Understanding Adjustment from an Information Theoretic Perspective

It is tried to understand the adjustment from an information theoretic viewpoint. Meth‍ods: Besides the concepts of covariance matrix, cofactor matrix and weight matrix, which are often introduced in “Surveying Adjustment” courses, we introduce the concept of Fisher information matrix (or simply information matrix) in this paper.

Several important properties of the information matrix are shown; the population and the sample information matrices are distinguished. The information matrix under the assumption of multivariate normal distribution is derived; the links among the population/sample information matrix, the population/sample covariance matrix and the cofactor/weight matrix are explained; and it is revealed that the weight matrix can be viewed as a normalized information matrix. Based on the information matrix, the concept of information vector is introduced; the adjustment in information domain is derived in which the information matrix and the information vector are calculated rather than the parameter vector and the covariance matrix calculated in standard adjustment. Though equivalent with the standard adjustment in terms of the results, the information-domain adjustment has simpler format and clearer structure; this provides a new perspective for understanding the traditional adjustment; more importantly the information-domain method has special merits in solving under-determined model and also in initializing the recursive adjustment method. Finally, the dynamic adjustment in information domain is derived for filtering a dynamic state space model; it is nothing but the information filter whih is equivalent to the standard Kalman filter.

Information theory provides new insights for comprehensively and thoroughly understanding the surveying adjustment theory.