The Local Modelling of the Gravity Field by Collocation

Abstract

The chapter aims at solving the problem of estimating the residual anomalous potential T _r from all available information, in particular in a certain area. Remember that here residual means that the long wavelength part as well as the short wavelength part of T have been at least reduced by means of the deterministic modelling described in Chaps. 3 and 4.

Appendix

1.1 A.1

We want to prove the relation (5.98), sending the interested reader to the literature Moritz (1980) and Sansò (1986) for the distribution of the vector T(ω).

We have

$$\begin{array}{rcl} \sum\limits_{m=-n}^{n}{T}_{ nm}^{2}(\omega )& =& \frac{1} {{(4\pi )}^{2}} \int d{\sigma }_{P}T({R}_{\omega }P)\int d{\sigma }_{Q}T({R}_{\omega }Q) \\ & & \cdot \sum\limits_{m=-n}^{n}{Y }_{ nm}({{\vartheta}}_{P},{\lambda }_{P}){Y }_{nm}({{\vartheta}}_{Q},{\lambda }_{Q}) \\ & =& \frac{2n + 1} {{(4\pi )}^{2}} \int d{\sigma }_{P} \int d{\sigma }_{Q}T({R}_{\omega }P)T({R}_{\omega }Q){P}_{n}(\cos {\psi }_{PQ}) \\ & =& \frac{2n + 1} {{(4\pi )}^{2}} \int d{\sigma }_{P \prime } \int d{\sigma }_{Q \prime }T(P \prime )T(Q \prime ){P}_{n}(\cos {\psi }_{P \prime Q \prime })\ ;\end{array}$$

(5.214)

the last equality is justified because ψ _PQ  = ψ _P′Q′ and the double integral over the sphere can be performed with any angular coordinates giving always the same result.

Now we organize the double integral in (5.214) as follows; first fix P′ and let Q′ circulate around P′ at a distance ψ _P′Q′  = ψ; then integrate in dσ _P′ ; then we finally let ψ to vary from 0 to π. We get, putting dσ _Q′  = sinψdψdα into (5.214), recalling also the definition (5.38) and using (5.94),

$$\begin{array}{rcl} \sum\limits_{m=-n}^{n}{T}_{ nm}^{2}(\omega )& =& \frac{(2n + 1)} {2} \int\limits_{0}^{\pi }d\psi \sin \psi {P}_{ n}(\cos \psi ) \cdot \\ & &\cdot \frac{1} {8{\pi }^{2}} \int d{\sigma }_{P \prime }\int\limits_{{\psi }_{P \prime Q \prime }=\psi }T(P \prime )T(Q \prime )d{\alpha }_{Q \prime } \\ & =& \frac{2n + 1} {2} \int\limits_{0}^{\pi }d\psi \sin \psi {P}_{ n}(\cos \psi )C(\psi ) = {c}_{n},\end{array}$$

(5.215)

as it was to be proved.

1.2 A.2

We want to prove formula (5.156), providing the explicit form of K _B (s, t) and K _− 2(s, t). We first expand (5.157) into the sum of fractions, with the identity

$$\begin{array}{rcl} \frac{n - 1} {(n - 2)(n + B)} \equiv \frac{B + 1} {B + 2} \frac{1} {n + B} + \frac{1} {B + 2} \frac{1} {n - 2}& & \\ \end{array}$$

so that we can write

$$\begin{array}{rcl}{ C}_{\Delta g\Delta g}(s,t)& =& A\left \{\frac{B + 1} {B + 2}\sum\limits_{n=3}^{+\infty } \frac{{s}^{n+2}} {n + B}{P}_{n}(t) + \frac{1} {B + 2}\sum\limits_{n=3}^{+\infty }\frac{{s}^{n+2}} {n - 2}{P}_{n}(t)\right \} \\ & =& A\left \{\frac{B + 1} {B + 2}{K}_{B}(s,t) + \frac{1} {B + 2}{K}_{-2}(s,t)\right \} \end{array}$$

(5.216)

We compute at first the last term:

$$\begin{array}{rcl}{ K}_{-2}(s,t)& =& {s}^{4}\sum\limits_{n=3}^{+\infty }\frac{{s}^{n-2}} {n - 2}{P}_{n}(t) \\ & =& {s}^{4}\int\limits_{0}^{s}\sum\limits_{n=3}^{+\infty }{\sigma }^{n-3}{P}_{ n}(t)d\sigma \\ & =& {s}^{4}\int\limits_{0}^{s} \frac{1} {{\sigma }^{3}}\left \{\sum\limits_{n=0}^{+\infty }{\sigma }^{n}{P}_{ n}(t) - 1 - \sigma t - {\sigma }^{2}{P}_{ 2}(t)\right \}d\sigma \\ & =& {s}^{4}\int\limits_{0}^{s} \frac{1} {{\sigma }^{3}}\left \{G(\sigma ,t) - 1 - \sigma t - {\sigma }^{2}{P}_{ 2}(t)\right \}d\sigma \\ & =& \frac{{s}^{2}} {2} [1 + 2ts - (3ts + 1){G}^{-1}(s,t)] - {s}^{4}{P}_{ 2}(t)\log \frac{1 - st + {G}^{-1}(s,t)} {2} \\ & & +{s}^{4}\frac{7{t}^{2} - 1} {4}. \end{array}$$

(5.217)

The last integral is calculated with the help of mathematical tables adjusting the integration constant in such a way that both members of (5.217), multiplied by s ^− 4, tend to 0 when s tends to 0. As for the first term one writes, assuming B > 0,

$$\begin{array}{rcl}{ K}_{B}(s,t)& =& {s}^{2-B}\sum\limits_{n=3}^{+\infty }\frac{{s}^{n+B}} {n + B}{P}_{n}(t) \\ & =& {s}^{2-B}\int\limits_{0}^{s}\sum\limits_{n=3}^{+\infty }{\sigma }^{n+B-1}{P}_{ n}(t)d\sigma \\ & =& {s}^{2-B}\int\limits_{0}^{s}{\sigma }^{B-1}\left \{\sum\limits_{n=0}^{+\infty }{\sigma }^{n}{P}_{ n}(t) - 1 - \sigma t - {\sigma }^{2}{P}_{ 2}(t)\right \}d\sigma \\ & =& {s}^{2-B}\int\limits_{0}^{s}{\sigma }^{B-1}G(\sigma ,t)d\sigma -\frac{{s}^{2}} {B} - \frac{{s}^{3}} {B + 1}t - \frac{{s}^{4}} {B + 2}{P}_{2}(t).\end{array}$$

(5.218)

Now the integrals

$$\begin{array}{rcl}{ I}_{B} =\int\limits_{0}^{s}{\sigma }^{B-1}G(\sigma ,t)d\sigma & &\end{array}$$

(5.219)

can be computed, for integer values of B, by exploiting a recursive relation, namely

$$\begin{array}{rcl}{ I}_{k+1} = \frac{{s}^{k-1}} {k} {G}^{-1}(s,t) + \frac{(2k - 1)} {k} t{I}_{k} -\frac{k - 1} {k} {I}_{k-1}& &\end{array}$$

(5.220)

which is derived from the identity

$$\begin{array}{rcl} \frac{\partial } {\partial s}\left [{s}^{k-1}{G}^{-1}(s,t)\right ] = \left [k{s}^{k} - (2k - 1)t{s}^{k-1} + (k - 1){s}^{k-2}\right ]G(s,t),& &\end{array}$$

(5.221)

integrating both members from 0 to s and re-arranging. In order to trigger (5.220) we need two initial values of I _k , for instance I ₁, I ₂. But I ₁ has already been given in (5.152) and I ₂ is easy to compute since, recalling (5.151),

$$\begin{array}{rcl}{ I}_{2}& =& \int\limits_{0}^{s}\sigma G(\sigma ,t)d\sigma =\int\limits_{0}^{s}(\sigma - t)G(\sigma ,t)d\sigma + t\int\limits_{0}^{s}G(\sigma ,t)d\sigma \\ & =& {G}^{-1}(s,t) - 1 + t{I}_{ 1}. \end{array}$$

(5.222)

The relations (5.216), (5.217), (5.220), (5.152) and (5.222) all together give the explicit form of the covariance function of Δg for every integer B. For a global use of this covariance the model (3.181) coming from the best fit of EGM08 degree variances between degrees 180 and 1,800, can be used, with the only warning that in (3.181) one has ${\overline{\sigma }}_{\mathcal{l}}^{2} = {c}_{\mathcal{l}}\left (\frac{T} {\gamma } \right )$, whereas we treat here c _n (Δg) related to the former by the relation

$$\begin{array}{rcl}{ c}_{n}(\Delta g) = \frac{{(\mathcal{l} - 1)}^{2}} {{\overline{R}}^{2}} {c}_{n}(T) = \frac{{(\mathcal{l} - 1)}^{2}} {{\overline{R}}^{2}}{ \left (\frac{GM} {{\overline{R}}^{2}} \right )}^{2}{\overline{\sigma }}_{ \mathcal{l}}^{2}.& &\end{array}$$

(5.223)

We notice by the way that also the improved model (3.178) transformed according to (5.223) can be added by applying exactly the same methods presented in the Appendix and the decomposition

$$\begin{array}{rcl} \frac{\mathcal{l} - 1} {(\mathcal{l} - 2)(\mathcal{l} + 4)(\mathcal{l} + 17)} = \frac{1} {114}\ \frac{1} {\mathcal{l} - 2} + \frac{5} {78} \frac{1} {\mathcal{l} + 4} - \frac{18} {247}\ \frac{1} {\mathcal{l} + 17}.& &\end{array}$$

(5.224)