A Simulation Study Of the Overdetermined Geodetic Boundary Value Problem Using Collocation
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Date
1989-03
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Ohio State University. Division of Geodetic Science
Abstract
The overdetermined geodetic boundary value problem is the object of this study. The problem is defined in general by the Laplacian and at least two different boundary conditions holding on a spherical boundary, or overlapping parts of it. The least-squares collocation method is applied to estimate spherical harmonic coefficients for the disturbing potential. An algorithm was developed and the methodology was tested using simulated gravity anomaly and undulation signals. Point as well as mean boundary values were used, both with and without random noise added. The OSU86F spherical harmonic coefficient set was used to generate the simulated data and the required covariances. By computing statistics of the input field recovery the applicability of the methodology is judged. Several problems have been encountered while applying least-squares collocation to this problem. A theoretical singularity of the covariance matrix, caused by the truncation in the summation of the covariance function at a finite degree, is found to be eliminated numerically when summing to degree 180. Instability and singularity due to the data distribution are treated successfully by applying Tikhonov regularization. Inversion of a large covariance matrix resulting from global data coverage limits the feasibility of the solution. To make the computation manageable the block-Toeplitz pattern of the auto-covariance matrices is exploited in forming and inverting them. A partitioned inversion procedure assists the assimilation of the influence of overlapping datasets into the inverse of the covariance matrix. Numerical experiments were conducted using a global mean gravity anomaly dataset and four additional mean undulation datasets referring to 10° and 5° equiangular blocks. With errorless data and a regularized auto-covariance matrix, the error of the estimated solutions to degree 18 and 36 increases with degree, ranging from 0.5% to a maximum of 10% for the first half of the spectrum, and reaches 40% near the Nyquist frequency. The influence of the individual data types is investigated through several one-data-type solutions. Incorporating data error into the solutions estimated from mean gravity anomalies increased the average error of the estimated coefficients by 25% and 30% for expansions to degree 36 and 18, respectively. Corresponding figures for the undulation solutions were on the order of 3%. The discretization error is studied by estimating expansions to degrees 18, 36, 60 and 90 from mean data referring to 10°, 5°, 3° and 2° equiangular blocks. When using 2° instead of 10° block data, the average error of the estimated coefficients to degree 18 is reduced by 95% for the anomaly solutions and 80% for the undulation solutions. Errorless mean anomaly data, of all block sizes, were used to estimate coefficients with an error of less than 1% for the low degrees, approaching the value of 10% for coefficients half the Nyquist frequency and reaching 40% near the Nyquist frequency. On the other hand, the undulation data recover the low degree coefficients with an error of 10% to 20%, increasing with degree, reaching 90% near the Nyquist frequency.