| Regularization is crucial to solve geophysical and geodetic ill-posed inverse problems. In
practice, we may combine different types of data to solve such an ill-posed inverse
problem; the accuracy of these different types of data may not be known precisely
and should be modelled by a number of unknown variance components. Although
the weighting factors, or equivalently the variance components, can significantly
affect joint inversion results of geophysical ill-posed problems, they have been
either assumed to be known or empirically chosen. No solid statistical foundation is
available yet to correctly determine the weighting factors of different types of data in
joint geophysical inversion. In this case, all regularization techniques may not be
proper to apply, unless techniques of variance component estimation are directly
implemented to determine the correct weighting factors for each type of data. In this
paper, we will solve ill-posed inverse problems by simultaneously determining the
regularization parameter and the weighting factors of different types of data, either by
using the criteria of mean squared errors or the cross validation. First we analyze
the biases of estimated variance components due to the regularization parameter
and then propose bias-corrected variance component estimators. We simulate two
examples: a purely mathematical integral equation of the first kind modified from the
first example of Phillips (1962) and a typical geophysical example of downward
continuation to recover the gravity anomalies on the surface of the Earth from satellite
measurements. Based on the two simulated examples, we extensively investigate
the MSE and the iterative GCV methods. The simulated results have shown that
these methods work well to correctly recover the unknown variance components
and determine the regularization parameter. In other words, our methods let data
speak for themselves, decide the correct weighting factors of different types of
geophysical data, and determine the regularization parameter. In addition, we derive
unbiased estimators of the noise variance by correcting the biases of the regularized
residuals. The two new estimators of the noise variance are compared with six existing
methods through numerical simulations. The simulation results have shown that
the two new estimators perform as well asWahba’s estimator for highly ill-posed
problems and outperform any existing methods for moderately ill-posed problems. |