A Method of Computing Fine Geoidal Heights from Vertical Deflections

Abstract

A rigorous program of adjustment for any geodetic network must have a useful file of geoidal heights in its own to adjust strictly observations by GPS for space vectors. The program, by which geoidal heights can be automatically computed without missing its fine undualtion at every point where vertical deflection is observed, was newly added to the Universal Program of geodetic net-adjustment. The conventional method to find geoidal heights successively along routes linking neighboring vertical deflection points seems not elegant mathematically, because selections of routes are too arbitrary. It ishighly mathematical to find out geoidal heights at all vertical deflection points badjusting simultaneously all observation equations of vertical deflections which are ex pressed as the function of their geoidal heights. Although it seems geophysically reason able that geoidal heights are expressed by a curved surface with the mathematical formula written by coordinates on the surface of the earth, we are afraid that it might miss local fine undulation of geoid. In order to adjust rigorously geodetic network including GPS-observations, it is desirable that every vertical deflection point has its more reliable geoidal height. Vertical deflection at a point is decided by a local shape of the geoid around it. Then we think a new method as follows : (1) We make a local geoid by using geoidal heights (rough height+small unknown correction) at several vertical deflection points inside a circle drawn around a vertical deflection point. (2) We make both observation equations for ξ (north-south component) and η (east-west component) of vertical deflection at thecentral point by considering the difference between geodetic longitude (latitude) and computed astronomical longitude (latitude) to be the direction of normal to the local geoid through the point. (3) We can find better geoidal heights at all vertical deflection points by solving simultaneously all observation equations by means of the method of least squares. How to derive a local geoid mathematically is the most important problem in abovementioned method. A curved surface with higher order terms requests many surrounding points. Reliability of local geoid made by using remote points deteriorates. Vertical deflection points are distributed locally in remarkable high and low densities. We adopted the following sophisticated way with much freedom after considering those peculiarities above. © 1989, The Geodetic Society of Japan. All rights reserved.