2.3 EOF analysis
Empirical orthogonal function (EOF) analysis has been widely applied for the analysis of the spatial and temporal variability of large multidimensional datasets (Zhao et al., 2012). The EOF, also known as a type of principal component analysis, decomposes the observed variability of a dataset into a set of orthogonal spatial patterns (EOFs) or a set of time series called expansion coefficients (ECs). This procedure is accomplished by transforming the original data set into a new set of uncorrelated variables, and then ordered in a manner so that the first few of the new variables explain most of the variation existing in the original data set. For example, it is possible to construct various second moment statistics linking one point to another in geophysical data maps. The resulting correlation matrix is real and symmetric, and therefore possesses a set of orthogonal eigenvectors with positive eigenvalues. If there are geophysical data maps that are time series with any m x n matrix, A , square or rectangular, there uniquely exists two orthogonal matrices, U and V and a diagonal matrix L such that,
A = U × L × V T (1)
where V T is the transpose of a matrix V . Note that L is padded with zeros to make the square diagonal matrix into an m x n matrix. This assumption also implies that L has at most M = min(m , n ) nonzero elements. The columns of U are called the EOFs of A and the corresponding diagonal elements of L are called the eigenvalues. Each row of V serves as a series of time coefficients that describes the time evolution of the particular EOF. The map associated with an EOF represents a pattern, which is statistically independent and spatially orthogonal to the others. The eigenvalue indicates the amount of variance accounted for by the pattern (Zhao et al., 2012).
While single soil moisture patterns might be affected by random processes (e.g., rainfall shortly before measurement), significant EOFs represent stable patterns of a dataset. The existing degree of randomness of a single soil moisture pattern is reflected by the associated EC, since the EC value represents the proportion of the significant EOF pattern within the soil moisture pattern of each date. In consequence, single soil moisture patterns (which might be random) were not used but the EOF patterns used for the subsequent correlation analysis. That is, the EOF patterns can be further correlated to the geophysical characteristics of the region to determine the dominant physical controls. For the EOF analysis, we used the spatial anomalies of the soil moisture dataset instead of the soil moisture which excludes the temporal variations from consideration (Perry and Niemann, 2007). The spatial anomalies are calculated by subtracting the mean soil moisture for a given sampling day from all the soil moisture observations collected on that day.