Canadian gravimetric geoid model 2010

Abstract

A new gravimetric geoid model, Canadian Gravimetric Geoid 2010 (CGG2010), has been developed to upgrade the previous geoid model CGG2005. CGG2010 represents the separation between the reference ellipsoid of GRS80 and the Earth’s equipotential surface of $W_0=62{,}636{,}855.69~\mathrm{m}^2\mathrm{s}^{-2}$. The Stokes–Helmert method has been re-formulated for the determination of CGG2010 by a new Stokes kernel modification. It reduces the effect of the systematic error in the Canadian terrestrial gravity data on the geoid to the level below 2 cm from about 20 cm using other existing modification techniques, and renders a smooth spectral combination of the satellite and terrestrial gravity data. The long wavelength components of CGG2010 include the GOCE contribution contained in a combined GRACE and GOCE geopotential model: GOCO01S, which ranges from $-20.1$ to 16.7 cm with an RMS of 2.9 cm. Improvement has been also achieved through the refinement of geoid modelling procedure and the use of new data. (1) The downward continuation effect has been accounted accurately ranging from $-22.1$ to 16.5 cm with an RMS of 0.9 cm. (2) The geoid residual from the Stokes integral is reduced to 4 cm in RMS by the use of an ultra-high degree spherical harmonic representation of global elevation model for deriving the reference Helmert field in conjunction with a derived global geopotential model. (3) The Canadian gravimetric geoid model is published for the first time with associated error estimates. In addition, CGG2010 includes the new marine gravity data, ArcGP gravity grids, and the new Canadian Digital Elevation Data (CDED) 1:50K. CGG2010 is compared to GPS-levelling data in Canada. The standard deviations are estimated to vary from 2 to 10 cm with the largest error in the mountainous areas of western Canada. We demonstrate its improvement over the previous models CGG2005 and EGM2008.

Appendices

Appendix A: The modified Stokes kernel

A general form of the modified Stokes kernel can be written as follows

$$\begin{aligned} S_\mathrm{M}(\psi , \psi _0) = \sum ^{\infty }_{n=2}\alpha _n(\psi _0){\frac{2n+1}{n-1}P_n({\cos }\,{\psi })} \end{aligned}$$

(20)

where $\alpha _n$ are called the SH degree-dependent spectral transfer coefficients (Wenzel 1982). For the truncated Stokes integration over a spherical cap, we can re-write the modified kernel in Eq. (20) as

$$\begin{aligned} {\overline{S}}_\mathrm{M}(\psi , \psi _0) = {\left\{ \begin{array}{ll} S_\mathrm{M}(\psi ), &{} 0 \le \psi \le \psi _0, \\ 0, &{} \psi _0 < \psi \le \pi . \end{array}\right. } \end{aligned}$$

(21)

Equation (21) can be expanded into the following Legendre polynomials

$$\begin{aligned} {\overline{S}}_\mathrm{M}(\psi , \psi _0) = -\beta _0(\psi _0) + \sum ^{\infty }_{n=2}\beta _n(\psi _0){\frac{2n+1}{n-1}P_n({\cos }\,{\psi })}\nonumber \\ \end{aligned}$$

(22)

where

$$\begin{aligned} \beta _n(\psi _0)=\alpha _n(\psi _0)-\frac{n-1}{2}Q^\mathrm{M}_n(\psi _0) \end{aligned}$$

(23)

$\beta _n$ are called the effective spectral transfer coefficients.

$$\begin{aligned} Q^\mathrm{M}_n(\psi _0)=\int ^{\pi }_{\psi _0} S_\mathrm{M}(\psi )P_n({\cos }\,{\psi })\sin \,{\psi }~\mathrm{d}\psi \end{aligned}$$

(24)

Using Eq. (21), the geoid residual computed over the spherical cap can be mathematically expressed as

$$\begin{aligned} dN(\varOmega )=\frac{R}{4\pi \gamma }\int _{\sigma } {\overline{S}}_\mathrm{M}(\psi , \psi _0) d g(\varOmega ^{\prime })~\mathrm{d}\sigma \end{aligned}$$

(25)

where $dg$ can be expanded into the SHs as

$$\begin{aligned} dg(\varOmega ^{\prime })=dg_0 + \sum ^{\infty }_{n=2}dg_n(\varOmega ^{\prime }) \end{aligned}$$

(26)

Substituting Eqs. (22) and (26) into Eq. (25), we arrive at

$$\begin{aligned} dN(\varOmega )=-\beta _0(\psi _0)R\frac{dg_0}{\gamma } + \frac{R}{\gamma }\sum ^{\infty }_{n=2}\beta _n\frac{dg_n(\varOmega )}{n-1} \end{aligned}$$

(27)

From Eq. (27), it is evident that the effective spectral transfer coefficients $\beta _n$ act as weights applied to the SH components of the terrestrial gravity anomaly residual.

Appendix B: Direct topographical effect

Let $t=H{/}R$, where $H$ is the orthometric height and $R$ is the mean radius of the Earth, the direct topographical mass potential effect (DTE) can be expressed as

$$\begin{aligned} \delta V_\mathrm{{DTE}}(r, \varOmega )&= V_\mathrm{A}(r, \varOmega ) - V_\mathrm{K}(r, \varOmega )\nonumber \\&= 2\pi G \rho R^2\sum ^{\infty }_{n=0}\frac{1}{2n+1} \left( \frac{R}{r}\right) ^{n+1}\nonumber \\&\quad \times \sum ^{n}_{m=-n} \overline{D}_{nm}\overline{Y}_{nm}({\varOmega }) \end{aligned}$$

(28)

where

$$\begin{aligned} \overline{D}_{nm}&= n(t^{2})_{nm} + \frac{(n+3)n}{3}(t^{3})_{nm}\nonumber \\&\quad +\frac{(n+2)(n+1)n}{12}(t^4)_{nm}\nonumber \\&\quad +\frac{2}{n+3}\sum ^{n+3}_{k=5}C^{n+3}_k(t^k)_{nm}\end{aligned}$$

(29)

$$\begin{aligned} (t^k)_{nm}&= \frac{1}{4\pi }\int _{\sigma } t^k{\overline{Y}}_{nm}({\varOmega ^{\prime }})~\mathrm{d}\sigma \end{aligned}$$

(30)