Proceedings of Research Works 1998 250 66 Zdiby 98, tel. 02/685 7375, fax: 02/685 7056, e-mail: odis@vugtk.anet.cz

Abstract. The purpose of this paper is to discuss apriori estimates (bounds) for the disturbing potential and its part representing higher degree harmonic components. An estimate was also obtained for a horizontal component of the gradient of the disturbing potential. The disturbing potential was treated as a solution of the geodetic boundary value problem and the estimates are considered in terms of the norm assigned to Lebesgue's space of square integrable functions on the boundary of the solution domain.

Abstrakt. Článek pojednává o apriorních odhadech (mezích) pro poruchovś potenciál a jeho çást representující vyšší harmonické komponenty. Odhad byl získán i pro horizontální komponentu gradientu poruchového potenciálu. Poruchovś potenciál je chápán jako řešení geodetického okrajového problému a odhady jsou uvaĹovány ve smyslu normy v Lebesgueovł prostoru funkcí integrovatelnśch s druhou mocninou na hranici oblasti řešení.

Introduction

Geodetic computations mostly have a quantitative character and a numerical realization of various quantities or functions represents a target to be achieved. Nevertheless any computation and also the computation of the (quasi-)geoid has also its functional-analytic aspect bearing a qualitative nature.

For illustration recall Stokes' problem in the exterior S_R of a sphere of radius R, i.e. in S_R ş {x Î R³; |x| > R}. Here |x| = (ĺ_{i = 1}³x_i²)^1/2 and x_i, i = 1,2,3, are rectangular Cartesian coordinates in Euclidean three-dimensional space R³. The solution of the problem is sought as a function T which is harmonic in S_R and such that

T(x) = T1 ć ç č x |x| ö ÷ ř + Ą ĺ n = 0,n 1  R n-1  Dgn ć ç č x |x| ö ÷ ř  ,

(3)

Let now ¶S_R be the boundary of S_R. Under the above notation we easily deduce that

ó ő ¶SR  T2 dS Ł R2 Ą ĺ n = 0,n 1  ó ő ¶SR  (Dgn)2 dS = R2 ó ő ¶SR  (Dg)2 dS

(4)

In case that a part of the solution represented by low degree harmonics is known (e.g. from global geopotential models), we can split T as T = T˘+ T˘˘, where

||dT˘˘||_L2(¶SR) Ł (R/N)||d(Dg)||_L2(¶SR), where dT˘˘ and d(Dg) are the incremental changes of T˘˘ and Dg, respectively. Recall, however, that similarly as above we have to suppose that d(Dg₁) = 0.

Finally, putting [ . ] = w^-1/2|| . ||_L2(¶SR), where w = 4pR² means the area of the sphere ¶S_R, we also have

[ dT˘˘ ] Ł (R/N) [ d(Dg) ] .

Example 1. The preceeding inequality contains some parameters. Numerically, taking e.g. [ d(Dg) ] = 10^-5 ms^-2 (i.e. 1 mgal), R = 6378 km and N = 100, we obtain that [ dT˘˘ ] Ł 0.64 m²s^-2. Moreover, dividing this inequality by g = 9.8 ms^-2, we arrive at [ dT˘˘ ]/g Ł 0.06 m, which enables us to interpret our apriori estimate of [ dT˘˘ ] in a more telling way, i.e. in terms of geoid undulations.

In reality (Molodensky's problem) the solution domain does not coincide with the exterior of the sphere and in the boundary condition the direction of the derivative is not perpendicular to the boundary. In consequence the orthogonal decomposition of T (i.e., T = T˘+ T˘˘) and of the right hand side Dg as used in (6) does not hold anymore. A more general approach is needed.

Let U stand for the potential of the normal gravity of the Earth and DW and Dg be the potential and the (vectorial) gravity anomaly, respectively. These anomalies are related to U and to an adopted model of the Earth surface. We will denote the exterior of this model by W and recall that for the boundary ¶W of W we usually use the term ``telluroid''. Moreover, we will assume that W˘ = R³ - [` (W)] (where [`(W)] means the closure of W) is a starshaped domain at the origin with the Lipschitz boundary such that áx,nń > 0 for almost all x Î ¶W and n denoting the outer unit normal of ¶W, see (Holota, 1996; 1997a,b).

Now we are ready to write that in general the problem is to find T which is harmonic in W and such that

h = - [M_ij]^-1grad U, provided that for

[M_ij] = [¶²U/¶x_i¶x_j] and x Î ¶W the Hessian: det [M_ij(x)] 0 . Note that for the standard choice of U the vector h is close to x/2.

As is well-known for an arbitrary constant vector c the function u = ác,grad Uń is harmonic in W and represents a non-trivial solution of (9) in case of f = 0. Recall, however, that this solution is ruled out by the following asymptotic condition

s = háh,nń^-1 , c = áh,nń^-1 , g = - fáh,nń^-1 , provided that áh,nń^-1 0 on ¶W, cf. (Holota, 1996, 1997a). Under this notation (9) turns into

Horizontal Component of the Gradient

In geodesy and in a number of related areas (e.g. in inertial navigation) we frequently need also information concerning the regularity of T along the boundary of S_R. In other words we need an estimate of

As a starting point we use the following identity

A Non-Spherical Case

Let T be a harmonic function in W such that it satisfies the boundary condition (9)˘. Suppose that S_R ş {x Î R³; |x| > R} is contained in W. Now in analogy to the spherical case, we split T into two parts: T˘, represented by a finite sum of low degree harmonics and T˘˘ which is harmonic in W and such that asymptotically

Thus T = T˘+ T˘˘. Assuming now that T˘ is known, we try to estimate T˘˘ in terms of the norm || . ||_L2(¶W) of the space L₂(¶W), i.e., our aim is to find an upper bound for

For this purpose we multiply equation (9)˘ by T˘˘ and integrate it over ¶W. We obtain

Remark 1. For the harmonic T˘˘ (24) represents a generalized version of Green's identity. Indeed, for s = n we have L = 0 and in this case (24) is a consequence of the usual Green's identity. In (Holota, 1996, 1997b) we can also find a geometrical interpretation of L.

For ((T˘˘,T˘˘)) we now have

K = - c- L/2 . In the sequel we will need a lower estimate of ((T˘˘,T˘˘)) and an upper estimate for the right hand side in (22).

A Lower Estimate

Putting D = W- [`(S)]_R, we can write

where

x Î ¶W

Here we suppose that in computing the essential supreme value we deal with a Lebesgue measurable function defined and bounded almost everywhere on ¶W. (Loosely speaking, we can say that k does not reflect extremes achieved on sets of a zero Lebesgue measure.) In addition from Green's identity, we deduce that

Our problem now is to examine (((T˘˘,T˘˘))). In S_R we expand T˘˘ in spherical harmonics, i.e.,

T˘˘(x) = ĺ_{n = N+1}^Ą(R/|x|)ⁿ⁺¹T˘˘_n(x/|x|), where the summation starts with n = N + 1 in view of (20). We know that

N + 2 - kR² > 0 and putting C(N) = (N + 2 - kR²)/R, we arrive at

Remark 2. In (Holota, 1996) the existence, uniqueness and stability of the solution of our oblique derivative boundary value problem have been proved for 3 - kR² > 0. Thus N + 2 - kR² > 0 for N > 1 is even a less restrictive assumption.

The last inequality together with (31) now yields

An Estimate of the Solution

As already mentioned we need an upper estimate of the right hand side of (22). It follows immediately from Hölder's inequality. Indeed,

An Improved Estimate of an Approximate Solution in a Finite Dimensional Space

Inspecting our computations, we can see that in (37) we have not made full use of the favourable value of the factor C(N). Indeed, the coefficients in (30) are not well balanced with C(N) and (1 - kR²)/3 in (36). Therefore, our aim is to modify (30).

In (29) we first try to estimate ň_D(T˘˘)² dx by means of ň_¶SR(T˘˘)² dS. This, however, is associated with the non-stability of the downward continuation problem. The estimate maybe obtained, but for T˘˘ smoothed up to a certain degree only. In order to smooth T˘˘ we will confine ourselves to a finite range of its spherical harmonic components, i.e., in D we will approximate T˘˘ by means of

T_n^* = (R/R₀)ⁿ⁺¹T˘˘_n , where R₀ is a greatest radius such that D is contained in S_R0 ş {x Î R³; |x| > R₀}. Thus,

In order to make the situation more transparent we give below several values of f(q,n) for R = 6378 km and R₀ = 6356 km, i.e. for q = 0.9965506 :

Inspecting the table of f(q,n) quickly and taking e.g. n_max = 1000, we can put c^* = 0.016 R. Hence for the mentioned parameters

Acknowledgements. The work on this paper has been supported by the Grant Agency of the Czech Republic through Grant No. 205/96/0956. This support is gratefully acknowledged.

References

Holota P. (1996). Variational methods for quasigeoid determination. In: Tziavos, I.N. and Vermeer, M. (eds.), Techniques for Local Geoid Determination, Proc. Session G7, EGS XXI Gen. Assembly, The Hague, The Netherlands, 6-10 May, 1996, Reports of the Finnish Geodetic Inst. No. 96:2, Masala, 1996, pp 9-21.

Holota P. (1997a). Variational methods for geodetic boundary value problems. In: Sansò, F. and Rummel R. (eds.), Geodetic Boundary Value Problems in View of the One Centimeter Geoid, Lecture Notes in Earth Sciences, Springer, Berlin etc., pp 468-481.

Holota P. (1997b). Coerciveness of the linear gravimetric boundary-value problem and a geometrical interpretation. Journal of Geodesy 71: pp 640-651.

Holota P. (1997c). Apriori estimates of the solution of the geodetic boundary value problem. Int. Assoc. of Geodesy Symposia, Springer, Berlin etc. (accepted, 1997).

Hörmander L. (1975). The boundary problems of physical geodesy. The Royal Inst. of Technology, Division of Geodesy, Stockholm, 1975; also in Archive for Rational Mechanics and Analysis 62(1976), pp. 1-52.

Moritz H. (1977). Recent developments in the geodetic boundary-value problem. Reports of the Dept. of Geod. Sci., Report No. 266, The Ohio State Univ., Columbus.