H:\EXCERC\GEOD98e.EX6.wpd

Department of Geophysics, Juliane Maries Vej 30, 2100 Copenhagen Ø.

Geodesy Course, Exercise 6.

The exercise has the purpose to show how a gravity anomaly is calculated, and to compare the value with the values calculated from a spherical harmonic expansion.

The values from exercise 5 are used at first. The gravity in the point in room 008 at Juliane Maries Vej is determined using the observed gravity difference and the known value in Buddinge. The altitude is found from a 2-cm map. The values of the reference gravity values are calculated using the gravity formulae for GRS80, see Torge p. 57. The gravity anomaly is then found by subtracting the reference gravity from the observed gravity.

We assume the geoid height is equal to the value calculated from the OSU91 series development (see exercise 4), 36.7 m. Calculate then the gravity disturbance in the point.

Then use the program geocol12 as in exercise 4. When you are asked for whether points are to be calculated in a grid answer no (F=False) and to the question whether a comparison with measured values is wanted answer yes (T=True).

Prepared job-files are also found in the directory /disk1/cct/geod.

geocol12

GEODETIC COLLOCATION, VERSION 1 SEP 1992, RELEASE 1 (UNIX)

Thu Mar 5 12:34:28 1993

NOTE THAT THE FUNCTIONALS ARE IN SPHERICAL APPROXIMATION

MEAN RADIUS = RE = 6371 KM AND MEAN GRAVITY 981 KGAL USED.

MAX NUMBER OF OBS= 3200, MAX NUMBER OF PARAMETERS=239

MAX NUMBER OF OBS IN GIVEN REF. FRAME = 200

SIZE OF NORMAL EQ. BLOCKS=19800, SIZE OF POT.COFF. BLOCK= 130322

INTERACTIVE INPUT (T/F) t

INPUT: LTRAN, TRUE IF NON-STANDARD REF. SYSTEM IS USED

LPOT, TRUE IF SPHERICAL HARMONIC EXPANSION IS USED

LTEST, TRUE IF TEST-OUTPUT IS NEEDED

LLEG, TRUE IF LEGEND IS TO BE OUTPUT

LPARAM,TRUE IF PARAMETERS ARE TO BE DETERMINED

LNCOL, TRUE IF COLLOCATION IS NOT USED

LIOSOL,TRUE IF SOLUTION IS STORED OR RECOVERED

f t f f f t f

ARE ALL PARAMETERS OK ? t

INPUT CODE FOR BASIC REFERENCE SYSTEM:

0: USER DEFINED, 1: ED50 NORTH SEA, 2: ED50/EDOC,

3: NAD1927 /NEW MEXICO, 4: GRS67, 5: GRS80, 6: NWL9D,

7: BEST CURRENT, 8: BEST CUR. FAROE ISL, 9: ED50 FOR SF,

10: IAG-75, 11: KRASSOWSKY, DDR, 12: GERMAN DHDN, BESS.

5

REFERENCE SYSEM: GRS1980.

A = 6378137.00 M

1/F = 298.2572221

GM= 0.3986005000E+15

REF.GRAVITY AT EQUATOR = 978032.6772 MGAL

POTENTIAL AT REF.ELL. = 62636860.8500 M**2/SEC**2

INPUT NAME OF POT.COEFF. SET OSU91A TO DEGREE 360

SOURCE OF THE POTENTIAL COEFFICIENTS USED:

OSU91A TO DEGREE 360

INPUT: GM, SEMI-MAJOR AXIS (M), C(2,0), MAX. DEGREE

LFM, TRUE IF COEFF. IN INPUT STREEM AND *1.0D6

LBIN, TRUE IF ON BINARY FORM

LFORM, TRUE IF FORMAT IS INPUT

LINT, TRUE IF STORED AS INTEGERS

3.986005D14 6378136.2 0.0 360 f f t f

GM A COFF(5) MAX.DEGREE

0.39860050E+15 6378136.2 0.0000 360

INPUT FORMAT (2I4,2D18.0) F.EX. (2I3,2D19.12)

INPUT NAME OF FILE HOLDING COEFF.

osu91a1f

NAME OF FILE HOLDING COEFFICIENTS: osu91a1f

COEFFICIENTS UP TO N=5

2 0 -0.484165533E-03 0.000000000E+00

2 1 0.857179552E-12 0.289607376E-11

2 2 0.243815798E-05 -0.139990175E-05

3 0 0.957139401E-06 0.000000000E+00

3 1 0.202968777E-05 0.249431310E-06

3 2 0.904648671E-06 -0.620437817E-06

3 3 0.720295507E-06 0.141470959E-05

4 0 0.540441630E-06 0.000000000E+00

4 1 -0.535373285E-06 -0.474065010E-06

4 2 0.350729847E-06 0.663967363E-06

4 3 0.991080200E-06 -0.202148896E-06

4 4 -0.190576532E-06 0.309704029E-06

INPUT: LGRID - TRUE IF COMPUTATIONS IN A GRID

LERR - TRUE IF ERROR ESTIMATES ARE TO BE COMPUTED

OR REPRODUCED IN OUTPUT

LCOMP- TRUE IF COMPUTED VALUES ARE SUBTRACTED FROM OBSERVED

f f t

INPUT DATA LINE AND OUTPUT SPECIFICATIONS

POSITION OF STATION NUMBER (0: NO NUMBER, -1: NO OUTPUT U6)

POSITION OF LATITUDE AND LONGITUDE (E.G. 2 , 3)

TYPE OF ANGULAR UNITS USED (1: DD MM SS.S, 2: DD MM.M 3: DD.D)

POSITION OF HEIGHT (0: NO HEIGHT)

POSITION OF OBSERVATION 1 AND 2 (0 IF NO OBS. 1 OR 2)

DATA OR COMPUTATION QUANTITY TYPE CODE (11: GEOID,

13: GRAVITY, 15: TZZ, 26: (KSI,ETA), NEGATIVE: REF.SUBTR.)

COORD.SYST. CODE, -1 INDICATE GLOBAL SYSTEM, +100 REVERSE TR.

HEIGHT (IN M OR KM), ONLY USED IF NO INPUT HEIGHT

LPUNCH - TRUE IF OUTPUT OF RESULT TO FILE

LWLONG - TRUE IF LONGITUDE POSITIVE EAST

LMEAN - OBS. OR COMPUTED QUANTITY IS A MEAN VALUE

LSA - TRUE IF ALL ERROR ESTIMATES ARE IDENTICAL

LKM - TRUE IF HEIGHT IN KM

LADMU - TRUE IF UNREDUCED OR CONSTANTS * OR +

STAT - TRUE IF STATISTICS OF RESULT WANTED

LAREA - TRUE IF DATA ONLY INSIDE SPECIFIC AREA ARE USED

LFORM - TRUE IF FORMAT OF DATA IS INPUT

LIN4 - TRUE IF DATA NOT IN INPUT STREAM (FROM FILE)

1 2 3 1 4 5 0 13 -1 0.0 f f f f f t f f f

INPUT MULTIPLICATIVE AND ADDITIVE CONSTANT AND

LMEGR, TRUE IF VALUE IS UNREDUCED 1.0 0.0 t

ALL SPECIFICATIONS OK ? t

SELECTED GEOCENTRIC SYSTEM USED.

0 NO LATITUDE LONGITUDE H DELTA G (MGAL)

D M S D M S M

POT OBS DIF

INPUT DATA RECORD, LSTOP

1 55 44 00.0 12 30 00.0 19.0 981550.0 t

1 55 44 0.00 12 30 0.00 19.0 -13.75 -6.35 -7.40

STOP ? t

In the actual exercise, you must input the correct position, height and gravity value.

Then you must extract gravity values (position, gravity anomaly, estimated error of the anomaly) from the gravity data-base from an area around Copenhagen. The database is a file with the name /disk1/cct/gravity.new. (Type: head gravity.new in order to see the first lines of the file).

The extraction is done using the program select. It must be used in mode 1, i.e. extracting from a rectangular area. For the variable IANG use 6, because the file is structured as an 80-character record. NDATA must be equal to 2, because we need two data-elements, the gravity anomaly and its error-estimate. The program is used as follows:

select

input: inputfile and outputfile (two lines):

gravity.new

gra.ex6

input: MODE (0:reformat, 1:select, 2:mean, 3:grid, 4:plot,

5:sel&rej, 6:sel&wndw, 7:sel&rej&wndw, 8:sel&noise)

IANG (1:deg, 2:dm, 3:dms, 4:alt, 5:bin, 6:80char, 7:grid,

8: alt1, 9: alt (NewEng.), neg:fmt, 0:dline)

NDATA

1 6 2

input: FI1,FI2,LA1,LA2,DFI,DLA (deg or m)

55.55 55.75 12.4 12.7 0.0 0.0

--- S E L E C T ---

total points: 183984

located within area: 54.0000 81.4090 3.0000 36.0025

no of output/selected points: 73, total poss. pixels: 0

selected data: mean std.dev. min max

no: 1 -13.59 2.22 -16.90 -9.30

no: 2 0.52 0.67 0.30 3.00

The output from the program explains how many records are found inside the area, their mean value and standard-deviation. Type head gra.ex6 in order to see the first lines. Note the identification numb er, which identifies the source of the data. The sources are found in the file gradoc.

Check whether the above calculated free-air anomaly agrees with the published values, i.e. has a value close to the one measured.

Use geoplot21 for viewing the values on a map, cf. Exercise 2. The values are put on the map by indicating that there are 3 data-elements, and that we want element no. 2 to be shown. (The input file from exercise 2 can be used as inspiration. Remember to change boundaries, name of data-file, coast-line).