Soil water content (SWC) is crucial to rainfall-runoff response at the
watershed scale. A model was used to decompose the spatiotemporal SWC into a
time-stable pattern (i.e., temporal mean), a space-invariant temporal
anomaly, and a space-variant temporal anomaly. The space-variant temporal
anomaly was further decomposed using the empirical orthogonal function (EOF)
for estimating spatially distributed SWC. This model was compared to a
previous model that decomposes the spatiotemporal SWC into a spatial mean
and a spatial anomaly, with the latter being further decomposed using the
EOF. These two models are termed the temporal anomaly (TA) model and spatial
anomaly (SA) model, respectively. We aimed to test the hypothesis that
underlying (i.e., time-invariant) spatial patterns exist in the
space-variant temporal anomaly at the small watershed scale, and to examine
the advantages of the TA model over the SA model in terms of the estimation
of spatially distributed SWC. For this purpose, a data set of near surface
(0–0.2 m) and root zone (0–1.0 m) SWC, at a small watershed scale in the
Canadian Prairies, was analyzed. Results showed that underlying spatial
patterns exist in the space-variant temporal anomaly because of the
permanent controls of

Soil water content (SWC) of surface soils exerts a major influence on a
series of hydrological processes such as runoff and infiltration
(Famiglietti et al., 1998; Vereecken et al., 2007; She et al., 2013a). Soil
water content in the root zone is, in many cases, linked to vegetative
growth (Wang et al., 2012; Ward et al., 2012; Jia and Shao, 2013). Obtaining
accurate information on the spatiotemporal SWC is crucial for improving
hydrological prediction and soil water management (Venkatesh et al., 2011;
Champagne et al., 2012; She et al., 2013b; Zhao et al., 2010). While remote
sensing has advanced SWC measurements of surface soils (<5 cm in
depth) at basin (2500–25 000 km

Decomposition of spatiotemporal soil water content (SWC) in different models.

Time stability of SWC, which refers to similar spatial patterns of SWC across different measurement times (Vachaud et al., 1985; Brocca et al., 2009), has been used for estimating spatially distributed SWC (Starr, 2005; Perry and Niemann, 2007; Blöschl et al., 2009). This method is conceptually appealing, but assumes completely time-stable spatial patterns of SWC.

The time-stable pattern does not explain all of the spatial variances in SWC, indicating the existence of time-variant components (Starr, 2005). In order to identify underlying patterns of SWC that have time-variant components, the spatiotemporal SWC was decomposed into a spatial mean and a spatial anomaly. The spatial anomaly of the SWC was further decomposed into the sum of the product of time-invariant spatial patterns (EOFs) and temporally varying, but spatially constant coefficients (ECs) using the empirical orthogonal function (EOF) (Fig. 1) (Jawson and Niemann, 2007; Perry and Niemann, 2007, 2008; Joshi and Mohanty, 2010; Korres et al., 2010; Busch et al., 2012). Spatially distributed SWC estimates based on the decomposition of spatial anomaly outperformed those based on time-stable patterns (Perry and Niemann, 2007).

Recently, the spatiotemporal SWC was also decomposed into a temporal mean
and a temporal anomaly (Mittelbach and Seneviratne, 2012) (Fig. 1). Previous
studies indicated that the contribution of the temporal anomaly to the total
spatial variance was notable (Mittelbach and Seneviratne, 2012; Brocca et
al., 2014; Rötzer et al., 2015). These studies, however, only focused on
surface soils at large scales (>250 km

To our knowledge, the importance of the space-variant term of the temporal
anomaly and its physical meaning at small watershed scales is not
well-known. Based on previous studies (Perry and Niemann, 2007; Mittelbach
and Seneviratne, 2012; Vanderlinden et al., 2012), we assume soil water
dynamics at watershed scales can be decomposed into three components (Fig. 1):
(1) time-stable pattern (i.e., temporal mean, spatial forcing): the

The static factors may be persistent in the space-variant temporal anomaly, and their impacts on the space-variant temporal anomaly likely change with time. Thus, we hypothesize that some underlying (i.e., time-invariant) spatial patterns exist in the space-variant temporal anomaly, and their impacts can be modulated by a time coefficient, both of which can be obtained by the EOF method (Fig. 1). If the hypothesis is true, the estimation of spatially distributed SWC utilizing the EOF decomposition may outperform the one suggested by Perry and Niemann (2007). This is because: (1) the spatial anomaly, which was decomposed using the EOF in Perry and Niemann (2007), lumped the time-stable pattern and space-variant temporal anomaly together (Fig. 1); (2) the underlying spatial patterns in the spatial anomaly may not fully capture both time-stable patterns and patterns in the space-variant temporal anomaly due to the possible nonlinear relations between these two terms.

Therefore, the objectives were (1) to test the hypothesis that underlying
spatial patterns exist in the space-variant temporal anomaly at small
watershed scales and (2) to examine whether the decomposition of the
space-variant temporal anomaly using the EOF has any advantages over the
decomposition of the spatial anomaly (Perry and Niemann, 2007) for
estimating spatially distributed SWC. Two steps were included in the
estimation of spatially distributed SWC. First, the spatial mean SWC was
upscaled from the SWC measurement at the most time-stable location using
time stability analysis. Following this, the spatially distributed SWC was
downscaled from the estimated spatial mean SWC. For the purpose of this
study, spatiotemporal SWC data sets at depths of near surface (0–0.2 m) and
root zone (0–1.0 m) from a Canadian Prairie landscape were used.
Spatiotemporal SWC of samples taken 0–0.06 m from a hillslope (100 m) in
the Chinese Loess Plateau and 0–0.15 m from the GENCAI network
(

Daily mean air temperature and precipitation during the study period.

This study was mainly conducted in the Canadian Prairie pothole region
(hereafter abbreviated as Canadian site) at St. Denis National Wildlife Area
(52

To further test the applicability of the new method, we compared its
performance at two other sites, covering both the hillslope and the large
watershed scale. Along a hillslope of 100 m in length in the Chinese Loess
Plateau, SWC of 0–0.06 m was measured 136 times from 25 June 2007 to
30 August 2008 by a Delta-T Devices Theta probe (ML2x) at 51 locations (Hu
et al., 2011). The hillslope was covered by

Spatiotemporal SWC at small watershed scales was decomposed into three components: time-stable pattern, space-invariant temporal anomaly, and space-variant temporal anomaly. This model was compared to the one that decomposed SWC into spatial mean and spatial anomaly (Perry and Niemann, 2007). Both the space-variant temporal anomaly and spatial anomaly were decomposed using the EOF method. The two models are termed the temporal anomaly (TA) model and the spatial anomaly (SA) model. Figure 1 displays the differences between the two models. Each component will be explained in detail later. The explanation of nomenclatures is listed in Table A1. Because we focus on estimating spatial distribution of SWC at any given time, only spatial variances of SWC were taken into account. Therefore, the variance or covariance denotes the quantity in space without specifications.

Perry and Niemann (2007) expressed SWC at location

According to Perry and Niemann (2007),

Spatial anomaly (

Usually, a substantial amount of variance can be explained by a small number
of EOFs. Johnson and Wichern (2002) suggested the eigenvalue confidence
limits method for selecting the number of EOFs. Once the number of
significant EOFs at a confidence level of 95 % is selected,

Mittelbach and Seneviratne (2012) decomposed the

If some underlying spatial patterns exist in

For estimation of spatially distributed SWC,

The Pearson correlation coefficient (

The TA model is more complicated than the SA model. In order to evaluate the
two models for parsimony, AICc values are calculated (Burnham and Anderson, 2002) as

Both cross-validation and split sample validation are used to estimate SWC distribution with both models. For the cross-validation, an iterative removal of 1 of the 23 dates is made for model development, and the SWC along the transect corresponding to the removed date is estimated iteratively. For the split sample validation, SWC from 14 dates of the first 2 years (from 17 July 2007 to 27 May 2009) is used for model development, and the SWC distribution of 9 dates in the second 2 years (from 21 July 2009 to 29 September 2011) is estimated.

Components of soil water content in

The Nash–Sutcliffe coefficient of efficiency (NSCE) is used to evaluate the
quality of estimation of spatially distributed SWC, which is expressed as

Many factors may affect the relative performance of spatially distributed
SWC estimation between the TA model and the SA model. First, the degree of
outperformance of the TA model over the SA model may depend on the amount of

Pearson correlation coefficients between time-stable pattern

The values of spatial mean (

The spatial patterns of spatial anomaly (

When SWCs of all 23 dates were used for model development, only EOF1 was
statistically significant (Fig. 4a), which accounted for 84.3 % (0–0.2 m)
and 86.5 % (0–1.0 m) of the variances in the

Figure 3b displays the three components in the TA model. The first component

The

The SWC variances and associated components (Eq. 8) also varied with time
(Fig. 5). Often, wetter conditions corresponded to greater

Spatial variances of different components in Eq. (8)
expressed in %

The time-invariant

The

Three significant EOFs of

The contribution of EOF1 to the space-variant temporal anomaly can be
examined through the product of the EOF1 and the associated EC1. The EC1
values tended to be positive during wet periods and negative during dry
periods (Fig. 6b); more positive EOF1 values were usually observed at
locations with greater

Depth to the CaCO

When all 23 data sets were used and only EOF1 was considered, the TA model had an AICc value of 4093 for the near surface and 562 for the root zone, while the corresponding values for the SA model were 6370 and 3460. This indicated that even when penalty for complexity was given, the TA model was better than the SA model. The two models in terms of spatially distributed SWC estimation are compared below.

The

Estimated soil water content (SWC) versus measured SWC for
three dates at different soil water conditions (23 August 2008, 27 October 2009,
and 13 May 2011 are associated with relatively dry, medium, and wet
days, respectively) using the TA model for

The Nash–Sutcliffe coefficient of efficiency (NSCE) of soil
water content estimation using the TA and SA models for

Estimated SWCs generally approximated those measured at different soil water
conditions during the cross-validation (Fig. 7). However, on 27 October 2009,
there were unsatisfactory overestimates at the 100–140 and 220–225 m
locations near the surface (Fig. 7a). Unsatisfactory NSCE values of

During the split sample validation, the TA model resulted in SWC estimations
with NSCE values ranging from 0.61 to 0.85 near the surface and from 0.32 to 0.92
in the root zone, with exception of 2 days (27 August 2009 and
27 October 2009 with NSCE values of

One significant EOF of

The difference in NSCE values between the TA and SA models for both
validations are presented in Fig. 9. Generally, the difference decreased as

Nash–Sutcliffe coefficient of efficiency (NSCE) difference
between the TA and SA models in terms of soil water content estimation using
both cross-validation (CV) and split sample validation (SV) as a function of
space-invariant temporal anomaly

Nash–Sutcliffe coefficient of efficiency (NSCE) difference
between the TA and SA models in terms of soil water content estimation using
cross-validation as a function of space-invariant temporal anomaly

On average, the

The

The

As Table 1 shows, both the depth to the CaCO

The outperformance of the TA model for estimating spatial SWC at the
Canadian site and Chinese site can be partly explained by the high
percentages (average of 19–118%) of the

The existence of underlying spatial patterns in the

The

The relationship between the

Therefore, the outperformance of the TA model over the SA model depends on
counterbalance among the variance of

In the real world, the relations between the

Previous studies on SWC decomposition mainly focus on near-surface layers
(Jawson and Niemann, 2007; Perry and Niemann, 2007, 2008; Joshi and Mohanty,
2010; Korres et al., 2010; Busch et al., 2012). This study decomposed
spatiotemporal SWC using the TA model for both the near surface and the root
zone. The results showed that the estimation of spatially distributed SWC at
small watershed scales was improved by the TA method that considers the

The TA model was used to decompose spatiotemporal SWC into time-stable
patterns

This study showed that outperformance of the TA model over the SA model is
possible when

This project was funded by the National Science Foundation of China (K305021308) and the Natural Sciences and Engineering Research Council (NSERC) of Canada. We thank Asim Biswas, Henry Wai Chau, Trent Pernitsky, and Eric Neil for their help in data collection. We thank the anonymous reviewers and the Editor for their constructive comments. Edited by: T. Blume