An improved Molodensky-type modification of the Stokes formula and reduced bias in the spectral combination and variance-minimizing ULSM

Abstract

A geoid is a reference surface for orthometric height computed by the Stokes integral, and the error caused by truncating the integral into a limit area can be regarded as a bias. Early on, Molodensky modified the Stokes formula by decreasing the numerical size of the truncation coefficient to reduce the bias, and Evans and Featherstone accelerated the bias convergence. In comparison, spectral combination and least-squares modification (LSM) account for terrestrial gravity and Earth gravity model (EGM) errors as a variance in modifying the Stokes formula. However, the spectral combination and unbiased LSM (ULSM) poorly consider the bias when minimizing the variance, and this issue can be overcome by introducing the two bias-reducing methods. A theorem, proven in this study, accelerates the convergence of the truncation coefficient and improves the Molodensky-type modification. The proposed spectral combination with reducing bias (SCRB) and variance-minimizing ULSM with reducing bias (VURB) mitigate the errors of terrestrial gravity and EGMs, and the bias converges faster by the theorem. Considering the internal correlation of gravity anomaly error, the regional solution is also derived for SCRB and VURB for local geoid determination. Additionally, two regularizers introduced in this study avoid the ill-conditioned problems of the coefficient matrices.

Appendix: Proof of the theorem

Proof

Definition the of operator.

$$\begin{array}{c}{\nabla }^{2}=\frac{\mathrm{d}}{\mathrm{d}x}\left[\left(1-{x}^{2}\right)\frac{\mathrm{d}}{\mathrm{d}x}{S}_{m}\left(x\right)\right]\end{array}$$

(41a)

and ${\nabla }^{m}{S}_{m}\left(x\right)={\nabla }^{m-2}{\nabla }^{2}{S}_{m}\left(x\right)$, so we can get ${\nabla }^{2}{P}_{n}\left(x\right)={\left[\left(1-{x}^{2}\right){P}_{n}^{\prime}\left(x\right)\right]}^{\prime}=-n\left(n+1\right){P}_{n}\left(x\right)$ and ${\nabla }^{m-1}{S}_{m}\left(t\right)={\nabla }^{m-1}{S}_{m}\left(-1\right)=0$ for $i=0,1,\dots ,m$. Thus

$$\begin{aligned}{\nabla }^{2k}{P}_{n}\left(x\right)& ={\nabla }^{2k-2}{\nabla }^{2}{P}_{n}\left(x\right)=-n\left(n+1\right){\nabla }^{2k-2}{P}_{n}\left(x\right)\\ &={\left[-n\left(n+1\right)\right]}^{k}{P}_{n}\left(x\right)\end{aligned}$$

(41b)

When $m$ is an odd number, integrating by parts produces

$$\begin{aligned} & {\int }_{-1}^{t}{\nabla }^{m+1}{S}_{m}\left(x\right){P}_{n}\left(x\right)\text{d}x \\ &\quad =-{\int }_{-1}^{t}\left(1-{x}^{2}\right){\left[{\nabla }^{m-1}{S}_{m}\left(x\right)\right]}^{\prime}{P}_{n}^{\prime}\left(x\right)\text{d}x\\ \\ &\quad ={\int }_{-1}^{t}{\nabla }^{m-1}{S}_{m}\left(x\right){\left[\left(1-{x}^{2}\right){P}_{n}^{\prime}\left(x\right)\right]}^{\prime}\text{d}x \\ &\quad={\int }_{-1}^{t}{\nabla }^{m-1}{S}_{m}\left(x\right){\nabla }^{2}{P}_{n}\left(x\right)\text{d}x\\ \\ &\quad={\int }_{-1}^{t}{S}_{m}\left(x\right){\nabla }^{m+1}{P}_{n}\left(x\right)\text{d}x \\ &\quad={\left[-n\left(n+1\right)\right]}^{\frac{m+1}{2}}{\int }_{-1}^{t}{S}_{m}\left(x\right){P}_{n}\left(x\right)\text{d}x\\ \\ &\quad={\left[-n\left(n+1\right)\right]}^{\frac{m+1}{2}}{Q}_{n}^{m}\end{aligned}$$

(42)

According to Taylor’s theorem,

$$\begin{array}{c}{S}_{m}\left(x\right)=\frac{{\left(x-t\right)}^{m+1}}{\left(m+1\right)!}\frac{{\mathrm{d}}^{m+1}S}{\mathrm{d}{x}^{m+1}}\left({t}_{0}\right)\end{array}$$

(43a)

where ${t}_{0}\in \left(t,x\right)$, we have

$$\begin{array}{c}{\nabla }^{m+1}{S}_{m}\left(x\right)=\frac{{\mathrm{d}}^{m+1}S}{\mathrm{d}{x}^{m+1}}\left({t}_{0}\right)\le \mathrm{max}\left\{\frac{{\mathrm{d}x}^{m+1}S\left(x\right)}{\mathrm{d}{x}^{m+1}}\right\}\end{array}$$

(43b)

Substituting Eq. (43b) into Eq. (42), we obtain

$$\begin{aligned}{Q}_{n}^{m}&={\left[-\frac{1}{n\left(n+1\right)}\right]}^{\frac{m+1}{2}}{\int }_{-1}^{t}{\nabla }^{m+1}{S}_{m}\left(x\right){P}_{n}\left(x\right)\text{d}x \\ & \le \frac{{I}_{n}\left(t\right)}{{n}^{m+1}} \mathrm{max}\left\{\frac{{\mathrm{d}x}^{m+1}S\left(x\right)}{\mathrm{d}{x}^{m+1}}\right\}\end{aligned}$$

(43c)

Therefore, the order of ${Q}_{n}^{m}$ is obtained:

$$\begin{array}{c}{Q}_{n}^{m}\sim O\left({n}^{-m-\frac{5}{2}}\right)\end{array}$$

(44a)

after using

$$\begin{aligned}{I}_{n}\left(t\right) & ={\int }_{-1}^{t}{P}_{n}\left(x\right)\text{d}x=-\frac{1}{n\left(n+1\right)}\left(1-{x}^{2}\right){P}_{n}^{\prime}\left(x\right)\\ & =O\left({n}^{-\frac{3}{2}}\right)\end{aligned}$$

(44b)

with ${P}_{n}^{\prime}\left(x\right)\sim O\left(\sqrt{n}\right)$.

When $m$ is an even number, integrating by parts yields

$$\begin{aligned}{\int }_{-1}^{t}{\nabla }^{m}{S}_{m}\left(x\right){P}_{n}\left(x\right)\text{d}x & ={\int }_{-1}^{t}{S}_{m}\left(x\right){\nabla }^{m}{P}_{n}\left(x\right)\text{d}x \\ &={\left[-n\left(n+1\right)\right]}^{\frac{m}{2}}{Q}_{n}^{m}\end{aligned} $$

(45a)

We obtain

$$\begin{aligned}{Q}_{n}^{m}&={\left[-\frac{1}{n\left(n+1\right)}\right]}^{\frac{m}{2}}{\int }_{-1}^{t}{\nabla }^{m}{S}_{m}\left(x\right){P}_{n}\left(x\right)\text{d}x \\ &={\left[\frac{-1}{n\left(n+1\right)}\right]}^{\frac{m}{2}}{\int }_{-1}^{t}{\nabla }^{m}{S}_{m}\left(x\right)\text{d}{I}_{n}\left(x\right)\\ & =-{\left[\frac{-1}{n\left(n+1\right)}\right]}^{\frac{m}{2}}{\int }_{-1}^{t}{\left[{\nabla }^{m}{S}_{m}\left(x\right)\right]}^{\prime}{I}_{n}\left(x\right)\text{d}x \\ &\le \frac{{J}_{n}\left(t\right)}{{n}^{m}} \mathrm{max}\left\{\frac{{\mathrm{d}x}^{m+1}S\left(x\right)}{\mathrm{d}{x}^{m+1}}\right\}\end{aligned}$$

(45b)

Therefore, the order of ${Q}_{n}^{m}$ is

$$\begin{array}{c}{\widetilde{Q}}_{n}^{m}\sim O\left({n}^{-m-\frac{5}{2}}\right)\end{array}$$

(46a)

After using

$$\begin{aligned}{J}_{n}\left(t\right)&={\int }_{-1}^{t}{\int }_{-1}^{t}{P}_{n}\left(x\right)\mathrm{d}x\text{d}x\\ &=-\frac{1}{n\left(n+1\right)}{\int }_{-1}^{t}\left(1-{x}^{2}\right){P}_{n}^{\prime}\left(x\right)\text{d}x\\ &=-\frac{2}{n\left(n+1\right)}{\int }_{-1}^{t}x{P}_{n}\left(x\right)\text{d}x\sim O\left({n}^{-\frac{5}{2}}\right)\end{aligned}$$

(46b)

with ${P}_{n}\left(x\right)\sim O\left(1/\sqrt{n}\right)$.

When $m=-1$, it satisfies ${Q}_{n}^{-1}={Q}_{n}$. The truncation coefficient can be estimated as follows:

$${Q}_{n}={\int }_{-1}^{t}S\left(x\right){P}_{n}\left(x\right)\mathrm{d}x\le {I}_{n}\left(t\right)\mathrm{max}\left\{S\left(x\right)\right\}\sim O\left({n}^{-\frac{3}{2}}\right)$$

Thus, ${Q}_{n}^{m}\sim O\left({n}^{-m-\frac{5}{2}}\right)$ also holds at $m=-1$.