the Creative Commons Attribution 4.0 License.

the Creative Commons Attribution 4.0 License.

# Model representation of the coupling between evapotranspiration and soil water content at different depths

### Wade T. Crow

### Jianzhi Dong

### Grey S. Nearing

Soil water content (*θ*) influences the climate
system by controlling the fraction of incoming solar and longwave energy
that is converted into evapotranspiration (ET). Therefore, investigating the
coupling strength between *θ* and ET is important for the study of
land surface–atmosphere interactions. Physical models are commonly tasked
with representing the coupling between *θ* and ET; however, few
studies have evaluated the accuracy of model-based estimates of *θ* ∕ ET
coupling (especially at multiple soil depths). To address this issue, we use
in situ AmeriFlux observations to evaluate *θ* ∕ ET coupling strength
estimates acquired from multiple land surface models (LSMs) and an ET
retrieval algorithm – the Global Land Evaporation Amsterdam Model (GLEAM).
For maximum robustness, coupling strength is represented using the sampled
normalized mutual information (NMI) between *θ* estimates acquired at
various vertical depths and surface evaporation flux expressed as a fraction
of potential evapotranspiration (fPET, the ratio of ET to potential ET).
Results indicate that LSMs and GLEAM are generally in agreement with
AmeriFlux measurements in that surface soil water content (*θ*_{s})
contains slightly more NMI with fPET than vertically integrated soil water
content (*θ*_{v}). Overall, LSMs and GLEAM adequately capture
variations in NMI between fPET and *θ* estimates acquired at various
vertical depths. However, GLEAM significantly overestimates the NMI between
*θ* and ET, and the relative contribution of *θ*_{s} to total
ET. This bias appears attributable to differences in GLEAM's ET estimation
scheme relative to the other two LSMs considered here (i.e., the Noah model
with multi-parameterization options and the Catchment Land Surface Model, CLSM).
These results provide insight into improved LSM model structure and
parameter optimization for land surface–atmosphere coupling analyses.

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Soil water content (*θ*) modulates water and energy feedbacks between
the land surface and the lower atmosphere by determining the fraction of
incoming solar energy that is converted into evapotranspiration (ET; Seneviratne et al., 2010, 2013). In water-limited regimes, *θ*
exhibits a dominant control on ET, and therefore exerts significant
terrestrial control on the Earth's water and energy cycles. Accurately
representing *θ* ∕ ET coupling in land surface models (LSMs) is
therefore expected to improve our ability to project the future frequency of
extreme climates (Seneviratne et al., 2013).

A key question is how the constraint of *θ* on ET and sensible heat
(*H*) varies as *θ* is vertically integrated over deeper vertical soil
depths. Given the tendency for the timescales of *θ* dynamics to vary
strongly with depth, the degree to which the ET is coupled with vertical
variations in *θ* determines the temporal scale at which *θ*
variations are propagated into the lower atmosphere. Therefore, in order to
represent *θ* ∕ ET coupling, and thus land–atmosphere interactions in
general, LSMs must accurately capture the relationship between vertically
varying *θ* values and ET. Unfortunately, their ability to do so
remains an open question.

Recently, land surface–atmosphere coupling strength has been investigated by
sampling mutual information proxies (e.g., correlation coefficient or other
coupling indices) between time series of *θ* and ET observations (or
air temperature proxies for ET). Results suggest that, even when confined to
very limited vertical support (e.g., within the top 5 cm of the soil
column), surface *θ* estimates retain significant information for
describing overall *θ* control on local climate (Ford and Quiring,
2014; Qiu et al., 2014; Dong and Crow, 2018, 2019). These
findings are in contrast with the common perception that ET is constrained
only by *θ* values within deeper soil layers (Hirschi et al., 2014).
Hence, it is necessary to examine whether LSMs can realistically reflect
observed variations of *θ* ∕ ET coupling strength within the vertical
soil profile.

Previous studies examining the *θ* ∕ ET relationship have generally been
based on Pearson product–moment correlation (Basara and Crawford, 2002; Ford
et al., 2014), which captures only the strength of a linear relationship
between two variables. However, the coupling between *θ* and ET is
generally nonlinear. Therefore, non-parametric mutual information measures
are generally more appropriate. Nearing et al. (2018) used information
theory metrics (transfer entropy, in particular) to measure the strength of
direct couplings between different surface variables, including soil water
content, and surface energy fluxes at short timescales in several LSMs. They
found that the LSMs are generally biased as compared with the strengths of
couplings in observation data, and that these biases differ across different
study sites. However, they did not look specifically at the effect of
vertical water content profiles or of subsurface soil water content on
partitioning surface energy fluxes.

Here we apply the information theory-based methodology of Qiu et al. (2016)
to examine the relationship between the vertical support of *θ*
estimates and their mutual information (MI) with respect to ET. Our approach
is based on analyzing the MI content between ET and *θ* time series
acquired from both LSMs, and ET retrieval algorithm – the Global Land
Evaporation Amsterdam Model (GLEAM) – and AmeriFlux in situ observations. MI
values are then normalized by entropy in the corresponding ET time series to
remove the effect of inter-site variations to generate estimates of
normalized mutual information (NMI) between *θ* and ET. Both surface
(roughly 0–10 cm) soil water content (*θ*_{s}) and vertically
integrated (0–40 cm) soil water content (*θ*_{v}) are considered to
capture the impact of depth on NMI results. AmeriFlux-based NMI results are
then compared with analogous NMI results obtained from LSM-based and
GLEAM-based *θ* and ET time series.

The AmeriFlux network provides temporally continuous measurements of
*θ*, surface energy fluxes and related environmental variables for
sites located in a variety of North American ecosystem types, e.g., forests,
grasslands, croplands, shrublands and savannas (Boden et al., 2013). To
minimize sampling errors, AmeriFlux sites lacking a complete 3-year summer
months (June, July and August) daily time series between the years of 2003
and 2015 (i.e., $\mathrm{3}\times \mathrm{92}=\mathrm{276}$ daily observations in total) of
*θ*_{s}, *θ*_{v} and latent heat flux (LE) are
excluded here, resulting in the 34 remaining eligible AmeriFlux sites
listed in Table 1. These sites cover a variety of climate zones within the
contiguous United States (CONUS). Table 1 gives background information on
these 34 sites including local land-cover information. Hydro-climatic
conditions in each site are characterized using the aridity index (AI), calculated using CRU (Climate Research Unit, v4.02) monthly precipitation
and potential evaporation (PET) datasets.

As described above, *θ* ∕ ET coupling assessments made using AmeriFlux
observations are compared with those using state-of-the-art LSMs including
the Noah model with multi-parameterization options (NOAHMP) and Catchment
Land Surface Model (CLSM). In addition, *θ* and ET retrievals provided
by the Global Land Evaporation Amsterdam Model (GLEAM) are also considered.
See below for details on all three approaches. To avoid any spurious
correlations between *θ* and ET due to seasonality, all NMI analyses
are performed on *θ* and ET time series anomalies acquired during the
period 2003–2015. The *θ* and ET anomalies are calculated by removing
the seasonal cycle – defined as 31 d window averages centered on each
day of the year sampled across all years of the 2003–2015 historical data
record – from the raw *θ* and ET time series data. The analysis is
limited to the CONUS during summer months (June, July and August) when
*θ* ∕ ET coupling is expected to be maximized.

^{a} Was 5 cm prior to 13 April 2005.
^{b} Was 25 cm prior to 13 April 2005.
^{c} Was 5 cm prior to 1 January 2006.
^{d} Was 0–30 cm prior to 2007.
^{e} Unavailable prior to 2007. NA = not available.

## 2.1 Ground-based AmeriFlux measurements

The Level 2 (L2) AmeriFlux LE and *H* flux observations are based on
high-frequency (typically > 10 Hz) eddy covariance measurements
processed into half-hourly averages by individual AmeriFlux investigators.
LE and *θ* observations at a half-hour time step and without
gap-filling procedures are collected from the AmeriFlux Site and Data
Exploration System (see http://ameriflux.ornl.gov/, last access: November 2018). The LE and *θ*
observations are further aggregated into daily (00:00 to 24:00 UTC) values, and
daily LE is converted into daily ET using the latent heat of vaporization.
Daily ET values based on less than 30 % half-hourly coverage (i.e.,
< 15 half-hourly observations per day) are considered not
representative at a daily timescale and are therefore excluded.

Soil water content measurements are generally available at two discrete
depths that vary between the AmeriFlux sites (Table 1). Here, the top (i.e.,
closest to the surface) soil water content observation is always used to
represent surface soil water content (*θ*_{s}). Since the depth of
this top-layer measurement varies between 0 and 15 cm (see Table 1), we
consider the surface-layer measurement *θ*_{s} to be roughly
representative of 0–10 cm (vertically integrated) *θ*. For more details on AmeriFlux sites utilized here, see Raz-Yaseef et al. (2015).

Given variations in the depth of the lower AmeriFlux *θ* observations
(see Table 1), we applied a variety of approaches for estimating vertically
integrated soil water content (*θ*_{v}). Our first approach,
hereinafter referred to as Case I, is based on the application of an
exponential filter (Wagner et al., 1999; Albergel et al., 2008) to
extrapolate *θ*_{s} to a consistent 40 cm bottom-layer depth.
Therefore, only *θ*_{s} is used to derive *θ*_{v} and the
bottom-layer (or second-layer) AmeriFlux *θ* measurement is neglected
in this case. The application of the exponential filter requires a single
timescale parameter *T*. Since *θ* measurements from the United States
Department of Agriculture's Soil Climate Analysis Network (SCAN) are taken
at fixed soil depth, we utilized this dataset to determine the most
appropriate parameter *T* at AmeriFlux sites. Following Qiu et al. (2014),
first, we estimated the optimal parameter *T* (*T*_{opt}) for the extrapolation of
*θ* measurements from 10 to 40 cm depth and established a global
relationship between *T*opt and site-based NDVI (MOD13Q1 v006, 250 m, 16 d; ${T}_{\mathrm{opt}}=\mathrm{2.098}\times \mathrm{exp}(-\mathrm{1.895}\times $ (NDVI +0.6271)) +2.766). Then, this global relationship (goodness of fit *R*^{2}: 0.85) is
applied to AmeriFlux sites to extrapolate 0–10 cm *θ*_{s} time
series into 0–40 cm *θ*_{v}.

Previous research has suggested that such a filtering approach does not
significantly squander ET information present in actual measurements of
*θ*_{v} (Qiu et al., 2014, 2016). Nevertheless, since
the quality of *θ*_{v} estimates is important in our analysis, we
also calculated two additional cases where 0–40 cm *θ*_{v} is
estimated using (1) the bottom-layer soil water content measurement acquired
at each AmeriFlux site (hereinafter, Case II) and (2) linear interpolation of
*θ*_{s}, and the bottom-layer AmeriFlux soil water content
measurement (hereinafter, Case III). The sensitivity of key results to these
various cases is discussed below.

## 2.2 LSM- and GLEAM-based simulations

Simulations are acquired from the NOAHMP (Niu et al., 2011) and CLSM (Koster et
al., 2000) LSMs embedded within the NASA Land Information System (LIS, Kumar
et al., 2006) and the GLEAM ET retrieval algorithm (Miralles et al., 2011).
Both NOAHMP and CLSM are set up to simulate 0.125^{∘} *θ*
profiles at a 15 min time step using North America Land Data Assimilation
System, Phase 2 (NLDAS-2) forcing data. A 10-year model spin-up period (1992
to 2002) is applied for NOAHMP and CLSM.

NOAHMP numerically solves the one-dimensional Richards equation within four
soil layers of thicknesses of 0–10, 11–30, 31–60 and 61–100 cm. Major
parameterization options relevant to *θ* simulation include options
for canopy stomatal resistance parameterization and schemes controlling the
effect of *θ* on the vegetation stress factor *β*. Here we
employed the Ball–Berry-type stomatal resistance scheme and Noah-type soil
water content factor controlling the *β* factor. The specific
expressions are as follows:

where *θ*_{wilt} and *θ*_{ref} are, respectively, soil water
content at the wilting point (m^{3} m^{−3}) and reference soil water content
(m^{3} m^{−3}), which is set as field capacity during parameterization.
*θ*_{i} and Δ*z*_{i} are soil water content (m^{3} m^{−3}) and soil depth (cm) at *i*th layer, *N*_{root} and *z*_{root} are total
number of soil layers with roots and total depth (cm) of root zone,
respectively.

Following the Ball–Berry stomatal resistance scheme, the *θ*-controlled *β* factor and other multiplicative factors including
temperature and foliage nitrogen simultaneously determine the maximum
carboxylation rate *V*_{max} as follows:

where *V*_{max25} is maximum carboxylation rate at 25 ^{∘}C (µmol CO_{2} m^{−2} s^{−1}), *α*_{vmax} is a parameter sensitive to
vegetation canopy surface temperature *T*_{v}, *f*(*N*) is a factor representing
foliage nitrogen and *f*(*T*_{v}) is a function that mimics thermal breakdown of
metabolic processes. Based on *V*_{max}, photosynthesis rates per unit leaf area index (LAI)
including carboxylase-limited (Rubisco limited, denoted by *A*_{C}) type and
export-limited (for C_{3} plants, denoted by *A*_{S}) type are calculated, respectively. The minimum of *A*_{C}, *A*_{S} and the light-limited photosynthesis
rate determines stomatal resistance *r*_{s}, and consequently affects ET over
vegetated areas. For the complete NOAHMP configuration, please see Table S1
in the Supplement.

CLSM simulates the 0–2 and 0–100 cm soil water content and evaporative
stress as a function of simulated *θ* and environmental variables. ET
is then estimated based on the estimated evaporative stress and
land–atmosphere humidity gradients. Energy and water flux estimates are
iterated with soil state estimates (e.g., *θ* and soil temperature) to
ensure closure of surface energy and water balances. For a detailed
explanation of CLSM physics, please refer to Koster et al. (2000).

GLEAM is a set of algorithms dedicated to the estimation of terrestrial ET
and root-zone *θ* from satellite data. In this study, the latest
version of this model (v3.2a) is employed. In GLEAM, the configuration of
soil layers varies as a function of the land-cover type. Soil stratification
is based on three soil layers for tall vegetation (0–10, 10–100 and
100–250 cm), two layers for low vegetation (0–10 and 10–100 cm) and only one
layer for bare soil (0–10 cm; Martens et al., 2017).

The cover-dependent PET (mm d^{−1}) of GLEAM is calculated using the
Priestley and Taylor (1972) equation based on observed air temperature and
net radiation. Following this, estimates of PET are converted into actual
transpiration or bare soil evaporation (depending on the land-cover type,
ET (mm d^{−1})), using a cover-dependent, multiplicative stress factor
*S* (–), which is calculated as a function of microwave vegetation optical
depth (VOD) and root-zone *θ* (Miralles et al., 2011). The related
expressions are as follows:

where *E*_{i} is rainfall interception (mm), *S* essentially represents the fPET
(see Sect. 2.3) estimated by GLEAM, *θ*_{c} (m^{3} m^{−3}) is the critical soil water content and *θ**ω* (m^{3} m^{−3}) is the soil water content of the wettest layer, assuming that
plants withdraw water from the layer that is most accessible. Based on Eq. (4), GLEAM *S* (or fPET) tends to become more sensitive to *θ* in areas
of low VOD seasonality (i.e., low differences between VOD and VOD_{max}).
As for bare soil conditions, *S* is linearly related to surface soil water
content (*θ*_{1}):

To resolve variations in the vertical discretization of *θ* applied by
each model, we linearly interpolated NOAHMP, CLSM and GLEAM outputs into
daily 0–10 and 0–40 cm soil water content values using depth-weighted
averaging.

## 2.3 Variable indicating soil water content and surface flux coupling

Soil water content–ET coupling can be diagnosed using a variety of
different variables derived from ET, e.g., the fraction of PET (fPET, the
ratio of ET and PET) or the evaporative fraction (EF, the ratio of LE and
the sum of LE and sensible heat). Since ET is strongly tied to net radiation
(*R*_{n}; Koster et al., 2009), both fPET and EF are advantageous in that they
normalize ET by removing the impact of non-soil water content influences on
ET (e.g., net radiation, wind speed and soil heat flux (*G*)). However, since
sensible heat flux is not provided in the GLEAM dataset, we are restricted
here to using fPET.

It should be noted that the applied meteorological forcing data for NOAHMP and CLSM are somewhat different from those used for GLEAM. Therefore, to minimize the impact of this difference, NOAHMP and CLSM fPET are computed from North American Regional Reanalysis (NARR) using the modified Penman scheme of Mahrt and Ek (1984), while GLEAM fPET is calculated using its own internal PET estimates. To examine the impact of the PET source on the results, AmeriFlux fPET calculations are duplicated using both GLEAM- and NARR-based PET values.

## 2.4 Information measures

Mutual information (MI; Cover and Thomas, 1991) is a nonparametric measure of
correlation between two random variables. MI and the related Shannon-type
entropy (SE; Shannon, 1948) are calculated as follows. Entropy about a
random variable *ζ* is a measure of uncertainty according to its
distribution *p*_{ζ} and is estimated as the expected amount of
information from *p*_{ζ} sample:

Likewise, 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 (KL)
divergence (Kullback and Leibler, 1951) between the conditional and marginal
distributions over *ζ*:

In this context, the generic random variables *ζ* and *ψ* represent
fPET and *θ* (soil water content), respectively. The observation space
of the target random variable fPET is discretized using a fixed bin width.
As bin width decreases, entropy increases but mutual information asymptotes
to a constant value. On the other hand, increased bin width requires a greater
sample size, which cannot always be satisfied. The trick is choosing a bin
width where the NMI values stabilize with sample size. After a careful
sensitivity analysis, we choose a fixed bin width of 0.25 [–] for fPET and
make sure that each AmeriFlux site had enough samples to accurately estimate
the NMI, and change of this constant bin width from 0.1 to 0.5 [–] will not
significantly alter our conclusions. Following Nearing et al. (2016), a bin
width of 0.01 m^{3} m^{−3} (1 % volumetric water content) for
*θ* is applied. Integrations required for MI calculation in Eq. (7) are
then approximated as summations over the empirical probability distribution
function bins (Paninski, 2003).

By definition, the MI between two variables represents the amount of entropy
(uncertainty) in either of the two variables that can be reduced by knowing
the other. Therefore, the MI normalized by the entropy of the
AmeriFlux-based fPET measurements represents the fraction of uncertainty in
fPET that is resolvable given knowledge of the soil water content state
(Nearing et al., 2013). Unlike Pearson's correlation coefficient, MI is
insensitive to the impact of nonlinear variable transformations. Therefore,
it is well suited to describe the strength of the (potentially non-linear)
relationship between *θ* and fPET.

Here, we applied this approach to calculate the MI content between soil
water content representing different vertical depths (as reflected by
*θ*_{s} and *θ*_{v}) and fPET at each AmeriFlux site. All
estimated site-specific MI are normalized by the entropy of the
corresponding AmeriFlux-based fPET measurements to remove the effect of
inter-site entropy variations on the magnitude of NMI differences. The
resulting normalized MI calculations between both *θ*_{s} and
*θ*_{v} and fPET are denoted as NMI(*θ*_{s}, fPET) and
NMI(*θ*_{v}, fPET), respectively.

The underestimation of observed *θ* ∕ ET coupling via the impact of
mutually independent *θ* and ET errors in AmeriFlux observations (Crow
et al., 2015) is minimized by focusing on the ratio between NMI(*θ*_{S}, fPET) and NMI (*θ*_{v}, fPET). Therefore, relative
comparisons between NMI(*θ*_{s}, fPET) and NMI(*θ*_{v},
fPET) are based on examining the size of their mutual ratio NMI(*θ*_{S}, fPET) ∕ NMI (*θ*_{v}, fPET). To quantify the standard error
of NMI differences between various soil water content products, we applied a
nonparametric, 500-member bootstrapping approach and calculated the pooled
average of sampling errors across all sites assuming spatially independent
sampling error.

Finally, we also examined the impact of potential nonlinearity in the
*θ* ∕ ET relationship by comparing non-parametric NMI results with
comparable inferences based on a conventional Pearson's correlation
calculation. The correlation-based coupling strength between *θ*_{s}
and fPET is denoted as *R*(*θ*_{s}, fPET) and between *θ*_{v}
and fPET as *R*(*θ*_{v}, fPET).

## 3.1 Comparison of NMI(*θ*_{s}, fPET) and NMI(*θ*_{v}, fPET)

Figure 1 contains boxplots of modeled and observed NMI(*θ*_{s},
fPET) and NMI(*θ*_{v}, fPET), i.e., the relative magnitude of fPET
information contained in surface soil water content and
vertically integrated (0–40 cm) soil water content estimated from Case I,
sampled across all the AmeriFlux locations listed in Table 1. According to
the AmeriFlux ground measurements, median values of NMI(*θ*_{s},
fPET) and NMI(*θ*_{v}, fPET; across all sites) are near 0.3 [–].
This suggests that approximately 30 % of the uncertainty (i.e., entropy at
this particular bin width of 0.25 [–]) in fPET can be eliminated given
knowledge of either surface or vertically integrated soil water content
state. This is consistent with earlier results in Qiu et al. (2016) who
used similar metrics to evaluate *θ* ∕ EF (evaporative fraction)
coupling strength. The sampled medians of NMI(*θ*_{s}, fPET) and
NMI(*θ*_{v}, fPET) estimated by the NOAHMP and CLSM models are
similar to these (observation-based) AmeriFlux values. With the single
exception that the CLSM predicts much larger site-to-site variation in
NMI(*θ*_{s}, fPET).

In contrast, NMI(*θ*_{s}, fPET) and NMI(*θ*_{v}, fPET)
values sampled from GLEAM *θ* and fPET estimates show positive biases
(with median *θ* of about 0.5 and 0.4 [–] for NMI(*θ*_{s},
fPET) and NMI(*θ*_{v}, fPET), respectively) with respect to all
other estimates.

Using the 34 AmeriFlux site-collocated samples pixels for a paired *t* test, both
LSMs and GLEAM overall exhibit significantly (at the 0.05 level) higher
NMI(*θ*_{s}, fPET) compared to NMI(*θ*_{v}, fPET), implying the surface soil water content observations contain more fPET
information than vertically integrated soil water content observations.
However, the observed difference between NMI(*θ*_{s}, fPET) and
NMI(*θ*_{v}, fPET) is less discernible in AmeriFlux measurements
(Fig. 1a).

Here, AmeriFlux observations are used as a baseline for LSM and GLEAM
evaluation. However, it should be stressed that random observation errors in
*θ* and fPET will introduce a low bias into AmeriFlux-based estimates
of both NMI(*θ*_{s}, fPET) and NMI(*θ*_{v}, fPET; Crow et
al., 2015) and thus their difference as well. To address this concern, Fig. 1b plots the ratio of NMI(*θ*_{s}, fPET) and NMI(*θ*_{v},
fPET), which effectively normalizes (and therefore minimizes) the impact of
random observation errors. As discussed above, these ratio results
illustrate the general tendency for NMI(*θ*_{s}, fPET) to exceed
NMI(*θ*_{v}, fPET). They also highlight the tendency for GLEAM to
overvalue *θ*_{s} (relative to *θ*_{v}) when estimating
fPET. A second approach for reducing the random error of *θ* and fPET
measurement errors is the correction based on triple collocation (TC) applied
in Crow et al. (2015). However, this approach is currently restricted to
linear correlations and cannot be applied to estimate NMI. Future work will
examine extending the information-based TC approach of Nearing et al. (2017)
to the examination of NMI.

## 3.2 Sensitivity of AmeriFlux-based NMI(*θ*_{s}, fPET) ∕ NMI(*θ*_{V}, fPET)

As mentioned in Sect. 2.1, an important concern is the impact of
interpolation errors used to estimate 0–40 cm *θ*_{v} from
AmeriFlux *θ*_{s} observations acquired at non-uniform depths. To
ensure that different methods for calculating AmeriFlux *θ*_{v}
values do not affect the main conclusion of this study, we configured three
cases for *θ*_{v} calculation, and compared their NMI(*θ*_{S}, fPET) ∕ NMI(*θ*_{v}, fPET) results in Fig. 2. Case I reflects
the baseline use of the exponential filter described in Sect. 2.1. However,
slight changes to AmeriFlux results are noted if alternative approaches are
used. Specifically, AmeriFlux-based NMI(*θ*_{v}, fPET) increases and
closes the gap with NMI(*θ*_{s}, fPET) if the bottom-layer soil
water content measurements are instead directly used as *θ*_{v}
(Case II) or if 0–40 cm *θ*_{v} is based on the linear
interpolation of the two AmeriFlux *θ* observations (Case III); the
impact of this modest sensitivity on key results is discussed below.

In addition, switching from GLEAM- to NARR-based PET when calculating fPET
for AmeriFlux-based NMI(*θ*_{s}, fPET) and NMI(*θ*_{v},
fPET) does not qualitatively change results and produces only a very slight
(∼6 %) increase in the median NMI(*θ*_{s},
fPET) ∕ NMI(*θ*_{v}, fPET) ratio.

## 3.3 Spatial distribution of NMI(*θ*_{s}, fPET) and NMI(*θ*_{v}, fPET)

Figure 3 plots the spatial distribution of NMI(*θ*_{s}, fPET) and
NMI(*θ*_{v}, fPET) results for each of the individual 34 AmeriFlux
sites listed in Table 1. The climatic regime is represented by AI (aridity
index) values plotted as the background color in Fig. 3. It can be seen in
Fig. 3 that NMI(*θ*_{s}, fPET) estimates from LSMs and GLEAM are
spatially related to hydro-climatic conditions, as NOAHMP and CLSM predict
that *θ*_{s} is moderately coupled with fPET (i.e., NMI(*θ*_{S}, fPET) of 0.3–0.5 [–]) in the arid southwestern USA (AI < 0.2) and only loosely coupled with fPET in the relatively humid eastern USA.
A similar decreasing trend of NMI(*θ*_{s}, fPET) from the
southwestern to eastern USA is also captured by GLEAM. However, as noted
above, GLEAM generally overestimates NMI(*θ*_{s}, fPET) and
NMI(*θ*_{v}, fPET) compared to NOAHMP, CLSM and AmeriFlux. In
contrast, a relatively weaker spatial pattern emerges in AmeriFlux-based
NMI(*θ*_{s}, fPET) results. In addition, spatial patterns for
NMI(*θ*_{s}, fPET) are less defined than for NMI(*θ*_{v},
fPET) in all four datasets.

Scatterplots in Fig. 4 summarize the spatial relationship between LSM- and
GLEAM-based NMI(*θ*_{s}, fPET) and NMI(*θ*_{v}, fPET)
results versus AmeriFlux observations across different land use types. While
observed levels of correlation in Fig. 4 are relatively modest, there is a
significant level (*p* *<* 0.05) of spatial correspondence between modeled and observed NMI results only over forest sites, motivating the
need to better understand processes responsible for spatial variations in
NMI results. In addition, stratifying NMI(*θ*_{s},
fPET) ∕ NMI(*θ*_{v}, fPET) ratio results according to vegetation type
(Fig. A1 in the Appendix) confirms that NMI(*θ*_{s}, fPET) slightly exceeds
NMI(*θ*_{v}, fPET) across all vegetation types (and thus all rooting
depths characterizing each vegetation type). This suggests that our analysis
is not severely affected by variations in the depth of *θ* measurements.
For further discussion on the impact of land cover on NMI results, please
see Appendix A.

## 3.4 Sensitivity of NMI(*θ*_{s}, fPET) ∕ NMI(*θ*_{v}, fPET) ratio to climatic conditions

Figure 5 further summarizes the NMI(*θ*_{s}, fPET) ∕ NMI(*θ*_{V}, fPET) ratio as a function of AI for all four products (NOAHMP,
CLSM, GLEAM and AmeriFlux). Error bars represent the standard deviation of
sampling errors calculated from a 500-member bootstrapping analysis. With
increasing AI, there is a significant decreasing trend in both NMI(*θ*_{S}, fPET) and NMI(*θ*_{v}, fPET) for all three simulations,
with a goodness of fit above 0.5 (figure not shown). For all cases, the
NMI(*θ*_{s}, fPET) ∕ NMI(*θ*_{v}, fPET) ratios are
consistently greater than unity under all climatic conditions. However, the
estimated NMI(*θ*_{s}, fPET) ∕ NMI(*θ*_{v}, fPET) ratios from
all three simulations (NOAHMP, CLSM and GLEAM) exhibit quite different
trends with respect to AI. The NMI(*θ*_{s}, fPET) ∕ NMI(*θ*_{V}, fPET) ratio for CLSM decreases with increasing AI, with a moderate
goodness-of-fit value of 0.28, while GLEAM estimates of NMI(*θ*_{s},
fPET) ∕ NMI(*θ*_{v}, fPET) shows an opposite increasing trend with
increasing AI. Conversely, there is relatively lower sensitivity of the
NMI(*θ*_{s}, fPET) ∕ NMI(*θ*_{v}, fPET) ratio to AI captured
in the AmeriFlux measurements.

Connecting these findings to the spatial distribution of NMI(*θ*_{s},
fPET) and NMI(*θ*_{v}, fPET; Fig. 3) confirms that the relative
magnitudes of NMI(*θ*_{s}, fPET) and NMI(*θ*_{v}, fPET) for
both LSMs and GLEAM are spatially related to hydro-climatic regimes. In
contrast, this link is weaker in the AmeriFlux measurements which, except
for a small fraction of very low AI sites, do not appear to vary as a
function of AI. These conclusions are not qualitatively impacted by looking
at NMI(*θ*_{s}, fPET) and NMI(*θ*_{v}, fPET) differences, as
opposed to their ratio as in Fig. 5, or by looking at *R*(*θ*_{s},
fPET) and *R*(*θ*_{v}, fPET) instead of NMI.

Since transpiration dominates the global ET (Jasechko et al., 2013),
deep-layer soil water content (*θ*_{v}) is generally considered to
contain more ET information than that of surface soil water content
(*θ*_{s}), given that plant transpiration is balanced by root water uptake from deeper soils (Seneviratne et al., 2010). However, this
assumption is rarely tested using models and/or observations. Here, we apply
normalized mutual information (NMI) to examine how the vertical support of a
soil water content product affects its relationship with concurrent surface
ET.

Specifically, using AmeriFlux ground observations, we examine whether
(NMI-based) estimates of LSMs and GLEAM *θ*_{s} versus ET and
*θ*_{v} versus ET coupling strength accurately reflect observations
acquired at a range of AmeriFlux sites. In general, compared to the baseline
case of exponential filter extrapolated 40 cm bottom-layer *θ*_{v},
LSMs and GLEAM agree with AmeriFlux observations in that the overall fPET
information contained in *θ*_{s} is slightly higher than that of
*θ*_{v} (Fig. 1). However, the sensitivity analysis showed this
difference between NMI(*θ*_{s}, fPET) and NMI(*θ*_{v}, fPET)
diminishes when using different methods for calculating *θ*_{v}
using AmeriFlux observations (Fig. 2). As a result, this result should be
viewed with caution.

While NOAHMP- and CLSM-derived NMI(*θ*_{s}, fPET) and NMI(*θ*_{V}, fPET) results are generally consistent with the AmeriFlux
observations, GLEAM overestimates NMI(*θ*_{s}, fPET), NMI(*θ*_{V}, fPET) and the ratio NMI(*θ*_{s}, fPET) ∕ NMI(*θ*_{V}, fPET) relative to observations. Although both LSMs and GLEAM are
based on the same classical two-section (soil water content-limited and
energy-limited) ET regimes framework (Sect. 2.2), they differ in two
fundamental aspects. First, the evaporative stress factor *S* is represented as
a more direct and strong function of soil water content in GLEAM – see Eqs. (4) and (5) – which leads to the overestimation of the *θ* ∕ ET coupling
strength. This is consistent with our results that GLEAM generally
overestimates NMI(*θ*_{s}, fPET) and NMI(*θ*_{v}, fPET)
consistently across all land covers, compared to AmeriFlux-based estimates.
On the other hand, NOAHMP and CLSM approximate ET in the manner of
biophysical models, and expresses biophysical control on ET through the
stomatal resistance *r*_{s}, which is a function of multiple limiting factors
including *θ*. Therefore, the more complex ET scheme employed by
NOAHMP and CLSM would seem to mitigate the overestimation of NMI(*θ*_{S}, fPET) and NMI(*θ*_{v}, fPET), as other relevant factors
besides *θ* (such as temperature, foliage nitrogen) are also
considered in determining maximum carboxylation rate *V*_{max} and stomatal
resistance *r*_{s}, and consequently more realistic actual ET.

Second, the stress factor *β* in both LSMs considers the cumulative
effects of *θ*conditions along different layers (Eq. 1), while the
corresponding factor *S* in GLEAM only uses the wettest soil layer condition,
which is top layer at most sites. This likely explains the overestimation of
the NMI(*θ*_{s}, fPET) ∕ NMI(*θ*_{v}, fPET) ratio by GLEAM.

Nevertheless, we would like to stress that all approaches considered in our
paper contain (at their core) a parameterized relationship between *θ*
and ET. While the implications of mis-parameterizing this relationship are
arguably more severe for a land surface model, we argue that the issue
remains relevant for any approach (such as GLEAM) that utilizes a water
balance (and/or data assimilation system) approach to estimate *θ*
and, in turn, uses *θ* to constrain ET. Regardless of the complexity
that a given approach employs, failing to accurately describe the
relationship between ET and (large number of potential) environmental
constraints should eventually degrade the robustness of the model, whether
it is employed as a retrospective, diagnostic or predictive manner. To
examine this issue directly, Fig. 6 plots the relationship between GLEAMS
bias in NMI(*θ*_{s}, fPET) ∕ NMI(*θ*_{v}, fPET) ratio versus
the RMSE of daily GLEAM ET simulations for a range of AmeriFlux sites. There
is a positive correlation between the two quantities, which suggests that
GLEAM overestimation of *θ* ∕ ET coupling during the summer may
undermine the accuracy of its daily ET retrievals. It should be noted that
GLEAM simultaneously overestimates both NMI(*θ*_{s}, fPET) and
NMI(*θ*_{v}, fPET); however, the impact of this mis-parameterization
impact on GLEAM ET accuracy is most obvious when plotted against the ratio
NMI(*θ*_{s}, fPET) ∕ NMI(*θ*_{v}, fPET).

Although the median values of NMI(*θ*_{s}, fPET) and NMI(*θ*_{V}, fPET) predicted by NOAHMP and CLSM are generally in line with
AmeriFlux observations, they are more spatially related to hydro-climatic
conditions (as summarized by AI) than their counter parts acquired from
AmeriFlux measurements. Seen from the plot of NMI(*θ*_{s},
fPET) ∕ NMI(*θ*_{v}, fPET) ratio as a function of AI (Fig. 5), the
modeled and observed median of NMI(*θ*_{s}, fPET) ∕ NMI(*θ*_{V}, fPET) ratio decreases with increasing AI, and the decreasing trend
is particularly clear when AI is lower than 1.0 [–]. In contrast, there is
relatively lower sensitivity to aridity exhibited in the AmeriFlux
measurements.

These results provide several key insights into future land–atmosphere
coupling analysis and LSMs, as well as ET algorithm development. First, all
the datasets – both model-based and ground-observed – indicate that
*θ*_{s} contains at least as much ET information as *θ*_{v}.
Hence, remote-sensing land surface soil water content datasets are suitable,
and should be considered, for analyzing the general interaction between land
and atmosphere, e.g., soil water content–air temperature coupling (Dong
and Crow, 2019) and the interplay of soil water content and precipitation
(Yin et al., 2014). Additionally, future generations of GLEAM may consider
more sophisticated evaporation stress functions, which may improve its
accuracy in representing the soil's control on local ET. This may, in turn,
improve the accuracy of the GLEAM ET product. Finally, our results demonstrate
that modeled *θ* ∕ ET is more sensitive to hydro-climates than the
observed relationship. Modifying the model structures to reduce such
sensitivity might be necessary for accurately representing the interaction
of land surface and atmosphere across different climate zones. This may lead
to more realistic projections of future drought-induced heat waves, when
coupled with general circulation models.

We performed an additional sensitivity analysis to explicitly demonstrate the
effect of different vegetation land-cover types and consequently different
rooting depths (or *θ*_{v} measurement depths) on the NMI(*θ*_{S}, fPET) ∕ NMI(*θ*_{v}, fPET) ratio, and plotted these results
in Fig. A1. The figure confirms that, consistent with AmeriFlux, both LSMs
and GLEAM predict that NMI(*θ*_{s}, fPET) is slightly higher than
NMI(*θ*_{v}, fPET) over most vegetation types, and GLEAM
overestimates NMI(*θ*_{s}, fPET) ∕ NMI(*θ*_{v}, fPET) for most
vegetation types.

Ground-based soil water content and surface flux data are available from https://ameriflux.lbl.gov/login/?redirect_to=/data/download-data/ (last access: February 2020) (Raz-Yaseef et al., 2015). GLEAM dataset is available from https://www.gleam.eu/ (last access: February 2020). LSMs simulations of NOAHMP and CLSM used in this study are available by contacting the authors.

The supplement related to this article is available online at: https://doi.org/10.5194/hess-24-581-2020-supplement.

JQ and WTC conceptualized the study. JD helped preparing the LSMs simulation. GSN assisted in the mutual information analysis. JQ carried out the analysis and wrote the first draft manuscript and WTC refined the work. All authors contributed to the analysis, interpretation and writing.

The authors declare that they have no conflict of interest.

This research has been supported by the National Natural Science Foundation of China (grant nos. 41971031, 41501450 and 51779278), the Natural Science Foundation of Guangdong Province (grant no. 2016A030310154).

This paper was edited by Dominic Mazvimavi and reviewed by two anonymous referees.

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*θ*) at different depths is key to investigating land–atmosphere interaction. Here we examine whether the model can accurately represent surface

*θ*(

*θ*

_{s}) versus ET coupling and vertically integrated

*θ*(

*θ*

_{v}) versus ET coupling. We find that all models agree with observations that

*θ*

_{s}contains slightly more information with fPET than

*θ*

_{v}. In addition, an ET scheme is crucial for accurately estimating coupling of

*θ*and ET.

*θ*) at...