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.
Expanded third-order Markov undulation model
An expanded form of the third-order Markov undulation model is developed which includes undulation of the geoid, vertical deflections, gravity anomaly, and the anomalous gravity gradients. First, the complete set of autocorrelation and cross-correlation functions, as specified by the autocorrelation of undulation, is obtained. Then, a time-domain approach is used to synthesize the corresponding linear system driven by white noise that represents the anomalous gravity field behavior along a straight horizontal track. The resulting nine-state Markov model provides a shaping filter for applying covariance analysis and optimal estimation techniques to inertial navigation and gravity gradiometry systems analysis. ONTINUED mechanical refinement has brought inertial navigation technology to the stage where unmodeled variations in the Earth's gravity field have become significant and sometimes dominant error sources.1 At the same time, inertial navigation system analysts have been facing the task of developing error models to evaluate the impact of gravity uncertainties on navigation system performance. Ad- ditionally, the development of gravity gradiometers has made it necessary to model gravity-gradient signals to specify sensor requirements and analyze gradiometer-aided inertial navigation or gradiometry survey performance. To apply the powerful tools of linear system theory and Kalman filtering, it has become common to consider gravity uncertainties random variables, and to model them as Markov processes. Following the first treatment of vertical deflections as a first-order Markov process,2 a succession of higher order Markov models was proposed culminating with the third- order Markov undulation model3'4 which is fully self- consistent; that is, the auto- and cross-correlation functions of undulation, vertical deflections, and gravity anomaly obey and are derived according to the mathematical relationships of physical geodesy. The motivation for using Markov models for gravity un- certainties is that they can be cast in the form of linear dif- ferential equations driven by white noise. This allows use of the model as a shaping filter in covariance analysis of linear dynamic systems, Kalman filters, and optimal smoothing algorithms. The expanded third-order Markov undulation model described in this paper is derived from the original version, but it has three differences. First, it was found that the original model could easily be extended to include all the anomalous gravity gradients by continued differentiation of auto- and cross-correlation functions. Second, the exact cross-correlation function between undulation and gravity anomaly contains modified Bessel functions.4 Modified Bessel functions are solutions to a class of differential equations whose coefficients are functions of the independent variable. To obtain a Markov shaping filter, it is necessary to substitute an approximate function which is the solution to a linear differential equation with constant coefficients. While the chosen exponential form did not approximate the exact Bessel function as well as at first hoped possible, it provides a suitable representation of the un- dulation-gravity anomaly cross-correlation.