ABSOLUTE ORIENTATION OF INDIAN GEODETIC SYSTEM BY GRAVIMETRIC METHOD

Abstract

More than a century ago, E/erest derived the dime nsions of a spheroid on which the Indian Geodetic System was based. Its orientation at the Kalianpur origin of the Indian triangulation system has been arbitrarily chosen at various times. Local fitting of the spheroid could lead to confli cting claims by neighbouring countries in the definition of their national boundaries. The absolute orientation of the geodetic system is therefore a prerequisite for the readju stment of the Indian triangulation net for use as a global geodetic system. The present work is the first long - awaited attempt to redefine the values at the orientation parameters at the initial point, with reference to the CiRs'67 Geodetic Reference System, 1967, by determining their absolute geocentric values. The classical gravimetric principle has been used as the principal tool to accomplish the task. The well - ' known Stokes' formula relates the gravity anomalies over the entire surface of the earth to the undulation of the geoid above a geocentric reference spheroid, as a solution of the third boundary-value problem of the earth's gravi tational potential. The Vening Meinesz' expressions sim ilarly provide the meridional and the prime vertical -XX11- oomponents of the deviations of the vertical. The results of these three global integrations of known gravity anoma lies weighted by functions of the spherical distance, are compared to the corresponding astro-geodetic values exist ing in terms of the local system to arrive at the required correction parameters. The irregularity of the gravity field over the earth's surface precludes the functional evaluation of the geoidal undulations, necessitating numerical discrete summation. The spherical surface is accordingly partitioned by finite elements with representative mean values of grav ity anomalies expressed over them. The grid divisions have been adopted in this work as being well-suited for automa tic computations. Furthermore, the nature of the Stokes' and Vening Meinesz' functions suggests that coarser grids may be used in the exterior regions without seriously affecting the accuracy of computation as long as compar atively finer meshes are used in the region of interest. Five-degree Squal-Area-Blocks have been used in the outer region, and further subdivisions of 1 , 0 .25, 0 .05, 0°.01 have been suggested for the interior region. The first part of the computational work started with evaluations of the contribution of a recent set of five-degree mean free-air gravity anomalies, extending -XX111- beyond a considerable margin around India. To suit machine evaluations on a digital computer, a number of analytical schemes have been developed in the formulation, such as, (a) matrix form of interstation separation, (b) non-dimensional forms of surface area and anomalies, (c) modification of the Vening Meinesz' function and rearrangement of the functions in algebraic forms. The geoidal parameters have been evaluated at the five-degree grid corners covering India and presented as an intermediate bye-product of the present investigation which may be useful for further work. The undulation ranges from -13 metres to -22 metres, whereas the deviation compo nents smoothly vary between +1". A bicubic spline inter polation technique was used to compute the values at any desired point. The next smaller size of mesh used is the onesquare degree Meridian-Parallel-Grid type unit. The availa ble data are nearly complete and updated. Gaps in farther areas have been filled up by a simplified procedure, keeping consistency of the average value over a block. For nearby unrepresented units, ho\\rever, a loeal covariance interpo lation has been used. The weighting functions which incr ease with decrease in distance, have been further norma lized to minimize Inaccuracies caused by exploding terms. After developing working formulae for computations from -XXIVgridded data, the partial geoid parameters have been comp uted at 1° corners within India using the one-degree mean free-air anomalies covering the interior region. The profil es are seen to be mutually consistent, whereas the slope components vary sharply. A combination of the void geoid and the partial geoid gives a pictorial representation of the one-degree mean free-air geoid in India. The variation is from -1+0 metres to -85 metres, with geoidal lows Vin the Himalayan region and in the Southern peninsula. The last part of the main objective has been acco mplished by completing the numerical algorithm using denser gravity details in the immediate neighbourhood of the com putation point, which includes a further modification of the intorstation vector to a differential expression. In order to estimate quarter-degree mean anomalies from point observations, simple average and patchwise surface-fitting have been used. For finer mesh sizes, a truncated pyramid window has been proposed. A few existing and suggested techniques for the evaluation of the effect of the inner most zone have-also been enumerated. The numerical work consists of using modified terrain-corrected free-air anomalies around the initial point for further precision in the determination. The final results obtained are, -XXV- 6 N = -59-0 metres, 6 £ = +0.65 arcsecond, 'O 5 t] = +2.6C arcseconds. Although the orientation through the initial point itself provides the most reliable and stable positioning, the formulation of the gravimotric method permits any other astro-geodetic station also to be considered as a computa tion point. A first-order triagulation station with a commendable distribution of gravity coverage all around, may even act as a supercontrol point. An invariant shift vector has been introduced to further generalize the proce dure and four zones at four geographical corners in India have been chosen for test computations. The limitations of availability and measurements of gravity data called for filling up some compartments of surrounding regions by prediction, for which a truncated local, covariance inter polation has been used. Despite all the defects and appr oximation in these stations, the various sets or orientation parameters provide consistent numerical checks. The variat ions of results between themselves as well as with those obtained at the initial point are of the order of 3 metres in 6N and 1" in 6£ or &ti , which are a little too o o o7 high for obvious reasons. An alternative proposition to obtain the absolute -XXV1- orientation parameters from the regional gravity data and the astro-geodetic geoid, forms the subject matt er of a subsequent chapter. The inherent errors in the available informations being of fluctuating nature, a practical solution of the orientation problem may be achieved by the logic of minimization of their noncoincidence in a least-squares sense. The matching of undulations seems preferable to the parallelism cond ition, and the shift vector formulations are further modified to simpler expressions. The existing astro geodetic geoid has been converted to one corresponding to the GRS 67 spheroid without changing the present orientation. The comparison of its undulations at some points with those of the gravimetric geoid obtained from one-degree mean free-air anomalies are made to frame condition equations and consequent normalization to yield optimal estimates of orientation parameters. The results differ by 1 metre in 6NQ , 1 m 6^0 and l" ^ in 6*1 from the gravimetric results at the -' o origin, showing thereby the possibilities of the exercise for further refinement. With a view to formulating an integrated strategy to tackle the orientation problem, another plausible solution without requiring the use of any gravity data directly, has been tested in this work. -xxvii- Similar to the astro-gravimetric geoid matching attem pted earlier ,the astro-geodetic geoid heights in this case have been compared with those obtained from the satellite-derived geopotential coefficients, on assu ming the apparent misfit to be solely duo to the local non~geocentric orientation of the former. To obtain a smoothened geoid a 7th-order surface has been fitted using a number of astro-geodetic deviations, and its comparison with the present geoid shows an average discrepancy of 3 to\ metres, the difference getting progressively increased with the distance from the origin. The other geoid is computed from the recent GSM 10 coefficients. Whilst the results obtained rev eal that further work is necessary in this regard to achieve a reliable solution, the present work contri butes all necessary formulations including the various recursion relations to optimize computer economy, which will be useful for future researchers. The concluding part of the thesis summarizes various outputs of the methods adopted in the present work and compares them among themselves as well as with the datum shift values supplied by the satellite research organizations in respect of their adopted ell ipsoids. All the sets fall within the reasonable limits -xxvuiof accuracy and the three alternative methods, viz., (I) the general astro-geodetic orientation, (ii) the least-squares coincidence approach, and even (iii) the astro-satellite matching provided quite useful checks. The linear shift components obtained from the pre sent determinations are, AX = 2^-3 metres, AY = 733 metres, AZ = I7J+ metres. The corrections to the existing geoidal heights, latitudes and longitudes, have been presented in functional, digital as well as graphical forms. Finally, the various contributions of the study have been enumerated to deline ate the scope of future advancement and further studies In this field.