A Modified Discrete Karhunen-Loeve Random Field Estimator
Based upon the modified Karhunen-Loeve model of the gravity disturbance vector proposed by us recently, a corresponding Karhunen-Loeve random field estimator is developed in this paper for a two-dimensional grid of gravity data in a local finite region.The new eigenvectors obey the "required" orthogonality relations on the chosen grid provided the KL expansion is now separated into odd and even integers. The closed-form solution of the estimator isthen obtained under the "diagonal" assumption for the gain coefficients. It is shown that this assumption, without which nosolution is possible, enables the estimator to reproduce the data atgrid points when the noise in the data is absent and the number of terms in the Karhunen-Loeve expansion are equal to the number of grid points.
Two-dimensional smoothing of a vector Laplacian random field with application to geodesy (Ph.D. Thesis abstr.)
A two-dimensional signal processing algorithm is developed to obtain smoothed estimates of a vector random field when observed in the presence of additive two-dimensional white noise. The vector random field consists of a two-dimensional scalar random field and its two orthogonal derivatives, each of which can be scaled arbitrarily by a constant. The signal model consists of the Laplacian operator acting on the two-dimensional scalar random field excited by two-dimensional white noise. Such a signal model that permits a nonstationary and nonisotropic random field is "spatially dynamic" as opposed to a static model specified by autocorrelation functions or power spectral densities. The model is also "bilateral" such that any given spatial point’s dependency upon all its neighboring points is preserved and no unnecessary limitation of causality is enforced on the two-dimensional system. To preserve the noncausality of the problem, the estimation algorithm must not order the two-dimensional data since there is no sense in which any data point can be said to be "prior" to any other data point. To this end, a two-dimensional smoothing algorithm is developed first in the continuous domain by making use of a two-dimensional Karhunen-Loeve expansion that makes no specific reference to either the geometry or the ordering of the parameter space.
Error Covariance Analysis of a Karhunen-Loeve Collocation Estimator
An error covariance analysis of a two-dimensional gravity compensation technique (KLC) employing a Karhunen-Loeve gravity disturbance model and the linear least-square collocation algorithm for its estimation is presented, without actually using any data. Its performance is compared with another gravity compensation technique (KLE), whose error covariance analysis was previously presented by Gupta. From the mismodeling analysis, KLC appears to be superior to KLE.