Statistical methods in physical geodesy - Physical Geodesy - page 339

9 Statistical methods in
physical geodesy

9.1 Introduction

Some of the most important problems of gravimetric geodesy are formulated
and solved in terms of integrals extended over the whole earth. An example
is Stokes' formula. Thus, in principle, we need the gravity g at every point
of the earth's surface. As a matter of fact, even in the densest gravity net
we measure g only at relatively few points so that we must estimate g at
other points by interpolation . In large parts of the oceans we have made no
observations at all; these gaps must be filled by some kind of extrapolation .
Mathematically, there is no difference between interpolation and extrap-
olation; therefore they are denoted by the same term, prediction .
Prediction (i.e., interpolation or extrapolation) cannot give exact values;
hence, the problem is to estimate the errors that are to be expected in the
gravity g or in the gravity anomaly ∆ g . As usual, gravity disturbances δg
are appropriately comprised whenever we speak of gravity anomalies.
Since ∆ g is further used to compute other quantities, such as the geoidal
undulation N or the deflection components ξ and η , we must also investigate
the influence of the prediction errors of ∆ g on N, ξ, η ,etc.Thisiscalled
error propagation , which will play a basic role.
It is also important to know which prediction method gives highest ac-
curacy, either in ∆ g or in derived quantities N, ξ, η , etc. To be able to find
these “best” prediction methods, it is necessary to have solved the previous
problem, to know the prediction error of ∆ g and its influence on the derived
quantities.
Summarizing, we have the following problems:

1. estimation of interpolation and extrapolation errors of ∆ g (or δg );
2. estimation of the effect of these errors on derived quantities ( N, ξ, η ,
etc.);
3. determination of the best prediction method.

Since we are interested in the average rather than the individual errors,
we are led to a statistical treatment. This will be the topic of the present
chapter.