UNIQUENESS IN INVERSION OF INACCURATE GROSS EARTH DATA
UNIQUENESS IN INVERSION OF INACCURATE GROSS EARTH DATA
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DOI：
10.1098/rsta.1970.0005
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发表时间：
1970-01-01
影响因子：
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通讯作者：
GILBERT, F
中科院分区：
文献类型：
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作者：
BACKUS, G;GILBERT, F
地球总基准面是描述整个地球的某些属性的单个可测量数字，例如质量、转动惯量或某些已识别的弹性-重力简正模式的振荡频率。我们假设已经测量了有限个地球总数据集G，并且这些测量是不准确的，并且可以估计测量误差的方差阵。我们证明，除了细微尺度的细节外，某些这样的测量集合G在一定的误差范围内决定了地球的结构。也就是说，从一些集合G可以计算出不同深度的地球结构的局域平均值。这些局域平均值将略有误差，随着其分辨率的缩短，它们的误差将会更大。我们展示了如何确定给定的一组测量的总地球数据是否允许这样的局域平均构造，如果是这样的话，如果从G计算局部平均的误差的方差小于指定量，如何找到G在特定深度给出局部平均结构的最短长度标度。我们将一般理论应用于线性问题，即从有限个简正波的观测阻尼率求出与频率无关的局部弹性耗散(Q)的深度变化。我们还将该理论应用于已知压缩和剪切速度的情况下，从总质量、力矩和简正波频率求密度与深度的非线性问题。
A gross Earth datum is a single measurable number describing some property of the whole Earth, such as mass, moment of inertia, or the frequency of oscillation of some identified elastic-gravitational normal mode. We suppose that a finite set G of gross Earth data has been measured, that the measurements are inaccurate, and that the variance matrix of the errors of measurement can be estimated. We show that some such sets G of measurements determine the structure of the Earth within certain limits of error except for fine-scale detail. That is, from some setsG it is possible to compute localized averages of the Earth structure at various depths. These localized averages will be slightly in error, and their errors will be larger as their resolving lengths are shortened. We show how to determine whether a given set G of measured gross Earth data permits such a construction of localized averages, and, if so, how to find the shortest length scale over which G gives a local average structure at a particular depth if the variance of the error in computing that local average from G is to be less than a specified amount. We apply the general theory to the linear problem of finding the depth variation of a frequency-independent local elastic dissipation (Q) from the observed damping rates of a finite number of normal modes. We also apply the theory to the nonlinear problem of finding density against depth from the total mass, moment and normal-mode frequencies, in case the compressional and shear velocities are known.