Transformation to equivalent dimensions—a new methodology to study earthquake clustering
Transformation to equivalent dimensions—a new methodology to study earthquake clustering
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等效维数变换——地震聚类研究的新方法
DOI：
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发表时间：
2014
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通讯作者：
S. Lasocki
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作者：
S. Lasocki
地震事件由参数空间中的点表示，并通过参数值向量进行量化。地震聚类的研究涉及考虑多维空间中这些点之间的距离。然而，地震参数的度量不同，因此多维参数空间中的度量不易定义。本文基于地震参数概率等价的概念提出了该度量问题的解决方案。在这个概念下，如果地震从任一区间取值的概率相同，则参数区间的长度是相等的。地震聚类是在等效维数空间而不是原始维数空间中进行研究，其中参数的等效维数（ED）是其累积分布函数。所有变换后的参数都具有 [0, 1] 区间内的线性尺度，并且由任意 ED 空间中的向量表示的地震之间的距离是欧几里德的。一般来说，未知的地震参数累积分布是通过无模型非参数核估计方法从地震目录中估计出来的。两个使用示例说明了向 ED 转换的潜力：查找时空中层次上最接近的邻居以及评估特定 4 维相空间中地震聚类的时间变化。
A seismic event is represented by a point in a parameter space, quantified by the vector of parameter values. Studies of earthquake clustering involve considering distances between such points in multidimensional spaces. However, the metrics of earthquake parameters are different, hence the metric in a multidimensional parameter space cannot be readily defined. The present paper proposes a solution of this metric problem based on a concept of probabilistic equivalence of earthquake parameters. Under this concept the lengths of parameter intervals are equivalent if the probability for earthquakes to take values from either interval is the same. Earthquake clustering is studied in an equivalent rather than the original dimensions space, where the equivalent dimension (ED) of a parameter is its cumulative distribution function. All transformed parameters are of linear scale in [0, 1] interval and the distance between earthquakes represented by vectors in any ED space is Euclidean. The unknown, in general, cumulative distributions of earthquake parameters are estimated from earthquake catalogues by means of the model-free non-parametric kernel estimation method. Potential of the transformation to EDs is illustrated by two examples of use: to find hierarchically closest neighbours in time–space and to assess temporal variations of earthquake clustering in a specific 4-D phase space.