CMG RESEARCH: Wavelet-Based Statistical Analysis of Multiscale Geophysical Data

小波是一种相对较新但日益流行的数学工具，用于分析地球科学中的数据。小波重新表示在时间跨度或空间区域上收集的数据，使得在时间/空间尺度上的变化被总结为小波系数。单个系数既取决于尺度又取决于时间/空间位置，因此小波是分析具有相互作用尺度的地球系统的理想方法。在这个项目中，研究人员开发了基于小波的统计方法来解决三个多尺度地球物理问题。第一个是表征来自NOAA热带大气海洋浮标阵列的大气压力时间序列的特定尺度的方差/协方差。由于这些序列中有空隙，研究人员构造了特殊的小波来计算统计上容易处理的小波系数。第二个问题是分析飞机收集的大气湍流测量数据。研究人员使用统计方法将小波系数与飞机高度相结合，以确定风的尺度/高度变化。第三个问题是评估地面雷达降雨测量的空间/时间变化。在这里，研究人员使用小波来提取空间梯度，评估面积平均降水量的变化，并使用基于小波的自举来确定估计量的变异性。该项目背后的动机是进一步科学地理解大气如何随着时间、高度和不同地区的变化而变化。更好地了解大气对人类来说很重要，因为大气作为地球气候的一部分发挥着关键作用。诸如NOAA和NASA(与日本同行NASDA合作)等联邦机构正在积极收集直接测量大气变化的数据集。这些数据集，以及其他在NSF赞助下收集的数据集，本质上是难以解释的。在这个项目中，研究人员使用小波来研究这些数据集，小波是解释复杂数据集中变化的数学显微镜。研究人员开发了利用小波研究大气变化的新方法。未来，其他研究人员可以将这些新方法应用于其他科学领域收集的数据集，如海洋学、天文学、农业和林业。

Wavelets are a relatively new - but increasingly popular - mathematical tool for analyzing data in the geosciences. Wavelets reexpress data collected over a time span or spatial region such that variations over temporal/spatial scales are summarized in wavelet coefficients. Individual coefficients depend upon both a scale and a temporal/spatial location, so wavelets are ideal for analyzing geosystems with interacting scales. In this project, the investigators develop wavelet-based statistical methodology to address three multiscale geophysical problems. The first is to characterize scale-specific variances/covariances of atmospheric pressure time series from NOAA's Tropical Atmosphere Ocean buoy array. Because these series have gaps in them, the investigators construct special wavelets for computing statistically tractable wavelet coefficients. The second problem is to analyze atmospheric turbulence measurements collected by an aircraft. The investigators use statistical methodology to combine wavelet coefficients with aircraft heights to determine scale/height variations of winds. The third problem is to assess spatial/temporal variations in ground-based radar rainfall measurements. Here the investigators use wavelets to extract spatial gradients, to assess variations in area-mean precipitation, and to determine the variability in the estimated quantities using wavelet-based bootstrapping.The motivation behind this project is to further a scientific understanding of how the atmosphere changes over time, over heights and over different areas. A better understanding of the atmosphere is important for mankind because of the key role that the atmosphere plays as part of the climate of the earth. Federal agencies such as NOAA and NASA (in cooperation with its Japanese counterpart NASDA) are actively collecting sets of data that directly measure how the atmosphere changes. These data sets, along with others collected under NSF sponsorship, are inherently complex to interpret. In this project, the investigators study these data sets using wavelets, which are a mathematical microscope for interpreting changes in complex data sets. The investigators develop new methods for using wavelets to study atmospheric changes. In the future, other investigators can apply these new methods to data sets collected in other areas of science such as oceanography, astronomy, agriculture and forestry.