Iterative generalized cross-validation for fusing heteroscedastic data of inverse ill-posed problems
Iterative generalized cross-validation for fusing heteroscedastic data of inverse ill-posed problems
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DOI：
10.1111/j.1365-246x.2009.04280.x
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发表时间：
2009-10
影响因子：
2.8
通讯作者：
Peiliang Xu
中科院分区：
文献类型：
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作者：
Peiliang Xu
广义交叉验证（GCV）方法已被广泛用于确定正则化参数，因为该标准最小化测量数据的平均预测残差，并且仅依赖于数据。只有当数据的方差-协方差矩阵可以表示为给定的正定矩阵和标量未知噪声方差的乘积时，数据驱动的优势才有效。在实际应用中，一些重要的地球物理反演不适定问题往往是通过组合不同类型的数据来解决的。在这种情况下，测量的随机模型包含许多不同的未知方差分量。虽然加权因子，或等效的方差分量，已被证明显着影响联合反演结果的地球物理不适定的问题，他们已被假定为已知的或经验选择。在地球物理联合反演中，如何正确确定不同类型数据的权重系数，还没有坚实的统计基础。我们扩展的GCV方法，以适应正则化参数和方差分量。扩展的GCV方法主要包括两个步骤，一个是通过固定正则化参数来估计方差分量，另一个是通过使用GCV方法并固定方差分量来确定正则化参数。我们模拟两个例子：从菲利普斯（1962）的第一个例子修改的第一类纯数学积分方程和从卫星测量恢复地球表面重力异常的向下延拓的典型地球物理例子。基于两个模拟的例子，我们广泛地比较了迭代GCV方法与现有的方法，这表明该方法可以很好地正确地恢复未知的方差分量和确定正则化参数。换句话说，我们的方法让数据自己说话，决定正确的加权因子的不同类型的地球物理数据，并确定正则化参数。此外，我们推导出一个无偏估计的噪声方差校正偏差的正则化残差。文中还给出了一个简化的计算公式，以节省计算时间。通过数值模拟，将这两种新的噪声方差估计与现有的六种方法进行了比较。仿真结果表明，这两个新的估计执行以及Wahba的估计高度不适定的问题，并优于任何现有的方法，中度不适定的问题。
SUMMARY The method of generalized cross-validation (GCV) has been widely used to determine the regularization parameter, because the criterion minimizes the average predicted residuals of measured data and depends solely on data. The data-driven advantage is valid only if the variance–covariance matrix of the data can be represented as the product of a given positive definite matrix and a scalar unknown noise variance. In practice, important geophysical inverse ill-posed problems have often been solved by combining different types of data. The stochastic model of measurements in this case contains a number of different unknown variance components. Although the weighting factors, or equivalently the variance components, have been shown to 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. We extend the GCV method to accommodate both the regularization parameter and the variance components. The extended version of GCV essentially consists of two steps, one to estimate the variance components by fixing the regularization parameter and the other to determine the regularization parameter by using the GCV method and by fixing the variance components. 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 compare the iterative GCV method with existing methods, which have shown that the method works well to correctly recover the unknown variance components and determine the regularization parameter. In other words, our method lets data speak for themselves, decide the correct weighting factors of different types of geophysical data, and determine the regularization parameter. In addition, we derive an unbiased estimator of the noise variance by correcting the biases of the regularized residuals. A simplified formula to save the time of computation is also given. 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 as Wahba's estimator for highly ill-posed problems and outperform any existing methods for moderately ill-posed problems.