多维分形克里格方法

时间序列与空间场信号往往是非规则分布的，经常需要将非规则分布的时空信号插值为规则分布的信号或估计某些未知点的值。如油气田、煤田以及金属矿山储量估算，工程地质参数估计，病虫害区域分布调查等都要求根据少量不规则数据点进行插值估算。估值方法中应用最为广泛的地质统计学方法（或(Krige)克里格方法）是一种低通滤波器，无法重建原始信号中的高频、局部与弱信号。开发的多维分形克里格方法可以将不规则分布的时间—空间（时空）信号插值为规则分布的信号；可以提取时空信号中高频、局部与弱信号，估计过程参数可以作为特征参数用于模式识别。利用褶积滤波理论定量导出了地质统计学的低通滤波特性，它在插值过程中丢失了高频、局部和弱信号。在定义了时空信号的度量尺度与测度后，实现了多维分形插值，多维分形插值保留了系统中更多的高频信息。将克里格方法与多维分形方法有机的结合起来产生了多维分形克里格方法，它具有克里格方法和多维分形插值的共同优点。用大洋钻探(ODP)184航次1143A孔的岩芯密度分析进行了插值试验，对比了插值结果及其功率谱。多维分形克里格插值比克里格插值、多维分形插值更为接近已知点值并保留更多的高频信息。还定量分析、对比了影响多维分形克里格插值的因素、厘清了估值问题中固有的测不准关系。 另外，多维分形克里格插值过程得到的局部奇异性、相关性和回归方差能有效地刻划高频、局部与弱信号。这样，多维分形克里格插值过程可以用于提取（非规则或规则网格）时空信号中的局部、高频与弱信号，用于信息提取、模式识别、找矿预测与信号增强等领域。

MULTIFRACTAL-KRIGE INTERPOLATION METHOD

The Multifractal Krige method developed in this study can not only interpolate irregular distributed values into regular distributed grids, but also extract the high frequency, local and weak signals, which are useful in feature retrieval or pattern recognition, from temporal spatial signals. The signal from observing science is often distributed irregularly and it is often critical to interpolate irregular distributed signal into regular grids or estimate values at some points. For examples, in reservoir, coal bed and mineral tonnage estimations or in engineering parametric estimations, regional harmonious insects inspections, the interpolation is necessary. The Krige method had been widely used even though it is a low pass filter and can not construct the high frequency, local and weak signals which are often play more important role in related study. The low pass filtering property of Krige method is studied from the filtering points of view in frequency domain and it was found that Krige method is a low pass filters. In contrary, Multifractal interpolation method can reconstruct part of these signals. To implement the fractal interpolation, which keeps more high frequency information, the measure and scale pairs are defined, formula and procedures are studied in this study. The integration of Krige and Multifractal method produced Multifractal Krige method that keeps benefits of both Krige and Multifractal interpolations. The core density data from Hole 1143A of Ocean Drilling Program (ODP) 184th cruise is used to test the algorithm. The interpolated results and power spectra are compared to show the benefits of Multifractal interpolation and Krige Multifractal method. The results proved that the Multifractal Krige interpolation approximates known points better and had richer high frequency frequencies than other methods. Factors that affect the method, such as uncertainty in the value estimation problems, had also been studied quantitatively. Further more, the local singularities, regression index and standard errors got from the interpolation procedure are good approximation of the high frequency, local and weak signals. So, Krige Multifractal interpolation method can also be used in other kinds of applications, such as information retrieval, enhancement and pattern recognition.