Single value decomposition (SVD) is a mathematical method of decomposing a matrix into three other matrices that have very important properties of the original matrix. One of these matrices provides us with the singular values of the original matrix. By analysing these singular values, we can determine whether an inverse problem has an ambiguous solution or whether the solution is stable. With this analysis we can also check whether the estimated solution gives an exact fit to the data, find out which combinations of parameters can be determined by observations and which cannot, and more. In this work, SVD was applied to assess the feasibility of a joint 3D inversion of marine magnetotelluric method (MMT) and marine controlled source electromagnetic method (MCSEM) data, as these methods should theoretically provide complementary information. The singular values were generated from the Hessian matrix of each method and from the joint implementation, and we compared them by considering the null space of each method. The analysis of the singular values proved to be an important step before the iterative inversion process, indicating the feasibility of the inversion without spending much time. However, we still need to spend the time to calculate the Jacobian and Hessian matrices.

中文翻译：

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MMT和MCSEM数据的3D联合反演问题中的奇异值分析

单值分解 (SVD) 是一种将矩阵分解为其他三个矩阵的数学方法，这些矩阵具有原始矩阵的非常重要的属性。这些矩阵之一为我们提供了原始矩阵的奇异值。通过分析这些奇异值，我们可以确定逆问题是否有模糊解或解是否稳定。通过这种分析，我们还可以检查估计的解决方案是否与数据精确拟合，找出哪些参数组合可以通过观察确定，哪些不能，等等。在这项工作中，SVD 用于评估联合 3D 反演海洋大地电磁法 (MMT) 和海洋可控源电磁法 (MCSEM) 数据的可行性，因为这些方法理论上应该提供补充信息。奇异值是从每种方法的 Hessian 矩阵和联合实现中生成的，我们通过考虑每种方法的零空间来比较它们。奇异值的分析被证明是迭代反演过程之前的重要一步，表明反演的可行性无需花费太多时间。但是，我们仍然需要花时间来计算雅可比矩阵和黑森矩阵。