Stability investigations of a discrete downward continuation problem for geoid determination in the Canadian Rocky Mountains

We investigate the stability of a discrete downward continuation problem for geoid determination when the surface gravity observations are harmonically continued from the Earth's surface to the geoid. The discrete form of Poisson's integral is used to set up the system of linear algebraic equations describing the problem. The posedness of the downward continuation problem is then expressed by means of the conditionality of the matrix of a system of linear equations. The eigenvalue analysis of this matrix for a particularly rugged region of the Canadian Rocky Mountains shows that the discrete downward continuation problem is stable once the topographical heights are discretized with a grid step of size 5 arcmin or larger. We derive two simplified criteria for analysing the conditionality of the discrete downward continuation problem. A comparison with the proper eigenvalue analysis shows that these criteria provide a fairly reliable view into the conditionality of the problem.The compensation of topographical masses is a possible way how to stabilize the problem as the spectral contents of the gravity anomalies of compensated topographical masses may significantly differ from those of the original free-air gravity anomalies. Using surface gravity data from the Canadian Rocky Mountains, we investigate the efficiency of highly idealized compensation models, namely the Airy-Heiskanen model, the Pratt-Hayford model, and Helmert's 2nd condensation technique, to dampen high-frequency oscillations of the free-air gravity anomalies. We show that the Airy-Heiskanen model reduces high-frequencies of the data in the most efficient way, whereas Helmert's 2nd condensation technique in the least efficient way. We have found areas where a high-frequency part of the surface gravity data has been completely removed by adopting the Airy-Heiskanen model which is in contrast to the nearly negligible dampening effect of Helmert's 2nd condensation technique. Hence, for computation of the geoid over the Canadian Rocky Mountains, we recommend the use of the Airy-Heiskanen compensation model to reduce the gravitational effect of topographical masses.In addition, we propose to solve the discrete downward continuation problem by means of a simple Jacobi's iterative scheme which finds the solution without determining and storing the matrix of a system of equations. By computing the spectral norm of the matrix of a system of equations for the topographical 5 × 5 heights from a region of the Canadian Rocky Mountains, we rigorously show that Jacobi's iterations converge to the solution; that the problem was well posed then ensures that the solution is not contaminated by large roundoff errors. On the other hand, we demonstrate that for a rugged mountainous region of the Rocky Mountains the discrete downward continuation problem becomes ill-conditioned once the grid step size of both the surface observations and the solution is smaller than 1 arcmin. In this case, Jacobi's iterations converge very slowly which prevents their use for searching the solution due to accumulating roundoff errors.