## 1. Introduction

As most of the impacts of soil moisture for the climate system are induced by its impact on evapotranspiration in water-limited regimes (Seneviratne et al. 2010), soil moisture information is important for the investigation of land–atmosphere coupling. However, it is typically assumed that the shallow vertical penetration depth of microwave-based surface soil moisture retrievals (generally considered to be constrained to the top 5 cm of the soil column) limits their relevance for such investigations. In response, various methods have been developed to vertically extrapolate information contained in surface soil moisture retrievals to recover deeper soil moisture information. The most popular of these approaches include both sophisticated data assimilation strategies (Reichle et al. 2007; Sabater et al. 2007; Reichle et al. 2008; Kumar et al. 2009; Bolten and Crow 2012) and the application of semiempirical low-pass filtering to historical surface soil moisture time series data (Wagner et al. 1999; Ceballos et al. 2005; Albergel et al. 2008; Brocca et al. 2011; Mo et al. 2011; Manfreda et al. 2014; Crow et al. 2015).

A key unanswered question, however, is exactly how the information content of soil moisture, with regards to land–atmosphere interaction studies (especially surface flux prediction) and ecosystem productivity forecasting, increases as soil moisture time series observations are integrated over a deeper vertical depth. Qiu et al. (2014) looked at whether the vertical integration of soil moisture information improves the skills to forecast vegetation states (with lead time up to 40 days). In particular, using long-term soil moisture observations from the U.S. Department of Agriculture’s Soil Climate Analysis Network (SCAN), they calculated mutual information (MI) between multiple soil moisture variables (with various vertical supports) and near-future vegetation conditions to examine the existence of drought information. Results suggest that, relative to superficial soil moisture observations

However, it is unclear if the results of Qiu et al. (2014) can be extended to other potential soil moisture applications. Climatologists (e.g., Budyko 1974; Eagleson 1978) have long developed different assumptions about the relationship between soil moisture (SM) and evaporative fraction (EF), that is, the ratio between latent heat flux (LE) and the sum of latent heat flux and sensible heat flux (SH). Among these assumptions, some research has interpreted the relationship as an exponential curve, whereas others have identified this relationship as piecewise linear function (or two-regime section), which transitions from a linear relationship under water-limited condition to an independent relationship under energy-limited condition (e.g., Komatsu 2003; Ford et al. 2014b). For instance, Koster et al. (2009) exploited the two-section evaporative regime to investigate the connection between seasonal meteorological drought and an increase in seasonal air temperature. Their results showed strong agreement between atmospheric general circulation models (AGCMs) and observation-based geographical patterns of drought-induced warming, which supports the idea that the same evaporative controls are also present in nature. Likewise, Ford et al. (2014b) examined seasonal variations on SM–EF interactions over the U.S. Southern Great Plains using both in situ observations and simulations from the Variable Infiltration Capacity model, and confirmed the well-identified two-section evaporative regime. More mechanistic studies include the proposal of a classical resistance-based method that links the soil evaporative efficiency with aerodynamic resistance and soil resistance, the latter of which can be expressed as a function of 0–5 cm soil moisture (Sellers et al. 1992). However, the impact of soil moisture vertical support on the SM–EF interactions over various land covers has received relatively little attention in the existing literature. In addition, the strength of the SM–EF relationship has generally been evaluated using a conventional Pearson product-moment correlation. However, since the relationship between SM and EF is certainly nonlinear, a linear correlation coefficient cannot fully capture the coupling strength between two variables. This issue motivates the use of a more integrative method (i.e., MI) to measure the SM–EF coupling strength in this study.

Here, we apply the methodology of Qiu et al. (2014) to examine the relationship between the vertical support of SM observations and their mutual information with respect to EF. In particular, we attempt to quantify the potential increase in information content, with respect to latent heat flux estimation, of surface soil moisture induced by integration over deeper vertical depths. This is based on analyzing the MI content between current/near-future surface energy flux time series and time series of three soil moisture datasets with various vertical supports: surface soil moisture, vertically integrated soil moisture, and a low-pass transformation of

## 2. Data and methods

AmeriFlux network provides continuous measurements of soil moisture, water vapor, energy fluxes, and related environmental variables, and the network covers a large variety of ecosystem types including forests, grasslands, croplands, shrublands, wetlands, and savannas (Boden et al. 2013). Before estimating mutual information between AmeriFlux SM and EF observations, we averaged datasets of observed SM

To remove the seasonal cycle and obtain anomaly time series, climatological averages for

### a. Soil moisture measurements

In situ soil moisture observations at a half-hour time step were collected from the AmeriFlux Site and Data Exploration System (see http://ameriflux.ornl.gov/). The level 2 (L2) standardized soil moisture data without gap filling were selected for this study. Soil moisture measurements at AmeriFlux sites were generally taken at two layers, and the measurement depths of top-layer and bottom-layer soil moisture vary between sites (Table 1). The sensitivity of results to these depth variations will be examined further in section 3a. Soil moisture observations were masked if they were uncorrected or did not fall within the (physically feasible) range of 0–0.6 m^{3} m^{−3} (Xia et al. 2015).

Attributes of selected 36 AmeriFlux sites.

The top-layer measurements were used to represent *θ*_{S} variations (as obtained, for example, from microwave remote sensing). In contrast, *θ*_{V} was calculated as the depth-weighted average of both

To reduce temporal sampling errors, only 36 AmeriFlux sites (four sites outside the contiguous United States were excluded) with over 5 years of two-layer soil moisture and surface flux observations were considered. These sites are located in a variety of climate zones within the contiguous United States. Figure 1 shows a map of these AmeriFlux sites using different symbols to represent landscape-scale land cover surrounding each plot-scale site.

### b. Surface fluxes and vegetation cover observations

*G*. The L2 AmeriFlux LE and SH observations are based on high-frequency (typically >10 Hz) eddy covariance measurements that are processed into 30-min averages by individual AmeriFlux investigators. Using these L2 products, we averaged LE and SH over a daytime period between 0700 and 1800 local standard time (LST) and calculated daily EF using these averaged LE and SH quantities. To ensure reliable surface flux observations, we utilized temporally coincident 30-min precipitation accumulation observations to exclude rainy conditions and masked days lacking any observations between 1130 and 1330 LST. In addition, we excluded days with energy budget closure rates

To describe the long-term vegetation cover conditions around each site, we derived 2001–14 multiyear leaf area index (LAI) time series from 4-day composites of the 1-km Moderate Resolution Imaging Spectroradiometer (MODIS) MCD15A3 product (version 5). The LAI dataset was quality checked by the LAI–Fraction of Photosynthetically Active Radiation (FPAR) general quality assessment field in the MCD15A3 product and only high-quality retrievals, that is, those categorized as “good quality,” “detectors apparently fine for up to 50% of channels 1,2,” “significant clouds NOT present,” or “main method used, best result possible” were utilized. In addition to the MCD15A3 product, an extra LAI dataset was obtained from 8-day composites of the 1-km Global Land Surface Satellite (GLASS) LAI product (http://glcf.umd.edu/data/lai/) that was generated from time series of AVHRR/MODIS reflectance data using general regression neural networks. The 2001–14 multiyear LAI mean for each site derived from both of these LAI datasets (i.e., MCD15A3 and GLASS) is listed in Table 1.

### c. SWI from low-pass filtering

*θ*

_{S}in order to create SWI to function as a proxy for vertically integrated soil moisture. Following Wagner et al. (1999), this filtering was based on the assumption that the water flux between two soil layers is proportional to the difference in soil moisture content between these two layers. The resulting solution takes the form of a semiempirical exponential filter that converts superficial

*t*−

*M*and

*t*into a proxy SWI estimate at time

*t*:where

*T*is a response time scale (parameter) and

*M*is the measurement length over which past

*M*is used for computation efficiency. In this study, the threshold was taken to be 3

*T*unless otherwise stated (Albergel et al. 2008; Qiu et al. 2014). The response time scale is controlled by regional hydroclimatic conditions, land-cover types, and the local soil characteristics, which impact the vertical hydraulic connectivity between various soil layers (Albergel et al. 2008; Ford et al. 2014a). Specific strategies for parameterizing

*T*are described later in section 2e.

### d. Information measures

Mutual information (Cover and Thomas 1991) is a nonparametric measure of correlation (here defined strictly as the lack of independence) between two random variables. It is a more rigorous measure compared to the commonly used metrics such as Spearman’s rank correlation coefficient and Pearson product-moment correlation coefficient, the latter being an approximation of MI under certain conditions (Nearing and Gupta 2015).

*ζ*can be interpreted as a measure of uncertainty according to its distribution

*p*

_{ζ}and is estimated as the expectation [

*E*(⋅)] of information from the

*p*

_{ζ}sample:MI between

*ζ*and another variable

*ψ*can be thought of as the expected amount of information about variable

*ζ*contained in a realization of

*ψ*and is measured by the expected Kullback–Leibler divergence

*D*(Kullback and Leibler 1951) between the conditional and marginal distributions over

*ζ*:

*ζ*and

*ψ*represent EF and soil moisture, respectively. The observation space of the target random variable EF was discretized using a Gaussian-optimized histogram bin width

*w*given by Scott (2004):where

*k*is the sample size of soil moisture and EF pairs. Following Nearing et al. (2016), a bin width of 0.01 m

^{3}m

^{−3}(1% volumetric water content) was applied for soil moisture. Integrations required for MI calculation in Eq. (4) are then approximated as summations over the empirical probability distribution function bins.

The rationale behind the application of this approach for SM–EF coupling strength is that the MI content between various soil moisture products and current/near-future EF can be used to quantify the value of each product for estimating/forecasting variations in surface energy fluxes and thus the general value of various soil moisture representations for land–atmosphere coupling applications. Specifically, we applied this approach to calculate the MI content between current soil moisture conditions (as reflected by

To quantify the standard error of NMI differences between various soil moisture products, we applied a nonparametric bootstrapping approach with 1000 replicates and calculated pooled average differences across all sites as an estimation of error bars, assuming spatially independent sampling error.

### e. Parameterization of T

Calculating SWI requires an estimate of the time-scale parameter. The general lack of information on microscale soil characteristics available at AmeriFlux sites necessitates a more generalized method to parameterize *T*, which involves the following three steps. The *T* was first optimized (i.e., *T*_{opt}) at each site so that the corresponding SWI time series had the highest possible MI content with EF time series at zero time lag (*τ* = 0). Then, the resulting (optimized) values of *T*_{opt} were used to establish a global relationship based on readily available observations from each site, including long-term (2001–14) annual mean LAI from MCD15A3 product, long-term (2001–12) annual mean LAI from GLASS product, annual mean precipitation (mm), annual mean net solar radiation (W m^{−2}), annual mean air temperature (°C), annual mean relative humidity (%), and the estimated De Martonne aridity index (De Martonne 1926). Stepwise-regression results indicate that long-term climatological air temperature is the only effective regressor of *T*_{opt} among this entire list of potential candidate predictors. Consequently, a single global linear relationship was established between the long-term mean air temperature at each site and *T*_{opt}. Estimations of *T* at all selected AmeriFlux sites based on this single fixed relationship were denoted as *T*_{reg}. It should be stressed that this regression relationship is treated here as a purely empirical result and that obtaining a clear theoretical motivation for its particular form is outside the scope of this study. In addition, to prevent unrealistic large estimates from this regression equation, we imposed an upper bound of 60 (days) on *T*.

Given that the success of SWI is likely to be dependent on the parameterization of *T*, we conducted the analysis using three different approaches for acquiring *T*: 1) *T* was individually optimized (i.e., *T*_{opt}) at each site, which represents an upper limit on the SWI performance; 2) *T* was globally regressed (i.e., *T*_{reg}) using the single linear relationship with annual mean air temperature as mentioned above, which seems to be a more realistic parameterization scheme; and 3) *T* was a constant value (i.e., *T*_{con}) for each site, which was taken to be the mean of *T*_{opt} from all selected AmeriFlux sites. The sensitivity of NMI(SWI) results to each of these parameterization approaches is discussed in section 3b.

## 3. Results

### a. Comparison between NMI( ) and NMI( )

Before examining the difference between *τ* = 0) reflect the skill of soil moisture product in diagnosing instantaneous land–atmosphere feedback, while positive lags (*τ* > 0) reflect the skill of prediction of future EF using current soil moisture conditions. It is seen that for anomaly time series of SM–EF pairs, the seasonal variation of *θ*_{V}) are notably higher in summer (June–August). For spring (March–May), summer (June–August), and autumn (September–November), the median of

Based on this, Fig. 3 plots the seasonal variation of NMI differences between *σ* sampling errors calculated from bootstrapping. Results in Fig. 3 demonstrate that, in general, vertically integrated soil moisture does not contain more information for diagnosing or predicting EF than surface soil moisture. This tendency is not sensitive to seasonal variations and is present regardless of whether the analysis is conducted on anomaly soil moisture and EF time series (Fig. 3a) or on the original time series containing seasonality (Fig. 3b).

Figure 4 plots the NMI difference between *σ* sampling error calculated using a 1000-replicate bootstrapping ensemble. Figure 4a illustrates that *σ* confidence interval for plotted differences typically contains zero). Furthermore, despite significant differences in

In terms of geographical distribution, symbol colors in Fig. 1 capture the *z* scores for *σ* sampling errors) for each of the selected AmeriFlux sites at lag *τ* = 0. In addition to confirming that

It is noteworthy that in Qiu et al. (2014), SCAN observations taken from fixed soil depths are employed to estimate the utility of multiple soil moisture variables in forecasting near-future vegetation conditions. However, in this study, the soil moisture measurements of AmeriFlux were taken from different depths (Table 1). To examine the potential impact of these variations, Fig. 5 plots the relationship between

### b. The added value of exponential filtering

In addition to the direct use of *T* parameter (i.e., *T*_{reg}, *T*_{opt}, and *T*_{con}; see section 2e), NMI(SWI) results for all land-cover types using observations from spring through autumn are plotted in Fig. 6.

Under the optimal parameterization conditions, SWI better diagnoses and forecasts EF than either *T*_{opt} case. However, in practice, such optimal parameterization will likely require either extensive site-specific calibration or a high level of site-specific information and will therefore be impossible to apply over spatially distributed domains. In contrast, for the cases of applying more practical globally parameterized and constant *T* approaches (*T*_{reg} and *T*_{con}), SWI is a comparable diagnostic EF monitor than *τ* = 0), and SWI does not improve upon *z* scores for NMI(SWI) minus *τ* = 0 do not demonstrate any clear geographical pattern.

### c. Sensitivity to temporal scales and evaluation metrics

#### 1) Sensitivity to temporal scales

To examine whether our previous conclusions are robust at multiple time scales, we also conducted the above analysis at a monthly time scale in comparison to the original 2-day time scale. Specifically, we selected AmeriFlux sites with temporal coverage of more than 10 years to conduct the analysis at monthly time scale. Overall, as the criterion of 10-yr coverage reduces the site number and reduces the sampling power, error bars are generally wider than those shown in Figs. 3 and 4 for 2-day results. However, outside of these differences, modifying the analysis to be based on a monthly time scale does not lead to any qualitative changes in results.

#### 2) Sensitivity to evaluation metrics

As discussed above, our use of NMI as an evaluation metric is based on the expectation that the relationship between soil moisture and EF will often be nonlinear and on the robustness of NMI in such cases. However, it is also instructive to consider the impact of other evaluation metrics. Therefore, in addition to NMI-based results in Fig. 6, we include results in Fig. 8 based on applying the Pearson correlation coefficient *R* to describe the strength of the linear relationship between soil moisture and current/near-future EF. The *R* between EF and three soil moisture time series *R*(SWI), respectively. To ensure consistency with NMI-based analysis in Fig. 6, sampled values of *R*(SWI) minus *T* parameterization schemes mentioned in section 2e, except that *T*_{opt} is now determined by maximizing the Pearson *R* (instead of MI) between SWI and EF at lag *τ* = 0.

Results based on the Pearson *R* are qualitatively similar to earlier NMI-based results. In particular, they reflect that surface measurements *θ*_{S} are equally, or even slightly more, valuable than deeper measurements *θ*_{V} for diagnosing and/or predicting EF. As with earlier NMI-based results, SWI performance is significantly better than *T* parameter is individually optimized (i.e., using *T*_{opt}) for each site. In contrast, for the constant *T* parameterization (i.e., using *T*_{con}), the transformation of *R*-based results in Fig. 8 suggest relatively better SWI results associated with the derivation of *T* from a global regression equation (i.e., using *T*_{reg}).

## 4. Conclusions

In this study, we calculated the normalized mutual information (NMI) between time series of three soil moisture products (with various vertical supports) and current/near-future EF time series to identify the impact of vertical support on information content of soil moisture for latent heat flux estimation. Specifically, we examined the surface energy flux information content contained in 1) vertically integrated soil moisture observations

Overall, results demonstrate that (surface only) *T* parameterization schemes. Using site-calibrated *T*, SWI is a significantly better monitor and forecaster of EF than *T* or/and constant *T*, SWI is comparable to *θ*_{V} and *θ*_{S} as a diagnostic EF monitor (Figs. 6, 7). Similar results are obtained when analyses are conducted at the monthly time scale and using evaluation metrics of Pearson *R* (Fig. 8).

These results are somewhat at odds with earlier results in Qiu et al. (2014) demonstrating that, using a parsimonious globally regressed *T* parameter, the application of exponential filtering (to convert *T* parameterizations with different levels of complexity should be adopted when targeting particular application. In this case, different target applications and different *T* parameterization approaches lead to different conclusions regarding the value of exponential filtering.

In summary, we see no clear evidence that the information content for surface energy flux predictions is enhanced by the vertical integration of superficial soil moisture observations. The results of this study suggest that it is unclear how soil moisture variations impact the surface energy balance and, therefore, how the land surface wetness measurements can (and should) be used to better constrain surface energy balances. Nonetheless, at the very least, we should reconsider the common assumption that the vertical support of satellite-based surface soil moisture retrievals must be increased before they can meaningfully be available to surface energy balance studies. However, it should be noted that our results are limited to just EF monitoring and prediction and the specific datasets that were examined here. To better understand how the soil moisture variations could be used to constrain surface energy balances, similar analyses using various sources of datasets (e.g., from both land surface models and remote sensing) should be addressed in future research.

## Acknowledgments

This work was supported by National Natural Science Foundation of China (Grant 41501450), Natural Science Foundation of Guangdong Province, China (Grant 2016A030310154), and the Fundamental Research Funds for the Central Universities (16lgpy06). We thank the anonymous reviewers for their helpful comments.

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