Geodetic Boundary Value Problem: the Equivalence between by Fernando Sansò, Michael G. Sideris

By Fernando Sansò, Michael G. Sideris

This e-book bargains a brand new method of analyzing the geodetic boundary price challenge, effectively acquiring the strategies of the Molodensky and Stokes boundary worth difficulties (BVPs) with assistance from downward continuation (DC) dependent tools. even supposing DC is understood to be an improperly posed operation, classical tools appear to offer numerically brilliant effects, and as a result it may be concluded that such classical equipment needs to in reality be manifestations of alternative, mathematically sound approaches.

Here, the authors first turn out the equivalence of Molodensky’s and Stoke's ways with Helmert’s aid when it comes to either BVP formula and BVP suggestions through the DC procedure. They then pass directly to exhibit that this isn't basically a downward continuation operation, and supply extra rigorous interpretations of the DC strategy as a metamorphosis of boundary strategy and as a pseudo BVP answer approach.

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Additional info for Geodetic Boundary Value Problem: the Equivalence between Molodensky’s and Helmert’s Solutions

Example text

In other words, it means that we are in reality solving a different problem, which leads to a solution close to the correct one, or we solve a problem, leaving some errors that are reduced to zero by convenient iterations. The formalization of the above “true” problems solved in geodesy will be the object of following chapters. Here we concentrate on understanding the properties of the DC either through the masses or in free air, namely a space free of masses. Indeed this item has been treated in geodetic and geophysical literature.

12) Pc and normal gravity γ (Pc ), we obtain using Eq. 13) 30 4 On the Equivalent BVPs of Stokes and Helmert … and therefore the equation to compute the Helmert gravity on the co-geoid from the measured gravity values is δV H ∂2T H (P) − γ H P − γ δ N H + δh ∂h 2 g H (Pc ) = g(P) + H˜ P . 14) Pc It should be noted here that, perhaps more appropriately, it is possible to derive at the same time the boundary condition in Eq. 9) with the Helmert gravity of Eq. 14) in its right-hand-side. To do this, we start by taking the vertical gradient of Eq.

19) or, equivalently, in Eq. 35): N (Po ) = N H (Pc ) + δ N (Pc ) = 1 ∂Δg H S{Δg H − γ ∂h Pc δV H (Pc ) . 42) We note that the “downward-continued” Helmert Molodensky solution of Eq. 41) is formally equivalent to the Helmert Stokes solution of Eq. 42) but the gradient of Δg H needs to be computed at different surfaces, namely the Earth’s surface and the co-geoid, respectively. Remark. If the more usual case of continuation to the geoid instead of the co-geoid is used, then the solution of the two ‘Helmertised’ BVPs will still be given by Eqs.

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