orbital elements. JOBX is the number of days in the data series. Subroutine ORBIT (XCEN, XSX, SPX, SHX, XP1X, XNX, OEX, T, XYER, NNN) This subroutine uses the value XYER to compute orbital elements (table 1) for the beginning of the century during which observations were taken. The elements are computed according to the expressions given by Newcomb and Schureman displayed in table 1. Values are computed in array OEX and transferred to the variables XSX, XPX, XHX, XP1X, XNX for convenience. XCEN, T, NNN are all internal variables. Subroutine TERPO This subprogram is a utility program designed to interpolate a table of figures in two directions. new array consists of the first value and every JJth value thereafter of the original series. KK is internal and dimensions the new array VX. Subroutine EDAT (MX, VIN, DIN, MA, LX, ICOR, CD, CV, LRD, XTEMP, INC) This routine is available only to MAP = 4. It is designed for cases where polar vectors are input as data from magnetic tape, but either simple substi- tutions or tilt corrections need to be applied to the data series. MX-MA + LRD VIN — Magnitudes (corrected) DIN — Directions (corrected) MA — Total number of values desired in final data series LX — Number of substitutions and inferences to be made within the series or at end of series ICOR — Interval number in the list MX which is to be corrected CD — Angle to be substituted for the value actually on tape CV — Magnitude to be substituted for the value actually on tape LRD — Number of records on tape to be skip- ped XTEMP — Temporary storage for TILT values INC — Integer indicating when tilt corrections are to be applied CAUTION: If a magnitude or direction for a given interval need correcting, both magnitude and direc- tion and interval number must be punched on a card. The program substitutes the entire record. For- mat for correction cards bearing ICOR, CD, CV is (15, F10.0, F10.2). The number of substitutions used may not exceed 300. If an inference of data is required to extend the series to the required length of 15 or 29 days, the magnitudes and directions must be punched on cards with the formats (24F3.2) and (24F3.0), respectively. These cards then follow any substitution cards in the data deck. Subroutine SORT (SORX, VK, KK, JJ) This is designed to sort values from an array (SORX) forming a new array (VX) such that the Subroutine XMEAN This is a utility routine to compute the arithmetic mean of an array. 10 Subroutine DAYXX Subroutine DETAZ Computes the day of the year and the Greenwich hour for the beginning and middle of the data series. Subroutine TEMP Provides temporary storage for any array so that operations can be performed on the array without loss of the original series. Subroutine FITAN This subprogram finds the angle for a given value of the arctangent function. This routine is designed to compute a mean major axis azimuth for a given approximate azimuth which is input on the master job contol card. Subroutine TILT This subroutine is designed for correcting errone- ous tidal current velocities for instrumental tilt. It should be applied only to instruments having Sa- vonius rotor current sensors with tilt meters. If the tilt reading exceeds 35 degrees, a manual inference will be required. Subroutine ONEPI This subroutine is designed to prevent the differ- ence between two angles from exceeding 180 degrees or becoming negative. Subroutine TWOPI This subroutine places a given angle in the proper quadrant. Subroutine AZIM (DX, VR, A, VN, K, J, COMPV) This subprogram is designed to compute the vector components along major and minor axes or to deter- mine north and east components. DX, VR — Angles, magnitudes of popular vector A — Azimuth (A = 0. North and east com- ponents) VN — New component series which is formed K — Indicates type of component formed K = 1. Major axis; north component K = 2. Minor axis; east component J — Number of data values in series COMPV — Magnetic variation in directions if the directions are not true readings REFERENCES Baird, A. W., A Manual for Tidal Observations and their Reduction by the Method of Harmonic Anal- ysis, Taylor and Francis, London, 1886. Doodson, A. T., "The Analysis of Tidal Observa- tions," Philosophical Transactions of the Royal Society, Vol. 227(A), pp. 223-279, 1924. Harris, R. A., "Manual of Tides," Appendix No. 8 to 1897 Annual Report of U.S. Coast and Geodetic Survey, U.S. Department of Commerce, Wash- ington, D.C., 1898. Newcomb, Simon, Astronomical Papers for the Amer- ican Ephemeris, Vol. IX, Part I, 1912. Schureman, P. W., Manual of Harmonic Analysis and Prediction of Tides, C&GS Special Publication No. 98, Revised (1940) Edition, U.S. Department of Commerce, Washington, D.C., 1941. Schureman, P. W., Manual of Harmonic Analysis and Prediction of Tides, C&GS Special Publication No. 98, Serial No. 244, U.S. Department of Com- merce, Washington, D.C., 1924. 11 PROGRAM FOR HARMONIC ANALYSIS AT TIDAL FREQUENCIES ENVIRONMENTAL SCIENCE SERVICES ADMINISTRATION - C.+ G.S. FOURIER - HARMONIC ANALYSIS PROJECT 13140? 29 DAYS PROGRAM CUR AN ( I NPUT • OUTPUT • TAPF9 ) DIMENSION VIN(9000) ' DIN<9000 ) 'CHSIpt ] 0) 'CRP( 10 ) ' COVU(25 ) 'CF( 25 > ' 1TXXIXU 5'20)' 7XXIX2 ( 5 ' 20 ) ' CK.APAP ( 20 ) ' CKAPI20)' THS I ( 20 ) ' CR ( 20 ) ' 2PELL ( 20 ) ' BI NGO( 20 > ' G( 5 ) ' PAK ( 5 > ' D( 24) ' W( 24 ) ' ZFTA(S)' CRC ( 5 > DIMENSION RES (4) ' ZEP( 4 ) 'XTEMP(9000) • VN(9000) ' ITEMPI9000) DIMENSION CV(300)' CD(300)' ICop(300>' IDBNSU8) DIMENSION CXX (30) ' DEG(40 ) ' TA&1(40)' TAB2I40)' TA63I40)' TAB4(40) DIMENSION TAB5(40)' TAB6 ( 40 ) ' AO (4 ) ' BQ(3>' OFX(^) DIMENSION TXXIXA( 20) ' TXXIXBI 20 > ' TXXIXCt 20) 'TXXIXDI ?0 > 'TXXIXEf 20) DIMENSION TXXiXF(2 0)'TXXIXG(2 0) , TXXIXH(2C) , TXXIXl(7 0) , TxXIXJ(20) COMMON / FOR T / AQ ' RQ/COSTX/ CXX' OEX/CONTT/NN COMMON / LEAPYR / DAY' MONTH SPEFDS OF CONSTITUENTS M ( ? ) ' N ( 2 ) ' S ( 2 ) ' ( 1 ) ' M 1 ) FOR 1 SAMPLE/HP. DATA SXM2'SXS2' SXN2'SXOl' SXK1/0. 16 10228' 0.1666667' 0.1579985 "0.0774 15 1613*0.08 3 5615/ 16 TABLE XXIX FROM SCHUREMAN' S.P. °8 (194]) ACCELERATION IN EPOCH OF K(l) DUE TO PC])' 20 DAYS DATA(TABl(JZ) , J2sl'18)/11.4 , 16.4'18.3 , 17.6 , I5.2 , 11.7 , 7.4 , 2.7'-2.2' 1 1-6.9' 11.3 , -14.9'-17.5 , -18.3'-16.7'-12.1'-4.7'3.9/ 2 RATIO OF INCRFASE IN AMPLITUDE OF K(l> DUE TO P(1)'29DAYS DATA(TAB2 (JZ) , JZsl , 18)/-0.26 , -0.17 , -0.06 , 0.04 , 0.14♦0.23•0.28'0.3 1 , 3 10.32' .29 •0.2?'0.15 I 0.05 '-0.06 '-0.16 ' -0. 25 ' -0. 30 ' -0.31/ 4 ACCELERATION IN EPOCH OF S(2) DUE TO K ( 2 ) ' 29DAYS DATA! TAB3 ( JZ) ' JZsl , 18)/5.9'Q.6 , I2.6'l4.6 , 15.0']- ; >.5*9.6 , ?.7 , -3.0 1 -9 5 l.l , -13.2 , -15.0 l -14.7'-l2.9 , -10.0'-6.4 , -2.3'1.9/ 6 PATIO OF INCREASE IN AMPLITUDE OF S(2> DUE TO K(2)'29DAYS DATA( TAG4( JZ ) , JZsl'18)/0.24 , 0.19 , 0.]2 , 0.04 , -0.05 l -0.14'-0.21' 2*-0 . 7 125' -0.21' -0.15 '-0.06 '0.03* 0.11' 0.18 '0.23 '2*0.26/ 8 ACCELFPATION IN EPOCH OF S(2) DUE TO T<2)* 29 DAYS DATA( TAB5 (JZ) , jZsl , 37)/-0.8 , -l.? , -1.8 , -2.2'-2.6 , -2.9 , -3.2 , 3--3.3 l - 9 ]3.] !, -2.8 , -2«5'-?.0'-'|.5 , -0.9 , -0.3 l 0.3'0.9'1.5'2«0 , 2«4'2.8 , 3.1 , ?*3. 10 23'^.2 , 3.0 , 2.7'2.^"l.'3 , 1.4 , ri.8'n.3'_n.2 , -C.8/ 11 RESULTANT AMPLITUDE IN S(2) DUE To T(2)' 29 DAYS DATA! TAB6( JZ) l jZsl , 37)/i.06 , 2*1.05'1.04 , 2*1.03 , 1.02 , i.01 , 1.00 , 0.99 12 l'0.98 , 0.97 , 0.96'2*0.95 , 4*0.94 l 2*0.95'0.9 6 , 0.97 , 0.98 , 0.99 , l.O0'1.0l 13 2 '1.02' 2*1. 03 '1.04' 2*1. 05' 4*1. 06/ 14 AMPLITUDE EFFECT OF CONSTITUENTS ON Ml?)' 29 DAYS DATA(TXXIXA( J ) ' J s 1 ' 20 ) / . ' . 050 • . 1 8 ' 2*0 . ' . 56 ' .0 50 ' . 049 ' 17 1.021' .060 '0. 096*0.0] 8'0. 096' 7*0.0/ 18 AMPLITUDE EFFECT OF CONSTITUENTS ON N(2>' 29 DAYS DATA(TXXIXB( J ) ' Js 1 ' 20 > / .0 50 ' .00 ' . 00 5 ' 2*0 . ' . 052 ' . 049 ' . 050 ' 19 10. 031 '0.021 '0.018' 0.096' 0.968' 7*0.0/ 2 13 PROGRAM FOR HARMONIC ANALYSIS AT TIDAL FREQUENCIES-Continued AMPLITUDE EFFFCT OF CONSTITUENTS ON S(2>' 29 DAYS DATA( TXXIXC.I J ) ' Js 1 ' ?0 ) /0 .0 1 8 ' .005 ' 3*0 .0 ' . 959 ' .096 ' .017 « 2*0 > 9 21 190'0. 50' 0.'018 '0.042 ' 7*0.0/ 22 AMPLITUDE EFFECT OF CONSTITUENTS ON 0(1)' 29 DAYS DATA(TXXIXD( J ) ' Js 1 ' 20 > /4*0 .0 ' .0 56 ' 8*0 .0 ' . 052 ' .065 ' .0 52 ' .0 1 8 23 1 ' 0.05 ' 0.049 ' 0.096/ 24 AMPLITUDE EFFECT OF CONSTITUENTS ON Kf 1 ) ' 29 DAYS DATA(TXXIXE( J ) ' J s ] ' 20 ) / 3 *0 . ' . 56 ' 9*0 . ' 2*0 . 050 * .0 56 ' . 9 59 ' . 25 152'0.49'O.Ol]/ 26 PHASE EFFECT nF CONSTITUENTS ON Mfp)' 29 DAYS DATA(TXXIXF( J ) ' Js 1 ' 20 ) /0 .0 ' 3 5 1 . ' 1 74 . ' 2*0 . ' 22 . ' 9 . ' 34 1 . ' 8 . ' 1 59 . ' 1 27 164. ' 186. • 196. ' 7*0.0/ 28 PHASE EFFtCT Of CONSTITUENTS ON N(2)' 29 DAYS DATA(TXXIXG( J )' Js 1 ' 20 ) /9 .' . ' 3 .' 2*0 . ' 32 .' 19 .' 35 1 •' 1 7 .' 1 69 .' 1 74 29 1 . ' ] 96. ' 25. ' 7*0.0/ 30 PHASE EFFECT OF CONSTITUENTS ON S<2>'29 DAYS DATA(TXXIXH) J ) ' Js 1 ' 20 ) / 1 86 . ' 357 . ' 3*0 .0 ' 29 . ' 196 . ' 348 . ' 14. ' 346 . ' 3 5 31 11.' 193. '202. ' 7*0.0/ ^2 PHASE EFFECT OF CONSTITUENTS ON 0(1)' 29 DAYS PATA(TXXIXI( J ) ' Jsl ' 20) /4*0.0 ' 22. ' 8*0.0 ' 32. ' 13. '44. ' 174. ' 351 . • 341 33 ] . ' ] 96./ 34 PHASE FFFFCT OF CONSTITUENTS ON K(l)' 29 DAYS DATA(TXXIXJ( J ) ' Jsl ' 20 ) /3*0.0 ' 338. ' 9*0.0 '9. ' 351. ' 22. ' 331. ' 328. ' 31 35 19. '3 54./ 3 6 LOADING OF TABLES INTO PPOPEP ARRAYS DO 80 8 MAXJ s 1 ' 20 TXX 1X1 ( 1 'MAXJ ) s TXXIXA(MAXJ) TXXlXl ( ?'MAXJ ) s TXXIXP(MAXJ) TXXIX1 ( ?'MAXJ ) s TXXIXT(MAXJ) TXXIX1 (4'MAXJ) s TXXIXMMAXJ) TXX IX] ( 5'MAXJ ) s TXXIXF(MAXJ) TXXIX21 1 'MAXJ ) s TXXIXF(MAXJ) TXXIX2 ( 2'MAXJ ) s T XX I XG ( M AX J ) TXXIX2I 3'MAXJ ) s TXXIXH(MAXJ) TXXIX2 (4'MAXJ ) s TXXIXI(MAXJ) 8008 TXXI X2( 5'MAXJ ) s TXXIXJ(MAXJ) CONTROL CARDS ENTERED READ 221 'NJ DO 1111 JN s 1'NJ READ 808' ( IDENSI JK) ' JK s 1'18> READ 76' AZI ' LSORT 1 Jj'N'MAP' K ' CVAR ' INC 'NSPH' TFAC'EAZI RFAD 2038' XYEP 'MONTH' DAY ' STT ' TM ' GONL RFAD 2039' AQ ' BO IF ( NSPH.EQ.O.OP.TFAC.EO.O.O) GO TO 8356 DATA INPUT LOOP BEGINS GO TO ( 101 ' 102 ' 103' 660 • 661 )' MAP 661 READ AQ,' (DIN(KX) 'VIN(KX) 'KX s 1'N) GO TO 104 101 READ AQ 1 ( VINI I ) ' I s 1 'N) GO TO 123 103 PFAD AQ' ( VIN ( I ) ' I s 1'N) IF ( AZI .EO.O.O.AND.K.FQ.O ) GO TO 123 READ BQ' (DIN< I ) • I s 1'NI GO TO 104 123 IF(LSORT - 1) 120'12]'l21 120 CALL TEMPI VIN'XTFMP'N) GO To 108 14 PROGRAM FOR HARMONIC ANALYSIS AT TIDAL FREQUENCIES-Continued 1?1 CALL SORT I VIN'XTFMP'N' JJ ) N s NN GO TO 108 10? READ(9'AQ) (DIN(I ) ' VIN(I)'I s l'N) IF(K.EQ.5) GO TO 120 GO TO 104 660 READ 222'MX ' LPD' LX IF(LX.FO.O) GO TO 666 RFAD 555 ' ( ICOR (LY) 'Cp(LY) 'CV( LY ) ' LY s 1 ' LX ) 666 CALL EDAT(MX' VIN'DIN'N'LX' ICOR'CD-' CV'LRD'XTFMPi INC) 104 NFCOM s 1 IF(K.FQ,3) NECOM s 3 IF(K.FQ.4) NECOM s 4 IF(MECOM.EG.3.0R.NECOM.E0.4) K s 1 IF ( NECOM. EQ. 4 ) GO TO 8155 CALL DETAZI DIN' AZI ' N • CvAR • VIN ' XTEMP' INC) FLX s AZI IFIFAZI. EQ.0.0) GO TO 8355 CALL DETAZI DIN' EAZI 'N'CVAR'VIN'XTEMP'O) FPX s FAZI DIPF s FLX - FBX AT7 s .0 PIPF s APS (DIPF) I F I P I P E . G T . 180.0) ATZ s PIPF - 180.0 IF( PIPE. LT.l 80.0) ATZ s 180.0 - PIPF APZ s ATZ/2.0 ZAP s AINT(ARZ) IF (DIRE. LT. 0.0 ) AZI s ELX + ZAP IF(DIRE.GT.O.O) AZI s FLX - ZAP PRINT 7359' AZ I ' DI R E • AR Z ' Z AP 8355 CALL AZIM ( DIN • VIM ' AZ I ' VM ' K 'N ' CVAR ' NFCOM ) I F ( LSORT - 1 ) 107 ' 106 ' 106 106 r A LL SORT ( VN ' XTfmp' N ' JJ) N s N M GO TO 108 107 CALL TEMP (VN ' XTFMP' M) FOURIER ANALYSIS PERFORMED FOR FIVF MAJOR TIDAL FREQUENCIES 108 XM? s TFAC*SXM2/FLOATf NSPH) XS? s TFAC*SXS2/FLOAT ( MSPH) XN? s TFAC*SXN2/FLOAT (NSPH) XO s TFAC*SX01 /FLOAT ( N Q PH ) XK s TFAOSXSO /FLOAT (N^PH) GO To 8^57 8356 PRINT 7358 CALL FXIT 83^7 CALL FOPAN ( XTEMP ' XM2 ' RFS ' ZEP ' N ' 4 ) CHSIPI 1 ) s ZEP(1 ) CHSIP(6) s ZEP12 ) CHSIPI 7 ) s ZEP( 3) CHSIP(8) s ZFPI-4) CPP ( 1 ) s PES ( 1 ) CRD ( m s RES ( 2 ) CPP(7) s R E S ( 3 ) CPP(8) s PES (4) 110 CALL FORAN(XTFMP ' XN2 ' PES'ZFP' N' 1) CHSIPf 2)s ZFP ( 1 ) CPP ( 2 ) s RFS ( 1 ) 111 CALL FORAN(XTEMP'XS2' RES'ZFP' N' 3) CHSIPI 3 ) s ZFP( 1 ) CHSIP(Q) s ZEP( 2) CHSIPI 10) sZEP( 3) CPP( 1) s RES! 1 ) CRP( 9) s PES( 2 ) CPP (lis RFS ( 3) CALL FORAN( XTEMP'XO" PFS' ZFP' M' 1) CHSIP(4) s ZFP(1 ) CPP (4) s RES( 1 ) CALL FORANtXTFMp ' XK' PES' ZFP' N' 1) CHS IP! 5 ) s ZEPI 1 ) 15 PROGRAM FOR HARMONIC ANALYSIS AT TIDAL FREQUENCIES -Continued CRP(5) s RES(l) CALL TWOPI (CHSIP' 10) IF(NEC0M.EQ.3.0P.NECOM.E0.4) GO TO 8057 R05R IF(K.FQ.2) GO TO 8059 GO TO 8058 DETERMINATION OF VO+ U FOLLOWS WITH LOGF AND ARGUMENTS R057 CALL DAYXXtMONTH' DAY'STT" TM' DAyB'DAyM' GRES'GPMS'CP DO 7135 JAN s 1'19 LXN s JAN + 18 TAP! ,'LXN) s TAB! (JAN) TAR2(LXN) s TAB?(JAN) TAB^(LXN) s TAB3(JAN) 71^5 TA«4(LXN) s TAB4UAN) CALL ASTPOIXYBR'DAYB' DAYM • GPBS ' GRM5 • 29) PM s CXX( 1 ) PL s CXX(2) <^L s CXX(3) PS s CXX(4) PLM s CXX(5) SKYN s CXX(6) VI s CXX19) V s CXX( 1 ) XI s CXX( ] 1 ) VP s CXX( 1 2 ) VPP s CXX ( 1 3 ) P s 0XXU4) AUL s CXX115) AUM s CXX (16) PAPA s 0.0 DO 3073 NAP s 1 '37 DFG(NAP) s PAPA ^073 PAPA s PAPA + 10.0 DETERMINATION OF EQUILIBRIUM ARGUMENTS (VO + U) 7777 TML s CP - GOML COM s SL + TML COVU( 1 ) s2.0* (CON - COVUt?) covU(3 ) COVU(4) COVU(5' C0VU(6) COVLU7) COvU(8 ) COVU(9 ) COVUt 1 COVU( n COVU( 1 2 COVU( 13 COVlM 14 COVU( 15 COVU(16 COVU( 17 COVUt 18 COVUt 19 ^OVU(2C COVUt 21 COVUt 22 COVUt 23 COVUt 24 COVU( 2 5 PM +XI -V ) PL) S COVUt 1 ) -( pm - s 2.0#(TML) s CON - V - 2.0* (PM - XI) - 90.0 s CON - VP + 90.0 s 2.0 -"CON - VPP S 2.0* CON - PM -PL + AUL s COVU(?) -(pm - PL) S SL - P c , + 180.0 + 2.0* (TML) s COVUC*) - (SL - PS) s COVLM i) -(2.0*(SL - PM)) - ( PM - PL) + 180.0 s COVLM 1 ) + 2.0* ( SL - PM) s COVLMl2> + (PM - PL) s CON + PM - PL - V + 90.0 S CON - PM + AUM s CON - V + 2.0* (PM - XI) + 90. C s TML + 270.0 - SL s C0VU(4) - (PM - PL> s COVIM18) - (PM - PL) s COVU( 18> + ?.0*( SL - PL ) s 2.0* (COVLM 1 ) ) s 3.0* (COVLM 1 ) ) s 4.C*(C0VU( 1 ) ) s 2. 0*(<~OVU( 3 ) ) s 3.0*(COVU( 3) ) CALL TWOPI (COV'J 1 2^ > 16 PROGRAM FOR HARMONIC ANALYSIS AT TIDAL FREQUENCIES-Continued DETERMINATION OF NODE FACTOR RECIPROCALS (F) V s V*. 0174533 P s P*. 01 745 3 3 VI sVI -'.0174533 CPA s 1.0/SQRT(1.0 - ]?.0*(SIN(.5*VI )**2/COS(.5*VI )**2)*COS(?.0*P) 1+ if.. *( SIN ( .S"VI ) »*4/00S ( ,S J "-VI )**4 ) ) COA s 1.0/SQRT(?.3lC + ] . 43 5* COS ( 2 . 0*P ) ) CF(1) S 1 .0/ ( r ^ c ( .5*VI )"*4/.91 ^44 ) CF(?) s CF(1) C F ( 3 ) s 1.0 C |r (4) s 1 .0/ ('SIM (VI )*COS( .5*VI )**2/. 37988 ) CF(5) s 1 ,0/SORT ( 0.8°65"- c INI 2.0*VI ) **2 + . 6001*SI N ( 2 .0*VI ) *COS ( V ) 1+0.1 6) CF(6) s 1.0/SQRTI 19. 0444*51 N < VI )**4 + 2 . 77C2*SI N ( VI ) **2*CCS ( 2 .0*V ) 1+ 0.0981) - 0.000 C J99 CF (7 ) s CF( 1 ) *CRA CF(8) s CF(1) CF(9) si .0 CF( 10) s 1.0 CFIllls CF(1) CF(12) s r F n ) CF( 13) s CF(1 ) CFU4) s 1.0/ (SIN(2«0*Vl ) /0. 72137) CF ( ] 5 ) s CF (4 )*COA CF(16) s 1.0/(SlN(VI)*SIN( .5*VI )**2/0. 016358) CF( 17) s 1.0 CF( 18) s CF(4) CF{ 19) s CF ( 18 ) CF ( 20) s CF( 19) CFI21) s CF(1)**2 CF(22) s CF( 1 )**3 CF(23) s CF ( 1 )**4 CF(24) s 1.0 CF(25) s 1.0 CKKF s 1.0/CFI6) AHNAT s CF(4)*CRP(4) C5F s CF(5 ) ENTER SECONDARY EFFFCTS AND DETERMINE KAPPA CALL SASTRfDEG'CjF'CKKF' SMAC 'PROD' ACCP' RESAM « 37 ' TAB1 « TAB2 ' TAB3 ' TAB 14 , TAR5'TAB6 , SL'VPP'P c, VP'OF'?5) 8059 DO 21 I s 1 ' c 2] CKAPAP(I) s CHSIP(I) + COVU(I) CALL TWOP I ( CKAPAP' 5 ) CKAPt] ' s CKAPAP ( 1 ) CKAP( 2 ) s CKAPAP ( 2 ) CKAP(3> s CKAPAP! 3) + SMAC CKAP(4) s CKAPAD(4) CKAP(5> s CKAPAP(5' + ACCP CKAP (6 ) S CKAP ( 3 ) DMN s CKAP( 1 ) - CAP ( ? ) CALL ONEP I ( DMN) CKAP ( 7 ) s CKAP ( 1 )+ D«N CKAP(8> s CKAPt?)- DMN CKAP (9> s CKAP (3) CKAP( 10 ) s CKAPf 3 ) SMD s CKAP( 3) - CKAP( 1 ) CALL ONEPI (SMD) CKAP(ll) s CKAP(l) + 0.464*SMD CKAP(12) s CKAP( 1 ) - SMD rKAP(13) s CKAP(l) - 0.866*D W N OKn s CKAPi5) - CKAP (4) CALL ONFPI (OKD) CKAPI14) s CKAP(5) + 0.5*OKD CKAPM5) s CKAP(5' - 0.5*OKD CKAP( 16 ) s CKAP( 5) + OKD CKAP(17)s CKAPI5) CKAPU8) s CKAP(5) - 1.5*OKD CKAP(19) s CKAPI5) - 2.0*OKD CKAPI2C) s CKAPI5) - 1.43*OKD CALL TWOP I (CKAP' 20 ) 17 PROGRAM FOR HARMONIC ANALYSIS AT TIDAL FREQUENCIES-Continued DETERMINATION OF ZETA(A) AND R(A) FOLLOWS DO 2 2 I s 1 ' 20 22 CHSI(I) s CKAP(I) - C0VU'( I ) CALL TWOPI ( CHSI • 20) CR ( 1 ) s CRP( 1 ) CR (2 ) s CRP( 2 ) CR ( 3 ) s CRP( 3 ) /PPOD CR (4) s CPP(4) CR(5) s CRP(5)/RFSAM CR(6) s 0.272*CP ( 3 ) /rF(6 ) CP(7) s 0.145*CP ( 2 ) /ORA CR(8) s 0.133*CR ( 2 > CP(o) s 0.008*CP<3) CR ( 1 0) s 0.059*C? ( 3 ) CP( ] ] ) s 0.007*CP( 1 ) CR(12) s 0.024*CR(1) CR( 1?) s 0.194*CP(2 ) CR(14) s 0.079*AHNAT/CF( 14) CR(15) s 0.071-*CR(4)/CQA CP(16) s 0.043*AHNAT/CF < 16) CR ( 17) s 0.331*CP ( 5)*CF( 5 ) CR (] 8 ) s 0.194*CP ( 4 ) CR ( 19) s 0.026*CP ( 4 ) CP(20) s 0.038*CP(4) ELIMINATION OF COMPONENT EFFECTS AND DETERMINE KAPPA AND H(A: DO 35 I s 1' 5 SUM s .0 SOO s .0 DO 20 J s 1'20 311 BELL(J) s CR ( J)*TXXIX1 ( I ' J) IF( BELL (J)-. 0005) 23' 24 '24 23 BELL(J> s 0.0 iu BINGO(J) s TXXIX?tI'J> - CHSI(J) + CHSIP(I) IF (BINGO (J) ) 2^' 2ft ' 26 25 BINGO(J) s < RINGO(J) + 360 .) *0 . 01 745^3 GO TO 29 26 IF(BINGO(J) - 360.0) 28'28'27 27 BINGn(J) s (BINGO(J) + 360.0 1*0.0174533 GO To 29 28 BINGO(J) s BI NGO( J 1*0.0174533 29 SUM s SUM + BELL ( J )*SIN( BINGOI J) ) ?0 SOO s SOO + BELL ( J>*COS( BINGO( J) ) QOS s CPP( I ) - SOO CALL TI TAN (SUM 'OOS 1 DCCH' 2 ) CDCH s DCCH*0 .017453^ ^pc ( i ) s o.o IF ( DCCH. EQ. 90.0. OR. DCCH. EO. 270.0) Go TO 34 CRC(I) sf^RPlI) - SOO) /COS(CDCH) 34 G( I ) s CRC( I ) *CF( I ) ZETA(I) s CHSIP(I) + CDCH*57. 29578 35 PAM I ) s CHSIP(I) + CDCH*57. 29578 + COVU(I) CALL TWOPI ( PAK ' 5 ) INFEPFNCE OF H(A) FPOM MAJOR CONSTITUENTS D( 1 ) s 0.079*G (4 ) D ( 7 ) s 0(5) D(3) s 0.272*0(3) DIM s 0. - O2 8*G ( 1 ) D ( 5 ) s . 7 ] * G ( 4 ) D ( 6 ) s G ( ] ) D(7) s CPP (6 ) *CF ( 21 ) D(8) s CPP ( 7 ) *CF ( 22 ) D(Q ) s CRP (8 ) *CF( 23 ) niiO) s G ( 2 ) D< 1 1 ) s 0.133*G(?J n ( i 2 ) s G ( 4 ) 18 PROGRAM FOR HARMONIC ANALYSIS AT TIDAL FREQUENCIES-Continued D(l^) s 0.043*G(4) Dt 14) s 0.331*G< 5 ) D(15) s 0.194*G(4) D(16) s 0.026*G(4) D(17) s 0.008*G( 3 ) D ( ] 8 ) s G ( 3 ) D< 1 ° ) s CRPI 9 ) *CF< 2 4 ) D(?0) s CRP ( 10)*CF(2S) DC?] ) s n.059*G( ? ) D(?2 ) s 0.007*01 1 ) D(?3) s 0.194*G(?) D(?4) s 0.038*G(4) INFERENCE OF KAPPA(A) FROM. MAJOR )NSTITUENTS 677 PRINT 813 PRINT 73' (CHSIP ( I ) 'I s 6' 10) PRINT 812 (CRPI I ) ' I s 6' 10) 19 PROGRAM FOR HARMONIC ANALYSIS AT TIDAL FREQUENCIES-Continued PRINT 5 PRINT 6' CKKF 'CPA'CQA ' AHNAT EQUILIBRIUM ApGU^ENTS AND NODE FACTOPS Dn INTED PPIMT PRINT PRINT PRINT PRINT PPI NT PR I NT PPI NT PPI NT PRINT PRINT PRI NT PRINT PRINT PRINT PRINT 7 811 814 815 816 814 815 817 814 815 818 814 815 813 814 815 !COVU( I (CF( I ) ' (COVU( I tCF(I) • (covut i (CF( I ) ' 'G(3) CRC(4)'ZETA(4)'PAK(4)'GU) CRC(5)'ZETA(5)'PAK(5) , G(5) COMPUTED VALUES nF H(A) AND KAPP.A(A) n PINTED FOP THE CONSTITUENTS M2 S2 N2 01 Kl M4 M6 M8 S4 S6 INFERRED VALUES OF H(A) AMD KAPPAIA) PRINTED FOP THE CONSTITUENTS K2 L2 2N R2 T2 LAMPDA MU2 NU2 Jl Ml 001 PI 01 ?01 PH01 PRINT 15 PPINT 16 PPIMT 71 ' ( W I I ) ' I s 1 5) PPIMT 72' ( D( I ) ' I s 1 5) PPINT 17 PPINT 7] (W( I ) ' I s 6 10) PRINT 72 ( D( I ) ' I s 6 10) 20 PROGRAM FOR HARMONIC ANALYSIS AT TIDAL FREQUENCIES-Continued 5899 1112 7111 PP I NT 18 PRI NT 71 ' (W( I ) ' I s n 1 5 ) PPI NT 72' ( D( I ) ' I s 11 1 5) PRINT 19 PPI NT 7 3 ' ( W ( I ) ' I s 16 20) PRINT 7 2-' ( D( I ) ' I s 16 ?0) PRINT 220 PRIN T 7 1 ' ( W ( I ) ' I s 21 24) PPI NT 7360 PPI NT 72' GO TO 5998 GO TO 5899 5998 K s 2 GO TO (5899'5 1H0' 1 OF 12. 4 1H0' 23HJOB 'TE 5X' F10.1 '4X' 1H0) K2> ' ' 2F1 BP I UM Y5I5 IMINA 5X' 4 RMONI RMONI RMONI '4HK ( •4HM( 1 4110 ( •4HR( ' 6HLA 5^10 5 F 1 . RIME ) • 2HP 5' F V + D 'IN ON C (A) ANAL ANAL CONS • 6X' •6X ' • 6X' '6X- da 1 5 /) //) 5F12 A' 8X ' 2HOA' AX • 6HHhO( 1 ) ) 10.4//) U) AND ELIMINATION (F) ARGUMENTS /) FERENCF //) F COMPONENT r FFFCTS // 1 5X ' 4HR ( A ) • 5X ) YSIS' 29-DAY.S ) YSIS' 15-DAYS' 2X'13HN(2) INFERRED) TANTS ' IX ' 13H(H) AND KAPPA //) 4HK ( 2 ) ' 6X ' 4HL ( 2 ) ' 6X *4HM ( 1 ) ) 4HM(6 ) • 6X'4HM(8 ) '6X*4HN(2 ) ) 4H(0O) ' 6X'4HP(1 ) ' 6X'4HQ< 1 ) ) 4HS(2) '6X'4HS(4) '6X*4HS(6) ) X' 5HN'J ( 2 ) ' 4X'6HRI-IO( 1 ) ) .3) '215' F10.8'F5.0) F4.2) 2) 26X 1 26X 1 1H0 ' 5X 1 5X 1 5X 1 5X 1 5X' • 9A8///) ' 4HN( 2 ) ' 8X'4HS(2) ' 8X'4HO( 1 ) '8X'4HK (1 ) ) PI ME)' SF1?.3//) ' 4HM( 6) ' 8X ' 4HM( 8 ) ' 8X '4HS (4) ' 8X'4HS (6 ) ) ) ' IX • 5F12.3) A) '2X' 5F12.5 // ) ' 4HL( 2) ■ 8X '4H( 2M) '8X'4HR(2) ' 8X'4HT ( ? ) ) 6X ' 5HMU ( 2 ) ' 7X ' 5HNU ( 2 ) ' 8X ' 4hJ ( ] ) ' 8X ' 4HM ( 1 ) ) X '4HP( 1 ) ' 8X '4HO( 1 ) 5F12.3) RX '4H( 2Q) ' 7X '6HRHO ( 1 ) ) TA" A) ' FlO FlO. 5' FlO. 5' F70.5 ' F70.5' 5F12. 1) 5F12.4 // ) 5' 2F10.2' 5' 2F10.2' 5' 2F10.2' 5' 2F10.2' 2F10.2 ' FlO. 5) FlO. 5) FlO. 5) FlO. 5) FlO. 5) ' F5.2' 2FA.2' ' 6F10.2) /// IX' 10F12.4) RMINATED '24H NSPH OR TFAC s (ZERO)) Fl0.1'4X' 2F10.1) 21 PROGRAM FOR HARMONIC ANALYSIS AT TIDAL FREQUENCIES-Continued SUBROUTINE FOPANISER' SP« PES' ZFP' N «" M ) FOUPIER ANALYSIS - DETERMINE ZETA (PR I ME ) AND O(PPI^E) DIMENSION SER(N)' RFS(M)' ZFP(M) DO 55 JR s 1' M PES(JR> s 0.0 ZFP(JP) s 0.0 C s 0.0 S s 0.0 DO 10 L s 1* N C s C COS(FLOAT(L-l)*<3.141593*(FLOAT 55 CONTINUE PFTIIPN FND 5URROUTINE SASTP(DEG C5 F ' CKK F ' ACS ' APS ' ACK ' Apr • I OT ' T ABl ' T AB2 ' TAB3 ' 1TAR4 ,T AB5'TAB6'SL'VPP , PS , VP , <'F , JOP) DETERMINE SECONDARY CONSTITUENT EFFECTS DIMENSION DEG(I0T) , TaBi(IOT)'TAB?(I0T) , TAB^(I0T) , TaB4(I0T)'TAB5(I0 IT)' TAB6( IOT ) • CF( JOP) ?043 FODMATI 1H0'4F10.5 ///) 2044 FORMAT!//// 6 X ' 4HSMAT • 6X ' 4HPR0D ■ 6X ' 4HACCP ' 4X ' 5HPF SAM ) ">^45 FORMAT!///// ?X' 6F10.4) AMT s SL - 0.5»VPP CALL TWOPI (AMT ' 1 ) CALL TEPDO(DEG , TAR3'^7'RAK'AMT) TAZ s RAK AMD s SL - PS CALL TWOPI (AMR ' 1 ) CALL TERPO! DEG ' TAB5 ' 37 ' RAL ' AMR ) TA71 s RAL ACS s RAK*CKKF + PAL CALL TEpPOfDEG' TAB4' 37'SKO' AMT ) TAZ2 s SKO CALL TEpPO s STOX APS s STOX*(1.0 + (SKO*CKKFn PALT s SL - 0»-5*VP CALL TWOPI (FALT 1 1 ) CALL TERPOfDEG' TABl ' ?7'PKX'FALT ) TAZ4 s PKX AC< s PKX*C5F CALL TERP0(DEG'TAB2'^7'PKXA'EALT) TA75 s PKXA APr s 1.0 + (C5F*PKXA) PRINT 2044 PRINT 2043' ACS' APS 1 ACk- 'APK PRINT 2045'TAZ' TAZ1' TAZ2* TAZ3' TAZ4 1 TaZ5 RFTURN FNP 1 22 PROGRAM FOR HARMONIC ANALYSIS AT TIDAL FREQUENCIES-Continued SUBROUTINE DAYXX( MONTH' DAY' ST T • TM ' DAYB' DAYM ' GRBS ' GRMS ' CP ) RPOO FOPMATf //// 5X' 8H DAyB s 'F5.0'10X' 8H DAYM s 'Fs.O /// 5X' 8H G 1HBS s l F7.2'10X' 8H GHMS s 'F7.2 /// 5X'17H ORIGINAL T.M. s 'F7.2' ?10X'18H rriRRECTFn T.M, s 'F7.2) GO TO ( 207' 209' 21.1 ' 213' 215' 21 7'219' 251 ' 223' 225' 227*229) 'MONTH 2^7 DAYP s DAY 207 GO TO 230 208 209 DAYB s DAY + 31. ?09 GO TO 230 2 10 21 1 OAYR s HAY + ^9. 211 GO In 2^0 2 12 21^ DAYB s DAY + on. 213 Go Tn 230 214 215 DAYB s DAY + 120. 215 GO To 2^0 216 217 DAYB s DAY + 151. 217 GO To 2 30 218 219 DAYB s DAY + 181 . 219 C,n To ?^n 2=,i HAYR s DAY + 212. 221 GO TO 230 222 22^ DAYP s DAY + 243. 223 GO To 230 224 225 DAYP s DAY + 273. 225 GO TO 2^0 226 227 DAYB s DAY + 104. 227 GO TO 230 228 2?9 DAYB s DAY + 334. 230 IF(STT.GT.12.0) GO TO 23 3 230 rp s T^ + ( ^TT#1 5 . ) IF(CP.LE. 180.0) GO Tn 235 ?33 CP s TM - (24.oo - 5TT)*15.0 233 DAYB s DAYB + 1. 234 2i5 GPP? s CP/15.0 2^5 GRMS s GPPS + 12.00 236 IF(ORMS.GT.24.0) GO to 240 237 DAYM s DAYB + 14.0 GO TO 242 239 240 GRMS s GRMS - 24.00 DAYM, s DAYB + 16.0 242 PRINT 8800' DAYB 'DAYM' GRBS' GRMS' TM' CP PFTUPN FND SUPPOI iT IMF AZIM ( DX' VR ' A' VN' < ' J' OOMpy ' MFC > COMPUTE MAJOR OR MINOR AXIS OR COMPLJTF NORTH AND EAST COMPONENTS DIMFNSION DX(J)' VP(J) 1 VN(J) do in i s 1 ' J ion VN ( I ) s (nx(i) + roMpv - a 1*0.0174^3'* Go to ( i ' 2 ) ' y 1 DO Aft I s 1 ' J 56 VM ( I ) s VR( I ) *COS ( VN ( 1 ) ) GO TO 8 8 2 I F ( NFC. EQ. 4. OR. NFC. Fo.i ) GO Tn 109 DO 7 7 I si' J 77 VM ( I ) s VR( I )*SIN( VN. ( I ) ) p E T U R N 88 DO 09 LM s 1 ' J TR s VN(|_M) TA s VR(LM) 09 VN ( LM) s SIGN ( TA • TR ) 109 PFTUPN FND 23 PROGRAM FOR HARMONIC ANALYSIS AT TIDAL FREQUENCIES-Continued SUBROUTINE ASTRO ( XYER ' DAYB • DAYM • GRBS ' GRMS ' JOBX ) DETERMINATION OF ORBITAL FLEMENTS FOR OBSERVED SERIES Common /COSTX/CXX (30) ' OFX( 5) COMMON / LEAPYP / hay 'month olTO FOPMATflHO' 3RHERPOP PAPAMFTFR (I) FXCFFDS LIMITS ) ?]1] FORMAT) 1H0' 15X'1HR' lOX'lHQ' 15X'9HU OF M(2)) 2112 Fopmak 8X 1 F10.3' 3X • F10.3' 9X ' Fin. 4) 7040 FOPMATI 1H1 ' 10F10.3//// IX • 10F10 .3/ /// 1X'10F10..3) TALL ORBIT (XCFN'XSX'XPX'XHX'XPIX'XNX'OEX'T 1 XYER '5) Do 30 NOE s 1 ' 30 TO CXX(NOE) s 0.0 XCFT s XCEN + 1.0 DORY s 0.25*(XYEP - XCFT ) AMI s XYFP - XCFM AMjT s AINT(DOBY) FA.pm s 0.25*(AMI) FAPX s FARM - AINT(FAPM) I F ( FAPX.FO. 0.0. AND. MONTH. Fq.t ) GO To ^5 1} IF ( FARX. FQ.O.O. AND. DAYB. LF. 60.0 > DAYB s DAYB - -j . o I F { FAPX .GT.O. I TiAYP s DAYP - 1.0 IF{ FARX. EQ. 0.0. AND. DAYM.LE. 60.0) DAYM s DAYM - 1.0 IF ( FAPX. GT.O. 0) DAYM s DAYM - l . o 32 CXX(l) s XSX + 129.38482032*AMI + 13. ] 763Q6768* (DAY8 + AMIT) + 0.5 lion] A^32*GRBS r XX(2) s XPX + 40.66?46584*AMI + . 1 11 4040 1 6* ( DA YB + AMIT) + 0.004 1 641 R'4*GPBS CXX(3) s XHX - 0.2387?498R*AMI + . 9 8 ^647' 2 P 8* ( DA Y R + AMIT) + 0.04 1 1 OA86^87*GPRS CXXI4) s XP1X + 0.01717836*AMl + .000047064* ( DAYB + AMIT) + 0.000 1 ^01 961*GPRS CXX<5) s XPX + 40.66246584*AMI + . 1 1 1 4040 16* ( DAYM + AMIT) + 0.C04 1641 834*GRMS CXX(6) s XNX - 19.328185764*AMI - . 05295 39336* ( DAYM + AMIT) - 0.0 102?064139*GRMS IF ( JOPX.LF.15.AND. JOBX.GT. ) GO TO 40 CALL TWOPI ( CXX '6 ) GO To 4] T5 I F( DAY.FO.l . ) GO TO ?? GO TO 31 40 CXXI7) s XPX + 40.66?46584*AM I + . 1 11 4040 1 6* ( DAYM + AMIT) + 0.004 i 641 B34*GRBS 0XX(8) s XNX - 19.3281 R5764*AMI - o .OS295 I9^f>* ( DAY" + AMIT) - 0.0 1022064139*GPBS CALL TWOPKCXX' 8) 41 AN s LXX(6')*0. 0174533 AX s CXXI6) FYF s .9136949 - . OT= 69 ?6*G0 S ( AN ) CXX(9) s ACOS(EYE)*57.?957795 IF(rxX(9).LT.l7.0.OP.rxX(O).GT.T0.0) PRINT 7110 CIC- s CXX( 9) *0. 0174533 IFtCIG. FQ.O.O) GO TO 2^0 I F ( AX.EO.O.O.OP.AX.Eo.l 80.0 ) GO TO ?30 VXX s .0896831*SIN I AN) /SIN(CIG) CXX (] ) s ASIN(\/XX)*57. 2957795 IFt AX. GT. 180.0. AMD. r XX ( 1 ) . GT . .0 ) CXX(IO) s -1.0*CXX(10) CVX s CXX ( 1 0) *0.01 74"' FXX s .2067274*51 N< AN)* ( 1 .0 - 0.0194926*COS ( AN ) ) IF ( FXX.EO.0.0 ) GO TO 202 FZZ s .99798^2 + 0.20^7274*COS( AN ) - . 00201 48*COS ( 2 . 0*AN ) IF ( FZZ. FQ.O.O ), GO JO ?02 EXFZ s EXX/EZZ IF( FXFZ.r,T.^4^o.o) Go TO ?0? CXX(ll) s ATA,\( FXFZ )*^7.2Q c ;779 c; I F ( AX.GT. 180.0. AND. CXX( 11 ) .GT.0.0 ) CXX'(ll) s -].0*CXX(11) rFy s CXX( 11 ) *0.0174^, ?n? vpx s SIN ( CVX ) / ( roc ( r\/x ) + o.3^4766/SIN(?.0*'"IG ) ) r XX(12) s ATAN + . n 7 2 6 ] 84 /S I N ( C I G ) **2 ) r XX(13) s ATAN(VPY)*57. 2957795 I F ( AX.GT. 1 80.0. AND. CXXI 1 3 ) .GT.0.0 ) CXX<13) 5 -1.0*rxX(1T) PVCP s GXX ( 1 3 )*n.01 7AS^3 24 PROGRAM FOR HARMONIC ANALYSIS AT TIDAL FREQUENCIES-Continued ?m dgx s rxx (5) - r xx (in tall TWOP I ( PGX ' 1 > xpn s PGX*n.0i7it;n rxx (14) s PGX PAX s SIN(2«0*XPG)/(rOS(0.?*riGl*'*?/(6.0*SIN(0.**CTG)**2 > ~ COS(2. T 0*XPG) ) PXX s ATA.N(RAX)*57. 2957795 UM? s 2.0*(CXX(11) - CXX (10)) UL? s UM2 - PXX IF(UL2.GT. 180.0) UL2 s UL2 - 180.0 IF(UL2.LT. 180.0) UL2 s UL2 + 1R0.0 CXX (15) s UL2 7F C ^ s ( 5.0*COS(CIG) - 1.0)*SIM(XPG) ?rr s ( 7»0*COS (CIG) + 1 .0 ) *("OS ( XPG ) r A|_L FITAN(ZFS'ZFr'OXX' 2 ) r PAV s 0.5*( MMi) + OXX + oo.n rALL TWOPI ( CRAV ] ) r X X ( 1 6 > s CRAV IF( J0BX.GT.15.AND. JORX.GT.O ) Go Tn 88 PGXX s GXX ( 7 ! - CXX ( 11 ) TALL TWOPI ( PGXX ' ] ) xx°r- s PGXX*0 .017453^" rxx(17) s PGXX PA,TX s rx.X ( 8 ) * n .ni 745^ FYFX s O.Pl^ft°4o - 0.0^56Q26*COS ( BATX) CXX(? ) s ACnc ( FYFX ) *R7.?o^77Qc; U M 7 X s 2.0*(CXX(11) - rxx(io)) fyit s rxx ( 20 ) *0.0] 74^?^ ZFXS s ( ^.0*COS ( FYI T ) - 1 . ) *S I N ( XXPG ) ZFxc s < 7.0*rrt^ ( fy it ) + i .o ) *cos( xxpr-, > CALL FITAN< ZFXc. ' ZFXr ' oxxx • ? ) CXXU9) s 0.5*(UM2X) + oxxx + on.n rpaVX s C X X ( 1 9 ) CALL TWOPI I <~PAVX • 1 ) CXX(IP) s C P. A V X qg no 33 i\]T s 1 ' 30 nPr-p s CXX( NT )*7O000. + 5.0 3 3 CXX (NT) s AINT(nPOD)*n.000l dpjmT ?04-O'(rxx(IAX)'IAX s I 1 30) PRINT 2111 PRINT 2112' RXX'OXX'iim? RETURN END 25 PROGRAM FOR HARMONIC ANALYSIS AT TIDAL FREQUENCIES-Continued SUBROUTINE ORBIT ( XCEN ' XSX • XPX • XHX • XP1X ' XNX ' OEX ' T » XYER » NNN ) DETERMINATION OF ORBITAL FLEMENTS FOR BEGINNING OF CENTURY DIMFNSION OFX (NNN) g fopmaT(]hO' 5F12.4' 5X' F5.0' 2 F6 . 1 ) S s 1^.1763968 P s 0.1114040 XH s .985647? D] s .0000471 XN s .0529539 XCAM s XYER*0.01 + 0.001 XCFN s AINT ( XCAN )*100.0 T s -3.0 YP s 2.5 GAT s 1600.0 no io jv s i' 30 GP s (GAT)*n. 01/4.0 + n. 00001 FPX s APMGP) PP s AIMT(FPX) COL s FDx - PP if (col.lt.o.ot o: go ^ n IF (GAT. FO. XCFN) GO To 12 YR s YR - 1.0 GO Tn 9 11 IF( GAT. EO. XCFN) GO TO 1? 9 GAT s GAT + 100.0 in CONTINUE 1? T s (GAT - 1900.0)*0.0l OFX(l) s 270.4374222? + ^07.892*T + . 00 25 25*T**2 + . OOOOO 1 89*T** 13 + YP*S OFX(2) s ^34.?2801944 + 1 09 . 3 220 5 5*T - . 1 n?4444*T**2 - 0.000012 15*T**3 + YP*P OFX(3) S 279.69667778 + 0.76892^*1" + . 000 30 ? 5*T**7 + YR*XH OFX(4) s 281.2208333? + 1.719175*T + . 00045 278*T**2 + 0.00000333* 1T**3 YR*P] OEX(5) s 259.1825333? - 1 34 . 1 42 3972*T + . 00 2 1 0556*T**2 + 0.000002 122*T**3 + YR*XM CALL TWOPI (OFX '5 ) XSX s OFX ( 1 ) XPX s OFX12) XHX s OEX(3) X^i X s 0EX(4) XNX s OFX(5) PRINT 8' (OEX(JX)'JX s 1'5) ' GAT' T' YP RFTUPM END 26 PROGRAM FOR HARMONIC ANALYSIS AT TIDAL FREQUENCIES-Continued SURPOUTINF TWOPI ( AUG' 10) C c C ELIMINATE NEGATIVE Op ANGLES GPEATEP THAN 360 DEGREES C DIMFNSION AUG (JO) no 114 mo s 1 ' 10 1?0 IF ( AUG(MO) ) H ? ' 1 1 4' 1 1 1 117 AUG(M^) s AUG(MO)+ 160.0 GO TO 120 111 IF (AUG (MO) - ^60.0 ) 1 14' 114 ' 1 1 5 116 AUG(MO) s AUG(MO) - 160.0 GO TO 120 114 COMTINUF RFTUPN END SUBROUTINE FITANI AUS ' AUC ' PT A ' JMAP ) C C C PFTFRMINF ARCTANGENT AND PLACE IN CORRECT QUADRANT C RXG s AUS/AUC IF DI VIDF CH^rk- 14' i =, 14 I F ( AUS) 16' 5 5 ' 17 16 PTA s 270.0 GO TO 88 17 RTA s 90.0 GO TO 88 15 RTA s ATAN(BXG)*57. 2957795 GO Tn ( 88 ' 38 ) ' JMAP 18 IF (AUS) 32 ' 32' 14 12 IF ( AUG) 11 ' 55' 17 n RTA s 180.0 + RTA GO TO 8 8 -54 I F ( AUT ) 1R ' 1 7 • R8 15 PTA s RTA + 180.0 GO TO 8 8 M RTA s PTA + 360.0 GO TO 8 8 R5 PTA s .0 88 CONTINUE RFTUPN END SUBROUTINE XMFAN ( VIN » FH' N ) C c C MEAN SERIES FOR NON-TIDAL CURRENT OR MEAN SEA LEVFL C DIMENSION VIN(N) 15 FORMAT!//// 1X'14H SUM OF SEP I ES • 3X • F 1 2 . 3 // IX' 7HD I V I SOR ' 1 2 X ' I 1 1 // 1X'14HMEAN OF SEP 1 ES • 5X • F 10 . 5 // IX'IOHTOTAL DATA'9X'I10 ) MA s N F H s .0 DO 10 MA s I'M IF( VIN( NA) .EO.0.0) MA s MA - 1 10 FH s EH + VIN(NA) BH s FH/FLOAT(MA) PRINT 15' EH' MA' BH' M RFTUPN FND h h 27 PROGRAM FOR HARMONIC ANALYSIS AT TIDAL FREQUENCIES-Continued SUBROUTINE TERPO ( ANG «TXX • LI D • ANSWER • AUGM ) INTERPOLATE TAPLFS 21 ' 22 ' 23 • 24 ' 25 ' AND 26 DIMFNSION ANG -(LIP) 1 TxxtLI 1 ^) 1 FORMAT ( ///23H JOB TERMINATED 26HFAILFD TO INTERPOLATE ANG. LIN s LID - 1 no 2 ION s l 1 LIN I F ( AUGM. GT.ANGI ION) .AND. AUGM.LT. ANG( ION+1 ) ) GO TO ?8 IF ( AUGM. FO. ANG ( ION > ) GO Tn ?g IF( AUGM.EQ.ANG( ION+] ) ) Go to 27 GO to 20n 26 ANSWFR s TXX ( ION ) GO TO 201 ?7 AN.SWFR s TXX ( ION+] ) GO Tr« 201 ?R RFW s AUGM* 0.1 WFP s REW - AINT(PFW) FPA s WFR*10.0 PFT s APS ( TXX (ION) ) TFX s APS ( TXX ( ION+1 ) ) IF ( PFT.GT.TFX ) GO TO 20 IF(PFT.LT.TFX) GO TO ^0 IF(PFT.FO.TFX) GO TO f\ 79 DIFP s (PFT - TFX)*0.1 IF ( TXX( ION) .GE. 0.0. AND. TXX( ION+i ) .LT.O.P ) DIFP s (PET + TEX)*0.1 I F ( TXX( ION ) .LT.n.O.AND.TXX ( ION + I ) .GF.0.0 ) DIFP s (pET + TFX)*0.1 DLFX s DIFR*FPA CHS s SIGN! DI FX ' TXX ( ION) ) ANSWER s TXX(ION) - CHS GO TO 201 30 DIFP s (TFX - PFT)*0.1 I F ( TXX ( ION ) .GE.O.O.AND.TXXI ION+1 ) .LT.ri.f) ) DIFP s (RET + TEX)*0.] I F ( TXX ( ION ) .LT .O.O.AND.TXX ( ION + l ) .GE.0.0 ) DIFP s (PET + TEX)*0.1 DIFX s DIFP#FPA CFK s SIGN ( DI FX' TXX ( ION+1 ) ) ANSWER s TXX( ION) + CH^. GO TO 2 01 ■>! IF ( TXX( ION) .GE. O.O.AND.TXX ( ION + 1 ) .GF.n. n ) GO TO 26 I F ( TXX ( ION ) .LT.O.O.AND.TXX ( ION + 1 ) .LT.0.0 ) GO TO 26 IF(TXX( ION) .GF. 0.0. AMD. TXX ( ION+1 ) .LT. 0.0) GO TO 32 IF(TXX( ION) .LT.O.O.AND.TXX ( ION + 1 ) .GF. 0.0 ) GO TO 3 2 PRINT 1 CALL FXIT ^2 DIFP s (TXXUON+1) - TXX ( ION ) ) *0.1 DIFX s DIFR*FPA ANSWFP s TXX(ION) + niFX GO TO 201 ?O0 CONTINUE 20] RETURN FN 1 ^ 28 PROGRAM FOR HARMONIC ANALYSIS AT TIDAL FREQUENCIES-Continued SUBROUTINE DETAZ (FIN ' AZO' NR' COMPV'VIN • XTEMP ' INC) DETERMINATION OF TRUF AZIMUTH THIS ROUTINE IS A SPECIAL PURPOSE PROGRAM FOP USE WITH TIDAL CURRENT ANALYSES IN THE COA^T AMD GEODETIC SURVEY DIMFNSION FI N ( NR ) 'VIM (MR) ' XTFMP ( NP ) 2 FOPMATf 1H1 ' 30H AZIMUTH OR MEAN FLOOD DIP. s ' F7 ,2 ' 5X ' I 5 • 5X ' Fl . 2 ) -3 FOP«AT( ]H0 • 14HERR0R IN DFTAZ ) ft FOPMATflH]. '28HOPERATOR STOP THIS JOB) LMM s SUMD s 0.0 AI7 s AZO IF( AZO. EO. 360.0) AIZ s 0.0 DO 10 KAZ s l'NR TDIP s FIN(KAZ) + COMPV IF( TDIP.LF.20.0) TDIR s TDIP + ^60. IF(ABS(TDIP - AIZ - 360.0). LE. 20.0) GO To on IF(ABS(TDIR - AZO).LF. 20.0) GO TO on IF(ABS(TDIR - AZn + ?60.0).LE. 20.0) GO To on GO In too oO LMM s LMM + 1 SUmD s SUMD + TDIR 100 CONTINUE AZO s SUMD / FLOAT(LMN) IF DIVIDF CHECK R' 5 8 PRINT 6 4 PRINT 3 RFTUPM ^ AZn s A I N T ( A Z O ) CALL TWOPI ( AZO' 1 ) PRINT 2' AZO' LMN' SUMD IF ( INC.FO.00) GO TO 7 CALL TI LT (VIM ' XTFMP' MP) 7 RETURN FMH 29 PROGRAM FOR HARMONIC ANALYSIS AT TIDAL FREQUENCIES-Continued ^UPPniiTINF EDAT(MX' VIM'DIN'MA «LX' ICOR 'CD'CV'LRD'XTFMP" INC) INFEP DATA CORRECTIONS To SERIES FROM MAGNETIC TAPF DIMFNMON ICOP(LX) ' PIN (MA) 'VIN(MA) • CDf LX ) 'CV(LX) ' XTFMPf MA ) COMMON/ FOP T/AO ( 4 ) 'RO(i) <-, format M qX ' A^ ) 6 FOdmaT ( 60X ' F? .0 ' ?4X ' F4.2 ) 7 foomAT ( 5X ' IS'^X'FS.O'SX'FR.^^X'I'i'^X'AM fodmaT ( ?4F3 .2 ) ■\ FopMAT (24F3.0 ) 1 ? FOPMAT ( 1 1 OX ' F^ .0 ) M FOPMAT IIQX'FR.O) REWIND ° IF( INC.FO.OO) GO TO 14 IF(LPD.EO.OO) GO To t? RFAn(9" 51 ) ( XTFMPf IM) • IM s l'LPD> 11 PFAD(9'P0) (XTFMPf IP) ' IP s l'MA) OF'- 1 1 MP <=> 14 IF s ] LAG s LEN s 1 ^O QR I s l'MX'LFN IF ( [_PP> QO ' 1 ' 2 2 IFf I L"n) W] 3 PFAD (P' p ) I TRASH GO TO 9 8 1 LAG s LAG + 1 I F ( IF.GT.LX ) GO TO 1] LP s ICOPf IE) IFfLD.FO. I )G0 TO 4 LDX s ICOR(IF) - 1 11 IFf IF.GT.LX ) I_PX s MX DO 40 IX s I ' LPX PFAD(P'AO) DIN(LAG)' VIN(LAG) I F ( IX.FO.LDX ) GO TO ^00 LAO s LAG + 1 i^nc oomTINUF I F ( LDX.FO.MX ) GO TO 07 LFN s LD - I GO TO 98 4 CFAD(9*^) ITRASH DIN (LAG) s CD(IF) VIM(LAG)s CV( IF) XTFMP(LAG) S 0.0 PRINT 7' I» DIM(LAG) 1 VlN(LAG)' I COP ( I F ) ' I TP A <^H LFN s 1 IF s IF + 1 08 CONTINUE 7 IF(LAG.LT.MA) GO TO q GO TO 9 9 8 LAO 5 LAG + 1 PFAD 01 (VIN(IN) 'IN s LAG'MA) PFAD 10' (DIMflD)' ID s LAG'MA) DO 3 MAG s LAG'MA ^00 XTFMP(MAG) s 0.0 00 IF(INC.EO.l) <"ALL T I I..T ( V I M ' XTEMP ' MA ) IF ( INC.EO.l ) INC s RETURN FMD 30 PROGRAM FOR HARMONIC ANALYSIS AT TIDAL FREQUENCIES-Continued SIIPPOIJTINF TILT (GIN' TFM'L IT) COPPECT DATA F0 1 ? INSTRUMENTAL TILT DIMFNSION GIN (LIT ) ' TFM ( L I T ) 7 FOPMATf 1H0' 16 ' ^X '?RHTIlT fcXCEEDS "* 5 DFGREF LIMIT'3X'I3) no 10 IP s 1 ■ L I t IF(TEM( IB) .EQ.0.0) GO TO 100 MIX s IFI X ( TEM( IP ) ) NIX s MIX/5 + 1 GO TO (100 l 10l I 102'103 , l'0 4 , 10 5'106 , l07 , 110 , 110 , 110'lir) l 110> i N j X 101 GIN (IB) s .03*GIN(IB) + GIN(IB) GO TO ICO 10? GIN (IP) s .06*6IN(IB) + GIM(IB) GO TO ]0O 10? GIN (IB) s .09*GIN(IB) + GIN (IB) GO TO 100 104 GIN (IB) s . 1?*GIN(IB) + GIN(IB) GO TO 100 1^5 g I N ( I P ) s . 1 6 *G I N ( I B ) + G I N ( I B ) GO To 100 106 GIN (IP) s . ?0*GIM(IB) + GIN(IB) GO TO 100 10 7 G I N ( I B ) s . 2 5*G I N ( I B ) + G I N ( I P ) GO TO 100 lOfl GIN (IP) s . 51*GIN(IB) + GIN(IB) GO TO 100 10Q GIN (IB) s .66*GIN(IB) + GIN(IB) GO TO mo 110 PRINT 2 ' IB'MIX 100 CONTlNUF RETURN END SUBROUTINE T FMP ( XSFP ' xTF^p ' I ) STOPF SEPIES FOP FOUPIFP ANALYSIS DIMENSION XSFP(I) • XTFMP(I) no i o j s l ' i 100 XTFMP(J) s XSFP (J) PFTUPN FNn SUB POUT INF c -OPT ( c >OPX • VX ' KK ' J J ) SORT DATA AND FORM NEW SFPIFS DIMENSION SOPXt K< )' VX( KK) COMMON /CONTT/NN 1 DO 10 Is l'KK'JJ NN s I/JJ + 1 ]O0 VX(NN) s SORX ( I ) PFTUPN PNH SURPOUTIMF ONFPK AUX) TEST FOR 180 DEGPEE DIFFERENCE BETWEEN TWO ANGLES IF(ABS(AUX) -i «0.0 ) 1 70' 1 "70' if.T 163 IF ( AUX )1 64 ' 1 64 ■ 1 65 164 AUX S AUX + 360. GO TO 170 IAS Ally s AUX - 360.0 170 PFTUPN FNn 31 ■6 U.S. GOVERNMENT PRINTING OFFICE: 197 1 O 420-627 o (Continued from inside front cover) C&GS 30. Cable Length Determinations for Deep-Sea Oceanographic Operations. Capt. Robert C. Darling, June 1966. Price $0.10. C&GS 31. The Automatic Standard Magnetic Observatory. L. R. Alldredge and I. Saldukas, June 1966. Price .$0.25 ESSA TECHNICAL REPORTS— C&GS C&GS 32. Space Resection in Photogrammetry. M. Keller and G. C. Tewinkel, September 1966. Price $0.15. C&GS 33. The Tsunami of March 28, 1964, as Recorded at Tide Stations. M. G. Spaeth and S. C. Berkman, July 1967. Price $0.50. *C&GS 34. Aerotriangulation: Transformation of Surveying and Mapping Coordinate Systems. Lt. Cdr. Melvin J. Umbach, July 1967. Price $0.25. C&GS 35. Block Analytic Aerotriangulation. M. Keller and G. C. Tewinkel, November 1967. Price $0.55. C&GS 36. Geodetic and Grid Angles — State Coordinate Systems. Lansing G. Simmons, January 1968. Price $0.10. C&GS 37. Precise Echo Sounding in Deep Water. G. A. Maul, January 1969. Price $0.25. C&GS 38. Grid Values of Total Magnetic Intensity IGRF— 1965. E. B. Fabiano and N. W. Peddie, April 1969. Price $0.60. C&GS 39. An Advantageous, Alternative Parameterization of Rotations for Analytical Photogrammetry. Allen Pope, April 1970. Price $0.30. C&GS 40. A Comparison of Methods of Computing Gravitational Potential Derivatives. L. J. Gulick, September 1970. Price $0.40. PENN STATE UNIVERSITY LIBRARIES in nun ii i A0Q0Q720nQA5