Estimating the expansion coefficients of a geomagnetic field model using first-order derivatives of associated Legendre functions.

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Abstract

The associated Legendre functions are defined on a closed interval. Thus, their derivatives do not exist at the endpoints of the interval. However, onesided derivatives may exist at the endpoints if the ordinary limit is replaced by a onesided limit. When computer models that evaluate associated Legendre functions at an endpoint are transformed by automatic differentiation, an approach is needed that is tailored to onesided derivatives. Rather than employing a blackbox approach of automatic differentiation, a hierarchical approach is introduced that is based on analytic firstorder onesided derivatives of the associated Legendre functions at the endpoints. This hierarchical approach is implemented in the automatic differentiation software ADiMat and its feasibility is demonstrated in a parameter estimation problem arising from geomagnetic field modeling.