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tsutterley avatar tsutterley commented on June 15, 2024

From Eq. 4 of Jacob et al. (2012) for spherical caps:

$$\left[\begin{matrix}\tilde{C}_{lm} \\ \tilde{S}_{lm}\end{matrix}\right] = \frac{3}{\rho_\textrm{ave}a}\frac{(1 + k_l)}{(2l + 1)^2}~F_l~\tilde{P}_{lm}(cos\theta)\left[\begin{matrix}\cos(m\phi) \\ \sin(m\phi)\end{matrix}\right]$$

From Eqs. 6 and 7 of Wahr et al. (1998) for integrating surface mass density anomalies over a sphere:

$$\left[\begin{matrix}\tilde{C}_{lm} \\ \tilde{S}_{lm}\end{matrix}\right] = \frac{3}{4\pi\rho_\textrm{ave}a}\frac{(1 + k_l)}{(2l + 1)} \iint \sigma(\theta, \phi)~\tilde{P}_{lm}(cos\theta)\left[\begin{matrix}\cos(m\phi) \\ \sin(m\phi)\end{matrix}\right]\sin\theta~d\theta~d\phi$$

The coefficients from Wahr et al. (1998) are a default attribute within the units class (i.e. factors.cmwe). But getting the units for a spherical cap (from Jacob et al., 2012) can be calculated from this default, which is the dfactor you find in the cap program.

from gravity-toolkit.

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