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.