Using data assimilation to identify diffuse recharge
mechanisms from chemical and physical data in the
unsaturated zone
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Citation
Ng, Gene-Hua Crystal et al. “Using Data Assimilation to Identify
Diffuse Recharge Mechanisms from Chemical and Physical Data
in the Unsaturated Zone.” Water Resources Research 45.9
(2009): 1-18. CrossRef. Web. Copyright 2009 by the American
Geophysical Union.
As Published
http://dx.doi.org/10.1029/2009wr007831
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American Geophysical Union
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Wed May 25 22:03:08 EDT 2016
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http://hdl.handle.net/1721.1/77891
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WATER RESOURCES RESEARCH, VOL. 45, W09409, doi:10.1029/2009WR007831, 2009
Using data assimilation to identify diffuse recharge mechanisms
from chemical and physical data in the unsaturated zone
Gene-Hua Crystal Ng,1 Dennis McLaughlin,1 Dara Entekhabi,1 and Bridget Scanlon2
Received 9 February 2009; revised 21 May 2009; accepted 17 June 2009; published 16 September 2009.
[1] It is difficult to estimate groundwater recharge in semiarid environments, where
precipitation and evapotranspiration nearly balance. In such environments, groundwater
supplies are sensitive to small changes in the processes that control recharge. Numerical
modeling provides the temporal resolution needed to analyze these processes but is highly
sensitive to model errors. Natural chloride tracer measurements in the unsaturated zone
provide more robust indicators of low recharge rates but yield estimates at coarse time
scales that mask most control mechanisms. This study presents a new probabilistic
approach for analyzing diffuse recharge in semiarid environments, with an application to
study sites in the U.S. southern High Plains. The approach uses data assimilation to
combine model predictions and chloride-based recharge estimates. It has the advantage of
providing probability distributions rather than point values for uncertain soil and
vegetation properties. These can then be used to quantify recharge uncertainty. Estimates
of moisture flux time series indicate that percolation (or potential recharge) at the data sites
is episodic and exhibits interannual variability. Most percolation occurs during intense
rains when crop roots are not fully developed and there is ample antecedent soil moisture.
El Niño events can contribute to interannual variability of recharge if they bring rainy
winters that provide wet antecedent conditions for spring precipitation. Data assimilation
methods that combine modeling and chloride observations provide the high temporal
resolution information needed to identify mechanisms controlling diffuse recharge and
offer a way to examine the effects of land use change and climatic variability on
groundwater resources.
Citation: Ng, G.-H. C., D. McLaughlin, D. Entekhabi, and B. Scanlon (2009), Using data assimilation to identify diffuse recharge
mechanisms from chemical and physical data in the unsaturated zone, Water Resour. Res., 45, W09409, doi:10.1029/2009WR007831.
1. Introduction and Background
[2] Characterization of groundwater recharge is critical
for sustainable management of water resources, control of
subsurface contamination, and better understanding of
hydrologic and ecological variability. It is well known that
the primary controls on recharge are meteorology, soil
properties, vegetation, and topography. These controls
interact to create the distinctive conditions that result in
recharge. Uncertainties about control mechanisms make it
difficult to predict long-term effects of climate change,
meteorological variability, and land use change on groundwater resources.
[3] De Vries and Simmers [2002] categorize recharge as
‘‘diffuse’’ (or ‘‘direct’’) when it originates from precipitation
that infiltrates vertically from the surface directly to the
water table. By contrast, nondiffuse (‘‘localized’’ or ‘‘focused’’) recharge travels laterally at (or near) the land
surface and then collects in streams or topographic depres1
Department of Civil and Environmental Engineering, Massachusetts
Institute of Technology, Cambridge, Massachusetts, USA.
2
Bureau of Economic Geology, Jackson School of Geosciences,
University of Texas at Austin, Austin, Texas, USA.
sions before it infiltrates. Both diffuse and localized recharge often travel via preferential pathways, such as
through cracks or root tubules, rather than exclusively
through the soil matrix. Preferential flow is especially
difficult to characterize or predict.
[4] Although diffuse recharge may occur infrequently in
arid and semiarid climates, it can still have significant
impact on the overall groundwater budget. Kearns and
Hendrickx [1998] point out that small amounts of diffuse
recharge spread over a large area can yield significant
volumetric contributions to groundwater in semiarid regions
of New Mexico. Also, in areas where deep-rooted vegetation has been replaced by shallow-rooted crops or pasture,
diffuse recharge rates can increase significantly [e.g., Cook
et al., 1989; Scanlon et al., 2007]. In fact, increases in
diffuse recharge induced by land use change have created
major environmental problems in Australia, where salt
trapped in the unsaturated zone has been flushed into
valuable ground and surface water reservoirs [Cook et al.,
1989; Leaney et al., 2003].
[5] In this paper we examine the interacting processes
that control diffuse recharge, in order both to better understand the relevant mechanisms and to provide a better basis
for predicting impacts of land use and climate change on
subsurface water supplies. Gee and Hillel [1988] observed
Copyright 2009 by the American Geophysical Union.
0043-1397/09/2009WR007831
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that diffuse recharge in semiarid regions is likely to occur
episodically rather than continuously. This has been documented in a number of settings [see, e.g., Zhang et al.,
1999a, 1999b; Lewis and Walker, 2002]. An understanding
of the origin and timing of discrete events is needed for
proper characterization of diffuse recharge. In particular, we
need to be able to determine the relative importance of
downward moisture movement driven by precipitation
events and upward moisture movement driven by evaporation and root uptake. If downward forces prevail for
even a brief period, infiltrating precipitation can move
from the surface past the root zone and eventually become
recharge. Otherwise, it is taken up by vegetation and/or
evaporated, and there is little or no moisture left to
recharge groundwater.
[6] Scanlon et al. [2002] provide a comprehensive review
of common methods for estimating recharge. One option is
to use numerical models of the unsaturated zone to predict
recharge from precipitation observations and various soil
and vegetation properties [e.g., Zhang et al., 1999b; Small,
2005; Keese et al., 2005]. Although numerical models are
undoubtedly useful, many studies [e.g., Gee and Hillel,
1988; Allison et al., 1994; Phillips, 1994] caution against
their application in semiarid environments. Because diffuse
recharge rates in such settings can be quite small relative to
precipitation and evaporation, they are very sensitive to
uncertain model parameterizations and input errors. For this
reason, tracer-based recharge estimation methods are favored by Gee and Hillel [1988] and Allison et al. [1994] in
semiarid environments. Natural tracers such as meteoric
chloride are particularly popular due to their ubiquitous
availability and increased sensitivity at lower recharge rates.
[7] Chloride concentrations in the unsaturated and/or
saturated zone are most often employed for recharge estimation using a simple chloride mass balance (CMB) approach, which assumes that, over the long-term, the average
rate of chloride deposition with precipitation balances the
rate of chloride flushed out of unsaturated zone soil water
(below the root zone) [Scanlon et al., 2002]. When surface
runoff/run-on is negligible (such as in low-relief areas), soil
water chloride is of meteoric origin, and chloride is fully
displaced via piston flow, this can be expressed as
P½ClP ¼ R½ClSW ;
ð1Þ
where P is the precipitation rate, [ClP] is the atmospheric
chloride concentration (accounting for wet and dry fallout),
R is the percolation rate below the root zone, and [Clsw] is
the soil water chloride concentration. Negligible amounts of
chloride are generally taken up by plants, making the above
an acceptable approximation. Although this steady state
mass balance approach can provide robust average recharge
estimates, it cannot resolve the fine time-scale dynamics that
link diffuse episodic recharge to transient surface conditions
(weather and changes in vegetation).
[8] Quasi-transient and generalized versions of chloride
mass balance (CMB) provide some temporal information by
associating particular portions of the unsaturated profile
with particular times in the past [Cook et al., 1992; Murphy
et al., 1996; Ginn and Murphy, 1997]. The chloride front
displacement method tracks solute flushing caused by
changes in land use [Allison and Hughes, 1983; Walker et
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al., 1991]. However, Cook et al. [1992] showed that
dispersion and mixing greatly limit the temporal resolution
of these alternatives. Chloride data do not appear to adequately capture seasonal effects, annual dynamics, or other
short-duration events responsible for episodic recharge.
[9] In summary, it seems fair to say that numerical
modeling provides temporally detailed, yet potentially inaccurate, recharge estimates, while chloride-based recharge
estimation methods produce robust, but less detailed, timeintegrated estimates when the CMB conditions cited above
are met. Although the two methods are complementary,
few studies have combined them to obtain recharge
estimates that have the temporal detail and accuracy
needed to fully describe episodic events. Some investigators [e.g., Cook et al., 1992; Flint et al., 2002; Scanlon et
al., 2003] use chloride measurements to validate numerical
models, but they do not actually combine the two sources
of information.
[10] In this paper we estimate recharge with a probabilistic data assimilation approach that integrates model predictions and unsaturated zone data while properly
accounting for sources of uncertainty. Data assimilation
methods, including Kalman filtering and variational least
squares methods, have been used in a number of other
hydrological applications [e.g., Kitanidis and Bras, 1980;
McLaughlin et al., 1993; Reichle et al., 2001; Margulis et
al., 2002; Reichle et al., 2002; Vrugt et al., 2005; Dunne
and Entekhabi, 2006]. Here we use a particular data
assimilation approach commonly known as importance
sampling (discussed more fully in section 2). This approach
generates populations of physically plausible model parameters (e.g., soil and vegetation properties), soil moisture
profiles, and recharge histories that are consistent with
model predictions and field observations. Importance sampling explicitly accounts for both model and measurement
uncertainty by weighting each source of information
according to its reliability.
[11] In the discussion that follows, we distinguish percolation, the moisture flux below the root zone (1.5 m in our
study), and recharge, the moisture flux at the water table.
The two are generally equivalent over the long term because
percolation, which we presume is beyond the influence of
evapotranspiration, eventually reaches the water table to
become recharge (with some delay). Recharge is the quantity of most interest from the perspective of groundwater
supply. However, its temporal dynamics are significantly
attenuated, especially if the water table is deep. Because
percolation responds more to episodic precipitation events
and other surface conditions, it provides a clearer picture of
the physical mechanisms that control recharge. For this
reason, our analysis is based on estimates of percolation
rather than recharge. However, our ultimate interest is in the
effect of land use practices and climate on recharge.
[12] This paper demonstrates how importance sampling
can be used to estimate episodic percolation in a semiarid
region affected by land use change and to provide the
insight needed to understand recharge mechanisms. Our
estimates of percolation and related quantities are derived
from physical and chemical field data collected at research
sites in the semiarid southern High Plains (SHP) region of
Texas. Over the past century, replacement of native grassland at these sites with rain-fed (dryland) cotton crops has
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led to increased flushing of unsaturated zone chloride,
resulting in increased diffuse recharge [Scanlon et al.,
2007]. This land use change has provided a strong chloride
signal that facilitates estimation of generally unobservable
soil and vegetation properties.
[13] Although CMB has been used in many past studies,
Gee et al. [2005] suggest that chloride-based recharge
estimates can be misleading when low chloride concentrations (due to increased flux) approach instrument detection
levels. Those errors can be further compounded when soil
water chloride is not fully flushed with piston flow. Good
agreement between CMB and chloride front displacement
calculations in the SHP indicate that these are not major
concerns and that measured chloride concentrations are
informative about flux rates at our study sites [Scanlon et
al., 2007].
[14] Our study of diffuse recharge mechanisms and related soil and vegetation properties begins in section 2 with a
brief review of Bayesian data assimilation concepts. Section
3 illustrates the application of importance sampling to the
diffuse recharge problem. Section 3 includes discussions of
the field sites, the unsaturated zone model, selection of
model inputs, and representation of uncertainty. Section 4
examines the percolation and property estimates obtained
by assimilating data from the SHP field sites. These
estimates indicate how episodic percolation is controlled
by a combination of meteorological, soil, and vegetation
factors. A comparison of percolation estimates with precipitation records and cropping schedule reveals the special
conditions that make episodic percolation, and thus recharge, possible at agricultural sites such as the two sites
considered here. These conditions are summarized in
section 5.
2. Data Assimilation and Importance Sampling
[15] The two main components of a data assimilation
problem are a numerical model and a set of measurements.
We assume these are described by a state equation model f
and a measurement model h, respectively:
x1:T ¼ f ðx0 ; p; b1:T ; e1:T Þ
ð2Þ
y1:T ¼ hðx1:T ; h1:T Þ;
ð3Þ
where x1:T is the vector of system states (prognostic
variables) defined over a time period of length T, x0 is the
initial state vector, p is the parameter vector, b1:T is the
boundary condition vector over time, e1:T is the state model
error vector not included in other inputs, y1:T is the
observation vector, and h1:T is the observation error vector.
[16] For the recharge problem of interest here, f represents
a combined soil moisture and solute transport model; x
includes soil moisture, chloride concentrations, and moisture fluxes at different depths; p includes soil and vegetation
parameters; b includes meteorological and solute deposition
forcing at the ground surface; and y includes soil moisture
and chloride concentration measurements in a vertical
unsaturated zone profile. Percolation (the moisture flux at
1.5 m) is a model output (diagnostic variable) that is
determined completely by the model inputs and states. In
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our study, measurements at each site are taken at only one
time, and thus (3) simplifies to
yT ¼ hðxT ; hT Þ:
ð4Þ
Many model calibration methods are designed to estimate a
single parameter set that provides a ‘‘best fit’’ to observations. When models and observations are highly uncertain,
it may be more useful to provide probabilistic estimates that
reveal the range of possible values rather than a single ‘‘best
estimate.’’ In this case, the objective of the estimation
procedure is to derive the conditional (posterior) probability
density p(Xjy) of an uncertain model input, state, or output
(indicated by the generic symbol X). This conditional
density conveys everything we know about X given
uncertain information from the model and measurements.
[17] Bayes’ theorem relates the conditional density p(Xjy)
to the unconditional (prior) probability density p(X) and to
the likelihood function p(yjX), expressed as a function of X
for a given measurement y:
pð X jyÞ ¼
pð yjX Þpð X Þ
:
pð yÞ
ð5Þ
It is sometimes possible to derive closed form expressions
for the unconditional probability densities of the model
states and outputs by combining the state equation given in
(1) with specified input probability densities. Such a closed
form derivation is not possible for our recharge problem.
Instead, we adopt a Monte Carlo approach and work with
samples from the relevant densities rather than the densities
themselves. Our approach to Bayesian data assimilation is
carried out in three steps:
[18] 1. Generate equally likely unconditional realizations
of uncertain model inputs (e.g., meteorological, soil, and
vegetation properties) from specified input probability density functions.
[19] 2. Use Monte Carlo simulation to derive a corresponding set of equally likely realizations of the unconditional model states (e.g., the soil moisture and chloride
concentration profiles) and model outputs (e.g., percolation). Each Monte Carlo sample includes a set of associated
inputs, states, and outputs.
[20] 3. Use importance sampling to reweight each Monte
Carlo sample to reflect its ability to match observations. The
reweighted samples provide the information needed to
construct approximate conditional probability densities for
the model inputs, states, and outputs.
[21] Monte Carlo simulation and importance sampling
represent the probability densities appearing in (5) discretely,
using a set of i = 1, . . ., N sample vectors (Xi) containing
input, state, and output variables, together with a set of
associated unconditioned weights (wiU) or conditioned
weights (wiC):
pð X Þ wiU d X X i
ð6Þ
i¼1
pð X jyÞ N
X
wiC d X X i
ð7Þ
i¼1
N
X
i¼1
3 of 18
N
X
wiU ¼ 1
N
X
i¼1
wiC ¼ 1;
ð8Þ
NG ET AL.: RECHARGE MECHANISMS
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Figure 1. Importance sampling example with unconditional probability distribution (black curve), observation
(dashed line), and conditional probability distribution (gray
curve). The set of points Xi is drawn at random from the
unconditional distribution and is indicated by the asterisks.
Unconditioned and conditioned particle weights are represented by black and gray circle sizes, respectively. The
points have equal unconditioned weights, but the conditioned weights are different, with higher values assigned to
points closer to the observation. The weighted average of
the Xi values is the mean of the conditional probability
distribution shown.
where d( ) is the Dirac delta function. The model input
variables of Xi are drawn at random from a population of
equally likely prior samples that conform to a specified
unconditional density p(X) (step 1 above). The corresponding model state or output component of Xi is derived by
running the simulation model with the particular set of
random inputs (step 2 above). Usually only some of the
inputs are treated as random variables, while the others are
assumed to be deterministic (i.e., uncertainty is not
accounted for). Because each sample generated in the
Monte Carlo procedure is equally likely, the unconditional
sample weights are all wiU = 1/N.
[22] The measurement conditioning process (step 3
above) is accomplished by adjusting the equal weights
associated with the Monte Carlo samples to reflect the
effect of measurements. In particular, the weights wiC of
the conditional density are given by substituting (6), (7), (8),
and wiU = 1/N into (5) to obtain
wiC ¼
pð yjX i Þ
:
N
X
p yjX i
ð9Þ
i¼1
For example, if measurement errors are additive, then yT =
h(xT, hT) = h(xT) + hT, and the likelihood function appearing
in (9) may be computed from
p yjxi ¼ pyjxi yjxi ¼ phT y h xiT ;
ð10Þ
where the subscripts on the probability densities refer to the
random variables of interest while the arguments refer to
the threshold values used to evaluate probabilities. The
measurement error probability density phT[hT] is usually
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presumed to have a convenient closed form, such as a
Gaussian or uniform density. This density should reflect the
effects of instrumentation and recording errors, as well as
representativeness errors that account for scale differences
between the observation support and model grid cells.
[23] The importance sampling approach to Bayesian
conditioning outlined above ensures that more weight is
given to samples that provide a good match to observations.
In our application, this has the effect of shifting the
probability densities of observed states toward the measurements. This is illustrated with a simple univariate example
in Figure 1.
[24] Importance sampling methods are commonly encountered in particle filters, which are sequential Bayesian
data assimilation algorithms. Particle filters are popular in
tracking and signal processing [Djurić et al., 2003], and
they have also been applied to hydrologic problems
[Moradkhani et al., 2005; Zhou et al., 2006; Pan et al.,
2008; Weerts and El Serafy, 2006; Smith et al., 2008]. Our
importance sampling procedure is a nonsequential (single
time) adaptation of the sequential importance resampling
(SIR) particle filter [Arulampalam et al., 2002].
[25] Note that importance sampling can be inefficient if
the unconditional probability density of the states is a poor
estimate of the conditional density. In this case, many
samples will be needed to obtain a few low-probability
realizations that match the data. This problem is exacerbated
when the state vector is high-dimensional, and generation of
samples is computationally demanding [Daum and Huang,
2003]. Because our recharge problem considers only a
single vertical profile and has a relatively small number of
uncertain states and inputs, importance sampling is computationally feasible, especially if care is taken to specify
physically reasonable unconditional probability densities.
Importance sampling has the distinct advantage of deriving
conditional samples directly from Bayes’ theorem without
relying on the implicit Gaussian and linearity assumptions
required in Kalman filtering and least squares variational
methods, or on heuristic measures used in calibration
approaches [McLaughlin, 2007].
3. Application of Importance Sampling to the
Recharge Problem
[26] The basic objective of our Bayesian data assimilation
procedure is to estimate the conditional probability densities
of percolation at various times and the conditional densities
of the uncertain model inputs used to predict percolation.
The desired conditional densities are constructed from a
large number of sample values and weights using the Monte
Carlo simulation and importance sampling procedures described in section 2. In our case, each Monte Carlo sample
includes values for uncertain model inputs (e.g., vegetation
and soil properties and meteorological variables), values for
the model states (e.g., soil moisture and chloride), and
values for model outputs derived from the inputs and states
(e.g., moisture fluxes used to quantify percolation). The
weights obtained from the importance sampling procedure
indicate the relative probability of each sample. The basic
components of the data assimilation process are shown in
Figure 2 and described in sections 3.1– 3.4.
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NG ET AL.: RECHARGE MECHANISMS
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Figure 2. Diagram of recharge estimation approach. Random samples of states and outputs are
generated via multiple model runs using different inputs, as represented by multiple arrows in the
diagram. Conditional statistics of the uncertain variables are obtained by associating the conditioned
weights obtained from importance sampling with the sample values.
3.1. Measurements at the Study Sites
[27] The measurements used in our data assimilation
study were obtained from study sites located in the southern
High Plains (SHP) region of the United States. This 75,000km2 region spans parts of northern Texas and eastern New
Mexico and is characterized by flat topography and about
16,000 draining playas or ephemeral lakes (see Figure 3).
The SHP is considered to have a semiarid climate with a
mean annual precipitation in the range of 375 – 500 mm.
About 75% of precipitation falls during May through
October in convective storms. In parts of the SHP with
native grassland and shrubland, recharge occurs almost
exclusively through playas [Wood and Sanford, 1995;
Scanlon and Goldsmith, 1997].
[28] Replacement of natural vegetation with shallowrooted crops (much of it rain-fed cotton) in the early
1900s led to increases in diffuse recharge. The natural
vegetation was mostly replaced with cotton, often grown
without irrigation in continuous monoculture. Over 2003 to
2006, Scanlon et al. [2007] drilled and sampled 20 unsaturated zone boreholes in rain-fed cotton areas of the SHP
(indicated in Figure 3) to investigate the effect of the land
use change on recharge. Scanlon et al. [2007] found
widespread occurrence of flushed meteoric chloride in their
profiles. This result is similar to Australian studies that
found low chloride concentrations in the unsaturated zone
following clearing of eucalyptus trees [Cook et al., 1989;
Kennet-Smith et al., 1994].
[29] This evidence of flushing and downward moisture
movement following land use change is in marked contrast
to the slightly upward moisture fluxes generally observed in
natural interplaya environments. The CMB method described by equation (1) should be applicable for the relatively flat study area. Simulations further support this with
insignificant runoff amounts. CMB calculated fluxes in the
SHP ranged from 4.8 to 70 mm/a in individual profiles
under cropland [Scanlon et al., 2007]. Such changes in
recharge above the natural level near zero have led to rises
in water table levels, which could, in turn, eventually lead to
waterlogging and possibly compromise water quality
through salinization [Scanlon et al., 2005].
Figure 3. Southern High Plains (SHP) map (adapted from
Scanlon et al. [2007, Figure 1]). Inset shows location of
SHP in Texas and New Mexico. Data sites used in this study
are fully flushed D06 –02 and partly flushed L05 – 01.
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Figure 4. Observed profiles for D06 – 02 (circles) and L05 – 01 (asterisks). Chloride and soil moisture
observations within root zone area, indicated in gray, are not included in the assimilation.
[30] Half of the chloride profiles examined by Scanlon et
al. [2007] exhibited low chloride concentrations throughout,
suggesting that they were fully flushed with postdevelopment moisture fluxes. The remaining profiles exhibited high
concentration bulges at depth, suggesting the presence of a
postdevelopment region affected by downward recharge
lying above a predevelopment dry region that experienced
little or no recharge. The positions of chloride bulges in the
partially flushed profiles convey important information
about the flux history, because deeper profile locations
reveal conditions further back in time. Scanlon et al.
[2007] used this profile information to derive chloride front
displacement estimates that corroborated CMB results. In
the analysis presented here, we examine a representative
profile from each group: the fully flushed profile D06 – 02,
and the partially flushed profile L05 – 01. Soil samples for
these profiles were collected on 14 February 2006 and 26
May 2005, respectively.
[31] Unsaturated zone profile measurements given by
Scanlon et al. [2007] include chloride concentration, soil
moisture, matric potential, and soil texture (percent clay,
silt, and sand). Chloride concentrations were obtained by
first drying soil samples, adding water, and shaking, and
then using ion chromatography to measure concentrations in
the supernatant. Soil moisture was measured gravimetrically.
Data from the National Atmospheric Deposition Program
(http://nadp.sws.uiuc.edu/) provided the basis for the atmospheric chloride deposition rates from precipitation, which
were doubled to account for dry deposition and correspond
to prebomb 36Cl/Cl ratios found in the region [Scanlon and
Goldsmith, 1997]. Further details on measurement methods
are provided by Scanlon et al. [2007].
[32] In this study, we assimilate soil moisture and chloride concentration data and use soil texture data to identify
unconditional soil property densities. Chloride and soil
moisture measurements in the root zone were omitted
because the numerical unsaturated zone flow and transport
model used in this study does not accurately simulate the
transient conditions in this region [Cook et al., 1992; Tyler
and Walker, 1994]. Matric potential measurements from the
SHP sites were not used because they appear to include
inconsistencies with soil moisture and chloride observations
in some partially flushed profiles [Scanlon et al., 2007].
Chloride and soil moisture measurements below the root
zone provided sufficient constraint on model parameters and
moisture fluxes. In summary, our assimilation study relies
on soil moisture and chloride profile measurements and
model predictions from several depths below the root zone,
representing snapshots in time at each site (14 February
2006 for D06 – 02 and 26 May 2005 for L05 – 01). These
measurements are plotted in Figure 4, and related information reported by Scanlon et al. [2007] is summarized in
Table 1.
3.2. Unsaturated Zone Flow and Solute Transport
Model
[33] The numerical model used in our data assimilation
study is the soil-water-atmosphere-plant (SWAP) model
version 3.0.3, which is described by Kroes and van Dam
[2003]. SWAP has been used in a wide range of studies for
various climate conditions [van Dam et al., 2008]. It is a
one-dimensional vertical unsaturated zone model of soil
moisture transport, solute transport, and vegetation. Soil
Table 1. Site Data From Scanlon et al. [2007]
Latitude, longitude
Water table depth
[Cl] in precipitation
Elapsed time since land use change (time used in this work)
Chloride mass balance (CMB) recharge estimate from Scanlon et al. [2007]
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D06 – 02
L05 – 01
32.63°N, 102.09°W
10 m
0.34 mg/L
67 – 75 years (71 years)
70 mm/a
33.93°N, 102.61°W
36 m
0.28 mg/L
70 – 85 years (76 years)
14 mm/a
NG ET AL.: RECHARGE MECHANISMS
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moisture is simulated in SWAP using a finite difference
solution of the well-known Richards’ equation
@q
@
@h
¼
K ðhÞ
þ 1 Ta ðt; zÞ:
@t @z
@z
ð11Þ
Solute transport is simulated using a finite difference
solution of the advection-dispersion equation for nonsorbing solutes, assuming no uptake by roots:
@C
@qC
@ ¼
q Ddif þ Ddisp
qC :
@t
@z
@z
ð12Þ
In these equations, q is volumetric soil moisture, h is matric
potential, K is unsaturated conductivity, Ta is water uptake
by roots, C is solute concentration in the pore water, Ddif
and Ddisp are molecular diffusion and dispersion coefficients, and q is moisture flux. The molecular diffusion
coefficient is set to 1 cm2/d for the solute diffusion
coefficient in free water, which is within the range of
values listed for chloride by Robinson and Stokes [1965];
the dispersion coefficient is derived from a dispersivity of
2 cm [Cook et al., 1992].
[34] Surface boundary conditions for (11) and (12) and
the sink term (transpiration) in (11) are determined from
daily meteorological inputs and vegetation parameters. The
simple (nondynamic) crop option of SWAP was implemented, for which vegetation parameters over the growing
season are user-specified. Soil evaporation is capped at
potential evaporation (Ep), which is calculated from the
Penman-Monteith equation [Monteith, 1981] assuming no
crop resistance and a decay factor based on crop foliage. An
empirical evaporation option in SWAP based on that of
Black et al. [1969] was selected to calculate actual evaporation (Ea) over a time step using
Ea ¼ min Ep ; EDarcy; Eemp :
ð13Þ
EDarcy is calculated from the average hydraulic conductivity
(K1/2) between the top node at z1 and the surface, and the
gradient between the top node soil water pressure (h1) and
soil water pressure that is in equilibrium with air humidity
(hatm):
EDarcy ¼ K1=2
hatm h1 z1
:
z1
ð14Þ
Eemp over a time step dt, at time tdry since a significant rain
event, is
Eemp tdry ¼ b
1=2
tdry
tdry dt
1=2 ;
ð15Þ
where b is an empirical parameter that describes evaporative
properties for the surface soil layer. Potential evapotranspiration (PET) is found using the Penman-Monteith equation,
including crop resistance. Potential transpiration is then
computed from PET less Ep. Actual transpiration in SWAP
depends on root density and solute and water stresses as well
as on potential transpiration. Since we assume no significant
interaction between the groundwater and the observed
profiles, we set a free gravity drainage bottom boundary
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condition below the measured depths (i.e., the bottom matric
potential gradient is set to zero).
[35] In our data assimilation procedure, the vector X of
values to be estimated includes SWAP model states (soil
moisture, chloride concentrations, and moisture fluxes over
the entire simulation period) and model inputs. The simulations began at the time of land use change (1935 for
D06 –02 and 1930 for L05 – 01) and ended at the measurement date (February 2006 for D06 – 02 and May 2005
for L05 – 01).
3.3. Model Inputs and Uncertainty
[36] In our approach to data assimilation, unconditional
probability densities of the model states are approximated
with a Monte Carlo approach that derives individual samples of the states from samples of uncertain model inputs.
Because the conditional state and parameter values derived
in importance sampling are just reweighted versions of the
unconditional sample values, it is especially important to
ensure that the unconditioned model inputs include values
that can reproduce observations to within measurement
error. The importance sampling procedure identifies these
values and gives them increased weight when it constructs
the conditional population. In our study, we found that a
sample population of about 30,000 was sufficient to yield an
adequate number of high weight samples.
[37] Although nearly any model input has some uncertainty, this does not imply that all inputs must be varied in a
Monte Carlo analysis. Since assumptions must be made
about the nature of randomness included in such an analysis, it is best to focus on uncertain variables that are known
to have an important effect on predictions so that the
number of probabilistic assumptions can be kept to a
minimum. We assume here that uncertainty in the model
is conveyed solely through parameter and boundary conditions, thus eliminating the residual model error term e in
equation (2). The uncertain model inputs include vegetation
and soil parameters and meteorological inputs. Whenever
possible, nominal input values and probabilistic assumptions are based on data reported in the literature. Some
decisions are made according to preliminary sensitivity
tests. The following paragraphs summarize the probabilistic
assumptions made for each type of uncertain model input.
The model’s initial solute and matric potential values at the
time of land conversion (
70 years before measurements)
are deterministic and reflect conditions currently found
under native grassland areas. Simulations showed little
sensitivity to the exact profiles used for the dry and saline
initial conditions.
3.3.1. Vegetation Parameters
[38] Because our simulations begin at the time of land use
change, vegetation parameters are only needed for cotton.
Because of the lack of detailed crop schedule data, we
assumed that crop emergence began every year on 15 May
and that harvest occurred on 19 October [Keese et al.,
2005]. For most vegetation parameters, time-varying nominal values over the growing season were chosen based on
the literature, and these were multiplied by an independent
time-invariant uniformly distributed random variable with
mean 1.0. Multiplicative uncertainty was held constant over
time to avoid averaging out perturbation effects over the
growing season.
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Table 2. Vegetation Parameters
Root depth
Leaf area index (LAI)
Nominal Value
Literature Basis
Unconditional Uncertainty
1 m (max)
Bland and Dugas [1989],
Sarwar and Feddes [2000],
Droogers [2000]
Bland and Dugas [1989]
time-constant multiplicative
uniform noise [0.5, 1.5]
Minimal crop resistance
90 s m1
Kroes and van Dam [2003]
Water stress parameters
h1 = 10, h2 = 25, h3H,L = 500, 900,
h4 = 16,000 cm
ECmax = 7.7 dS/m, ECslope = 5.2% dS1 m1
Sarwar and Feddes [2000]
time-constant multiplicative
uniform noise [0.75, 1.25]
time-constant multiplicative
uniform noise [2/3, 4/3]
multiplicative uniform
noise [2/3, 4/3]
none
Kroes and van Dam [2003]
none
Crop height
Solute stress parameters
5 (max)
0.9 m (max)
Askew and Wilcut [2002]
[39] The vegetation parameters used in our study are
summarized in Table 2. Sensitivity tests indicated that root
depth has the greatest impact on percolation. Consequently,
this parameter was assigned a reasonably high level of
uncertainty. Although the cited sources often report maximum rooting depths greater than the nominal 1 m used here
[e.g., Bland and Dugas, 1989; Sarwar and Feddes, 2000],
smaller values were specified to reflect the fact that most
moisture uptake is in the shallower active root zone.
[40] The root density profile was assigned a deterministic
set of values based on those of G. L. Ritchie et al. (see
http://pubs.caes.uga.edu/caespubs/pubs/PDF/B1252.pdf),
with greatest root density near the surface. In effect, the
large uncertainty in rooting depth accounts indirectly for
uncertainties in the depth profile. Water and solute stress
parameters were also assigned deterministic values from
Sarwar and Feddes [2000] and the SWAP manual [Kroes
and van Dam, 2003], because they have minimal impact on
percolation in our simulations.
3.3.2. Soil Parameters
[41] Soil texture data in the study region provided the
basis for the soil parameters used in the simulations. SWAP
uses the van Genuchten-Mualem soil moisture retention and
unsaturated hydraulic conductivity functions given by van
Genuchten [1980]. The six soil parameter needed for these
functions were determined from soil texture using the
pedotransfer model Rosetta [Schaap et al., 2001]. Because
Keese et al. [2005] found soil layering to significantly affect
recharge, we used heterogeneous parameters that vary
between layers centered at the soil texture measurement
locations for each study site.
[42] Uncertainty for the vertically heterogeneous soil
parameters was attributed to two sources: (1) the texture
values entered into Rosetta and (2) the pedotransfer functions imbedded in Rosetta. Because nearby soils can be
expected to be similar, percent clay and silt profile densities
in the SHP were assumed to be stationary Gaussian firstorder autoregressive (AR-1) random processes over depth.
Length-scale (aAR1) and standard deviation (sAR1) parameters were based on texture data collected from 24 boreholes
in the SHP [Scanlon et al., 2007]: aAR1 = 0.9964 and
sAR1 = 1.7 for percent clay, and aAR1 = 0.9956 and sAR1 =
1.9 for percent silt, using a 1-cm depth resolution.
[43] The percent clay and silt profile samples used for our
data assimilation study were conditioned on the point texture
observations obtained from D06 – 02 and L05 – 01 (shown in
Figure 4), assuming additive Gaussian measurement errors
with standard deviations of 5% clay and silt. The conditional
random sample profile was averaged over each soil layer
interval to give a single layer value. The resulting layer
values for percent clay, silt, and sand were entered into
Rosetta, which provided corresponding layer values for the
van Genuchten-Mualem soil parameters. Possible errors in
the Rosetta pedotransfer function were represented by AR-1
depth-correlated perturbations added to the parameter values
produced by Rosetta. Perturbations are scaled according to
parameter uncertainty values produced by Rosetta.
[44] The prior value for the evaporation model b in (15)
was drawn from the uniform density over [0.2, 0.6] cm/d1/2,
based on the values given by Ritchie [1972]. Although the
evaporation parameter is undoubtedly related to soil type, its
prior density was assigned independent of texture because
literature values were too sparse to infer correlations.
3.3.3. Meteorological Inputs
[45] Meteorological inputs needed for SWAP are the
chloride concentration in precipitation, daily precipitation,
minimum and maximum air temperature (Tmin and Tmax),
solar radiation, vapor pressure, and wind speed. The chloride concentrations in precipitation were obtained from
Scanlon et al. [2007] (see Table 1). The four meteorological
stations with long-term historical daily data closest to our
study sites are in Amarillo, Lubbock, Lamesa, and Midland,
Texas (see Figure 3). Information on the daily meteorological data available at these four stations is provided in
Table 3. Daily (or even finer) precipitation data are needed
to resolve the episodic precipitation events that largely
control diffuse recharge in semiarid areas.
[46] The closest station to D06 – 02 is Lamesa, which is
15 km away. The closest station to L05 – 01 is Lubbock,
which is a relatively distant 77 km away. Data for each
study site were taken from the closest station when possible.
Missing data at the closest station were reconstructed from
precipitation and temperature records at the other stations,
or by sampling seasonally appropriate historical records for
the station with missing data. Further details on the reconstruction of daily meteorological series for each study site
are provided by Ng [2008]. Preliminary simulations showed
that recharge response is relatively insensitive to nonprecipitation meteorological values. As a result, only precipitation was represented as uncertain.
[47] Uncertainty was included in the daily rainfall intensity to account for (1) differences between rainfall at the
study sites and the meteorological stations, and (2) observation errors at the meteorological station. Uncertainty in
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Table 3. Micrometeorological Data Availability and Sources for 1930 – 2005 in the U.S. Southern High Plains
Station (MAPa)
Precipitation
Temperature
Solar Radiation/Vapor Pressure/WIND
Data Sourceb
Amarillo (470 mm)
Lubbock (450 mm)
Lamesa (470 mm)
Midland (350 mm)
1948 – 2005
1930 – 2005
1930 – 2005 (
750 missing)
1948 – 2005
1948 – 2005
1930 – 2005
1930 – 2005 (
850 missing)
1948 – 2005
1961 – 2005
1961 – 2006
none
1961 – 2005
SAMSON, NSRDB
SAMSON, NSRDB
NCDC
SAMSON, NSRDB
a
Mean annual precipitation for 1961 – 2005.
SAMSON, Solar and Meteorological Surface Observational Network 1961 – 1990, available with GEM weather generator [Hanson et al., 1994] data
set; NSRDB, National Solar and Radiation Data Base 1991 – 2005 update [National Renewable Energy Laboratory, 2007]; NCDC, daily surface data
inventory (National Climatic Data Center; see http://www.ncdc.noaa.gov/oa/ncdc.html).
b
the timing of individual precipitation events was not included because it did not significantly influence percolation
characteristics over the approximately 70 year study period,
from which recharge mechanisms were inferred. Instead,
uncertainty is included only in the daily rainfall intensities
on days that are recorded as rainy at the meteorological
stations.
[48] A simple random rainfall intensity model was constructed to produce different temporal daily rainfall intensity
means and standard deviations for the different samples
generated in the data assimilation procedure. The resulting
rain series from the model have the same pattern of rain
occurrence as that recorded at the meteorological stations,
but the daily rain intensities on rainy days are randomized to
account for uncertainty. While such a model cannot generate
the true rainfall series at the observation sites, it makes it
possible to include in the analysis uncertainty in temporal
rainfall statistics. Details are given by Ng [2008].
[49] The chloride mass flux rate, which is the product of
[Cl]p and the precipitation P, is uncertain. Since available
data are insufficient to warrant an explicit distinction
between chloride concentration and precipitation uncertainties, we account for the net effect of these uncertainties on
the chloride mass flux through the precipitation term alone.
3.4. Measurement Error
[50] The measurement error probability density plays a
key role in importance sampling because it determines,
through the likelihood function, the conditional weight wiC
assigned to each Monte Carlo sample. For this reason, care
should be taken to properly account for all the factors that
contribute to measurement error. In our study, gravimetric
soil moisture measurements (g/cm3) were assumed to have
additive zero-mean Gaussian observation errors wqg that are
independent at different depths. The associated volumetric
soil moisture values used for data assimilation can be
written
yq ¼ rb yqg ¼ q þ rb wqg ;
ð16Þ
that bulk density is deterministic and constant. All
uncertainty related to soil moisture measurements is
represented through the gravimetric soil moisture observation error. We selected a standard deviation of sqg = 0.03 for
wqg. This value is intended to account for errors introduced
during the soil collection, weighing, and oven-drying
process.
[51] As described in section 3.1, measurements of chloride concentration in the pore water yC were derived from
the measurement of chloride concentration in the sample
supernatant yS, using the following relationship:
yC ¼
RE yS
RE yS
¼
;
yqg
qg þ wqg
ð17Þ
where the extraction ratio RE is the mass of water added to
the dried sample per mass of dried soil. For the same reason
as bulk density, exact RE values used for the results by
Scanlon et al. [2007] were implemented as deterministic
values. As a controlled experimental input, actual RE should
have little uncertainty. Additive and multiplicative Gaussian
chloride measurement errors were used to account for the
±0.1 mg/L instrument error for ion chromatography
[Scanlon, 2000] and the 9% mean difference between split
tests for the chloride data [Scanlon et al., 2007],
respectively. This gives the following expression for the
supernatant concentration measurement:
yS ¼ S þ wS1 þ SwS2 ;
ð18Þ
where S (=Cqg/RE) is the true supernatant concentration and
wS1 and wS2 are independent with means of 0 and 1,
respectively, and standard deviations of sS1 = 0.1 mg/L and
sS2 = 0.1.
[52] Evaluating p(yq,yCjC,q) is problematic because (17)
is nonlinear in wqg, making yC non-Gaussian. We deal with
this problem by expanding (17) about wqg = 0, substituting
(18) for yS, and ignoring higher order terms. This gives
where the bulk density rb is set to 1.6 g/cm3, the value used
for the results reported by Scanlon et al. [2007]. We selected
a standard deviation of sqg = 0.03 for wqg. This value is
intended to account for errors introduced during the soil
collection, weighing, and oven-drying process. Note that
treating bulk density as an uncertain parameter would result
in volumetric soil moisture and chloride concentration data
values that differ from those reported by Scanlon et al.
[2007]. To maintain consistency with that work, we assume
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yC ¼ C þ Ce1 þ e2
wqg wS2 wqg
qg
qg
ð20Þ
RE wS1 RE wS1 wqg
:
qg
q2g
ð21Þ
e1 ¼ wS2 e2 ¼
ð19Þ
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The conditional mean and covariance of the errors e1 and e2
are then
E½e1 jC; q ¼ E ½e2 jC; q ¼ 0
ð22Þ
r2 s2q
r2b s2S2 s2qg
s2e1 ¼ E e01 e01 jC; q ¼ E ½e1 e1 jC; q ¼ s2S2 þ b 2 g þ
q
q2
ð23Þ
r4b R2E s2C1 s2qg
r2 R 2 s 2
s2e2 ¼ E e02 e02 jC; q ¼ E ½e2 e2 jC; q ¼ b E2 C1 þ
q
q4
ð24Þ
0 0
ð25Þ
E e1 e2 jC; q ¼ E½e1 e2 jC; q ¼ 0
(the primes represent the mean-removed variables). To
facilitate our likelihood function calculations, we assumed
that e1 and e2 were Gaussian with the means, variances, and
covariance given in (22) – (25). Then, after determining the
conditional covariance between soil moisture and chloride
concentrations, the Gaussian likelihood function for a
particular depth can be described with the following mean
and covariance:
E
Cov
yC C
C;
q
¼
yq q
2
2 2
2
6 C se1 þ se2
yC 6
C; q ¼ 4
yq Cs2qg
q
ð26Þ
Cs2qg
3
7
q 7:
5
ð27Þ
r2b s2qg
4. Results and Discussion
4.1. Soil Profiles and Long-Term Recharge
[53] The importance sampling procedure provides a set of
weighted samples that may be used to estimate conditional
probability densities, statistics, and other quantities that can
shed light on the mechanisms that control episodic diffuse
recharge. The importance sampling results obtained for our
study sites are summarized in Figure 5 (top), which shows
the conditional weights assigned to each of the simulated
samples. These weights reflect the quality of fit between
each sample and the soil moisture and chloride measurements, as determined by the likelihood function.
[54] The black lines plotted in Figure 5 (bottom) show the
chloride and soil moisture profiles for the 100 most highly
weighted conditional samples (wc > 8 104 for D06 – 02
and wc > 0.002 for L05– 01). The gray lines show a random
sample of 100 equally weighted unconditional samples. It is
apparent that the importance sampling procedure selects,
out of the large sample of unconditional profiles, the
relatively small number that gives the best fit to the
observations. Each of these high weight samples corresponds to a particular model simulation based on a particular set of physically plausible inputs.
[55] It is useful to compare percolation values from our
Monte Carlo and importance sampling procedure with
traditional chloride mass balance recharge values derived
from the same data sets. For this purpose, we computed
long-term average percolation for each Monte Carlo sample
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by dividing the cumulative percolation over the approximately 70 year simulation period by the simulation time
(i.e., the time from land use change until the observation
time at the study site). Over such time scales, long-term
average percolation and recharge are essentially equivalent.
Figure 6 compares the long-term recharge probability densities obtained from the conditional (unequal) and unconditional (equal) weights. Negative values indicate downward
fluxes. The unconditional densities have long tails, demonstrating high uncertainty. By contrast, the conditional densities are narrower, with peaks that are very close to the steady
state CMB estimates. The conditional peak for the fully
flushed D06 – 02 site is shifted noticeably toward greater
downward flux values. The partially flushed L05 – 01 site has
a conditional density that does not differ as dramatically from
the unconditional density as D06 – 02, yet its uncertainty is
substantially reduced. The significant shifts and uncertainty
reductions in the conditional percolation densities demonstrate that chloride measurements can contribute significantly
to unconditional Monte Carlo predictions derived solely from
a numerical model.
4.2. Model Parameters
[56] Our data assimilation approach has the advantage of
providing information on unobservable model inputs, such
as soil and vegetation properties, as well as on model states
and recharge values. Conditional parameter estimates can be
used to make deterministic or probabilistic recharge predictions under different climate conditions or agricultural
practices. These predictions are more robust than those
derived from unconditional parameter estimates because
they are constrained by available observations.
[57] The strongest impact of observations on the conditional probability distributions were for maximum rooting
depth, certain shallow soil parameters, and the evaporation
parameter b. Beyond providing estimates of individual
parameters, the weights assigned by importance sampling
can be used to identify the groups of parameter values that
are most consistent with observations. This makes it possible to describe the distinctive conditions needed to generate
episodic diffuse recharge. The basic idea can be illustrated
by considering the unconditional and conditional bivariate
probability densities (Figure 7) for two particular model
inputs: the maximum root depth and the evaporation parameter b. The diagonal strip of higher probability apparent
in the conditional bivariate density plotted in Figure 7
indicates that deep roots are most likely to occur when the
parameter is low. Equation (15) suggests that evaporation is
greater when the evaporation parameter is large. A greater
rooting depth yields more transpiration. The observed
profiles are possible only if these two elements, which
compete for moisture, are properly balanced. This balance
is achieved by selecting parameter values in the highprobability region of Figure 7.
[58] Data-conditioned relationships among other vegetation parameters and soil parameters over depth are given by
Ng [2008]. Many weakly related unconditional parameters
show notable conditional correlations. Such probabilistic
assimilation results provide a way to identify key interactions among the soil, vegetation, and meteorological factors
that control recharge.
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Figure 5. (top) Conditioned (posterior) samples have unequal weights, unlike the equiprobable
unconditioned samples. (bottom) Model samples with significant conditioned weights (black) are
compatible with observations (symbols); unconditioned model results are in gray.
4.3. Recharge Dynamics and Mechanisms
[59] The major result of our data assimilation study is a
set of high-resolution conditional percolation samples that
are constrained by field observations. Such information is
not available from either numerical modeling or traditional
CMB alone. The fine temporal resolution provided by our
approach is especially useful because it provides insight
Figure 6. Probability densities for unconditional (dashed)
and conditional (solid) long-term recharge; chloride mass
balance (CMB) estimate is indicated in gray. These and all
subsequent probability densities shown were obtained from
a kernel density estimator with a Gaussian kernel function.
about the conditions that control episodic percolation.
Significant uncertainty shown earlier in unconditioned
parameters and long-term recharge indicate that percolation
time series produced with unconditioned parameters can be
Figure 7. (top) Unconditional and (bottom) conditional
bivariate distributions for evaporation parameter and root
depth. The two parameters have are independent and
uniformly distributed before conditioning. Importance
sampling reduces uncertainty, creating a region of higher
probability in the space of possible parameter values.
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Figure 8. Conditional precipitation (blue) and percolation (red) starting from the time of land use
change until observation date; color intensities of time series scale with conditional weights (i.e., brighter
line is more probable). Percolation is highly variable and episodic. Recorded El Niño – Southern
Oscillation (ENSO) episodes include winters of 1940– 1941, 1957 – 1958, 1965 –1966, 1972 – 1973,
1982– 1983, 1986 – 1987, 1991– 1992, 1997 – 1998, and 2002 –2003. The time periods bracketed by
arrows and dotted cyan lines are shown in greater detail in Figure 9.
misleading. Here we show that conditioning percolation
time series on data even reveals some differences in
temporal patterns between the two study sites, despite
compatible meteorology and similar range of unconditional
percolation simulations (Figure 5). This underscores the
importance of data conditioning in identifying recharge
dynamics and mechanisms.
4.3.1. Recharge Time Series
[60] Conditional estimates of weekly precipitation and
flux below the root zone are shown in Figures 8, 9, and 10,
each of which focuses on a progressively narrower range of
the simulation period. Figure 8 provides information on the
frequency and magnitude of percolation events over the
entire historical period, while Figures 9 and 10 show details
that help reveal recharge mechanisms.
[61] Figure 8 demonstrates that flux below the root zone
in SHP has been highly episodic with large interannual
variability over the 70 year test period. Some years, such
as 1941, with the annual highest precipitation, were dominated by very high recharge, while other periods, such as
the early 1950s, which had a long-term drought, and the
mid-1990s had virtually no recharge. At D06 – 02 there was
very little recharge between major events. At L05 – 01 there
was some upward flux at the reference 1.5 m depth. This
suggests that the finer soil in L05 – 01 was more likely to
retain moisture for roots in the top 1 m of the soil.
[62] Interannual recharge variability at our study sites
may be related, at least in part, to known climatic processes.
Kurtzman and Scanlon [2007] found a statistically significant correlation between the summer Southern Oscillation
Index (representing El Niño) and wet winters in the SHP
(P < 0.001). Gurdak et al. [2007] analyzed historical
rainfall records and groundwater levels in the High Plains
for known climate cycles and identified signatures of both
the Pacific Decadal Oscillation (10 – 25 years) and
El Niño– Southern Oscillation (ENSO) (2– 6 year) in their
data. Our data assimilation procedure enables us to directly
estimate and compare percolation in the SHP for known
El Niño years.
[63] During the historical period considered in this study
(1930 – 2006), there have been nine recorded El Niño
episodes (Climate Prediction Center; see http://www.cpc.
ncep.noaa.gov/products/predictions/threats2/enso/elnino/
tx_bar.html), including the winters of 1940 –1941, 1957 –
1958, 1965 –1966, 1972 – 1973, 1982 – 1983, 1986 – 1987,
1991 – 1992, 1997 – 1998, and 2002 – 2003. In the SHP, El
Niño events typically bring cooler and wetter winters. Our
comparison of major percolation events and El Niño years
reveals that the two are sometimes, but not always,
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Figure 9. Sample time periods of conditional precipitation and percolation during time ranges indicated
in Figure 8; the growing seasons are shaded yellow. Color intensities of time series scale with weights
(i.e., brighter line is more probable). Insets are histograms of weekly percolation at indicated time slices;
date indicates end of week. The time period bracketed by arrows around the percolation peak in D06 – 02
is shown in greater detail in Figure 10.
coincident. A number of high-intensity percolation events
directly followed El Niño episodes, such as in 1941 and
1992 for both D06 – 02 and L05 –01 and in 1986 for D06 –
02. Although other El Niño years did not coincide with
intense percolation events, some of the years did coincide
with longer-duration, low-intensity events. These include
1973 and 2003 for D06 – 02 and 1958 and 1986 for L05 –
01. No large percolation events occurred in 1997– 1998,
even though this is generally considered to be the largest
El Niño year of the twentieth century. We need to look at
finer temporal resolution to understand how wet El Nino
winters may or may not affect recharge in the SHP.
[64] Figure 9 shows that percolation at our study sites
varies greatly within the year. Although annual precipitation
might provide a convenient and tempting indicator for
recharge, strong percolation events are clearly not determined by rainfall on that time scale. For example, 1991 and
1992 had almost identical total precipitation amounts at
D06 – 02, yet only 1992 experienced significant percolation.
Comparable annual rainfall fell at L05 –01 in 1961 and
1962, with only the former yielding significant percolation.
It appears that high rainfall intensity is a better predictor of
recharge than high annual rainfall. This is demonstrated
during 1985 and 1992 at D06 – 02 and 1957 and 1967 at
L05 – 01.
[65] However, high weekly rain intensity alone is not
sufficient to guarantee high-percolation events. Heavy rainfall events in 1991 at D06 –02 and in 1965 and 1966 at
L05 – 01 failed to produce much recharge. A close examination of Figure 9 reveals that high-intensity rainfall events
were generally unable to trigger recharge during the middle
to late part of the growing season. This applies to the three
examples cited above. It appears that heavy rainfall cannot
escape plant uptake during periods when the root system is
mature. Thus episodic recharge is most likely to occur in
response to intense rainfall outside the growing season.
[66] This reasoning is supported by the observation that
the relatively small precipitation events occurring during the
winter generally yield some low, but discernible, percolation. This can be seen in 1984 – 1985, 1986– 1987, and
1991 – 1992 for D06 – 02 and in 1957 – 1958 and 1960 –
1961 for L05 – 01. Some of the years coincide with wet
El Niño winters. Although these small winter percolation
events may not make a significant contribution to total
recharge, they often precede larger percolation events that
occur when more intense rainfall arrives at the start of the
growing season. This probably occurs because small winter
rainfall events increase antecedent soil moisture. This raises
soil hydraulic conductivity which, in turn, causes more
water to be transported through the root zone during
subsequent early summer rains. Similar behavior has been
observed in southeastern Australia, where fallow years
appear to increase episodic recharge by boosting root zone
moisture [Zhang et al., 1999b]. Such conditions explain the
large percolation events at D06 – 02 during 1987 and 1992
shown in Figure 9, as well as the L05 –01 percolation events
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percolation in the SHP as an episodic phenomenon.
Figure 10 focuses on an even smaller time range, showing
the evolution of a particular high-intensity recharge event
over several weeks in the summer of 1987. At this resolution, uncertainty about the exact timing and magnitude of
the percolation peak is significant. This is reflected in the
percolation histograms shown in Figure 10 (bottom). However, all of the histograms and samples are consistent with
the observation that a major event with percolation values
over 10 mm per week occurred. This event followed a
moderate rainstorm occurring at the end of an unusually wet
spring and before maturation of the crop root system. The
mechanisms generating this event and other events in the
historical record are consistent with the discussion presented
above.
Figure 10. (top) Percolation event during the time range
between arrows in Figure 9 for D06 – 02. Darker lines indicate
estimates with greater weights. (bottom) Weekly percolation
probability densities at the times indicated with triangles in top
plot. Although there is uncertainty in the peak weekly
percolation magnitude and arrival date, it is highly probable
that a major percolation event occurred during this time.
in 1958 and 1961 following wet winters. In fact, the
presence of winter rain preceding strong spring rains seems
the key difference between 1991 and 1992 at D06 – 02 and
1961 and 1962 at L05– 01. These years have similar annual
rainfall histories, but much different recharge responses.
[67] The above discussion indicates that antecedent moisture resulting from above-average winter rains, often seen
during ENSO events, creates the conditions needed to
obtain high recharge during intense spring rains. By contrast, dry conditions that prevail during the growing season,
when crops take up most available moisture, make highpercolation events unlikely during the late summer and
early fall, even when rainfall is intense. Late-year percolation events are rare at L05 – 01, where finer soils facilitate
evapotranspiration of available moisture.
[68] The spread of values shown in Figures 8 – 10 reveals
the uncertainty in the percolation time series generated by
our probabilistic approach. The flux histograms shown at
selected time slices in Figure 9 show that there is a high
certainty that both sites experienced negligible downward
flux between isolated high-percolation events, with some
probability of a slight upward flux at the end of the growing
seasons at L05 – 01. This indication that downward moisture
flux is very likely to be zero for extended periods between
intense percolation events reinforces our characterization of
4.3.2. Summary Statistics
[69] Although diffuse recharge in semiarid regions is
dominated by relatively short-term episodic events, monthly
trends and other summary statistics are helpful for characterizing seasonal patterns. Figure 11 shows box plots that
reveal monthly variations in the conditional probability
densities of monthly average rainfall, rainfall intensity,
and recharge at the two study sites over the approximately
70 year simulation period. These plots show that May –July
generally contributed most of the recharge at both sites,
even though the rainy season extends from May through
October. This confirms observations from the sample time
series in Figures 8 – 10, which indicate that little moisture
escapes fully developed roots. Although both sites have
similar seasonal flux patterns, only the coarser-textured
D06 – 02 site experiences winter recharge. In fact, L05– 01
on average experiences a small net upward (positive) flux
during the months of September, October, and November.
Less than 25% of the recharge at our SHP sites occurs
during the three driest months of the year (December –
February). By comparison, 20– 50% of episodic recharge in
Western Australia is believed to occur during the dry season
[Lewis and Walker, 2002].
[70] The ensemble spread shown in the box plots of
Figure 11 also reveals that percolation uncertainty scales
with precipitation rather than percolation magnitude. For
example, late growing season percolation values exhibit
greater spread than values during the less rainy winter
months, even though the median percolation values are
close to zero. Uncertainty in precipitation intensity appears
to have a greater impact on percolation uncertainty during
the summer when there are more rainy days. Precipitation
uncertainty is further amplified by soil and vegetation
parameter uncertainty in the root zone, which is most active
during the rainy growing season.
[71] The SHP monthly average percolation values tend to
lag monthly average precipitation, reflecting the travel time
through the root zone soil column to 1.5 m depth. This
effect varies with season, as indicated by the correlation
results plotted in Figure 12. Figure 12 confirms a time lag of
a month or more during most times of the year. However,
note the abrupt halt in correlation between rainfall and
recharge starting in August, when the system memory is
effectively reset. This reflects the fact that nearly all
moisture from previous months has been extracted by roots.
Figure 12 suggests that May – July recharge at L05– 01 is
controlled mostly by rainfall occurring after March, and
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Figure 11. Box plots of conditional mean monthly precipitation, daily precipitation intensity, and
percolation. Box lines indicate the first, second (median), and third quartiles; whiskers show data within
another 1.5 times the interquartile range; and dots represent outliers. Spread shown is over ensemble of
simulations, not over time. Most percolation occurs in late spring/early summer at the study sites.
May –July recharge at D06 – 02 may be affected by rainfall
from as early as the previous November. Overall, May
rainfall seems most important for determining May– July
recharge, because it is the only high-rainfall month (in terms
of total amount and intensity) without significant root
density. Correlations between rainfall and recharge are
generally stronger at D06 – 02, a result which suggests that
soil at this site is more conductive and responsive.
[72] The cumulative plots in Figure 13 show how much
of the percolation at each site comes from high-intensity
Figure 12. Correlation between conditional monthly precipitation and percolation in same and later
months. Abbreviated month labels start with September. Off-diagonal correlations indicate time lag
between precipitation and resulting percolation. Note that correlations are negative because downward
percolation is defined as negative.
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Figure 13. (top) Cumulative fraction of percolation originating from percolation events of given peak
intensities and stronger. (bottom) Cumulative number of events of given peak intensities and stronger.
Results are for conditional percolation time series estimates over the historical period (71 – 76 years).
Median and first/third quartiles over ensemble are indicated. For example, about half the total percolation
at D06 – 02 originated from events with at least 9-mm/week peak magnitudes; these 9-mm/week
events occur over only about 20– 30 events during the 71-year historical period.
episodic events. Figure 13 (top) shows the fraction of
cumulative percolation over the historical period associated
with events having a peak weekly percolation more negative
than a given value. Figure 13 (bottom) shows the number of
events having a peak weekly percolation more negative than
a given value. We define a recharge event period as the time
interval between successive inflection points (changes in the
sign of the second derivative) in the weekly percolation time
series. The recharge event magnitude is the peak weekly
percolation value during the event period. We performed a
cumulative percolation analysis for every sample in our
conditional simulation distribution. Figure 13 plots the
median of all percolation fractions or events counts as well
as the first and third quartiles.
[73] Figure 13 shows that 50% of the recharge at D06 – 02
comes from percolation events of about 9 mm per week
strength or greater, which only occurred about 20–30 times
over the entire 71 year period, or typically about once every
2.5 years. About 50% of the recharge at L05 – 01 originates
from events of about 7 mm per week. These occurred even
more rarely than at D06 – 02, only about once every 6 years.
Recharge at L05– 01 can be characterized as more episodic
than at D06 – 02, considering that percolation events were
about 40% less common than at D06 – 02. Furthermore, the
single most extreme recharge event at L05 – 01 (in 1941)
surpassed that of D06 – 02 in magnitude, and it contributed
nearly 10% of the total L05 – 01 percolation over the 76 year
historical period. It should be noted, however, that D06 – 02
had more occurrences of moderately intense percolation
events (>20 mm per week peaks). Monthly tallies of
percolation events confirm that most high-magnitude events
occur in the spring, with occasional large events in the
autumn at D06 – 02.
5. Summary and Conclusions
[74] This paper presents a probabilistic recharge estimation approach that integrates numerical modeling, which is
sensitive to errors in uncertain inputs, and unsaturated zone
moisture and chloride measurements, which do not contain
enough information to provide fine time-scale recharge
estimates. By combining these two imperfect sources of
information, we are able to identify the episodic events that
account for most of the diffuse recharge at our semiarid
southern High Plains study sites.
[75] The Monte Carlo and importance sampling approach
adopted here provides a probabilistic characterization of
recharge and related model inputs and states. Probability
densities are inferred from a large set of samples, each
corresponding to a model simulation based on a particular
set of randomly generated inputs. Unconditional (or prior)
densities are computed by weighting all samples equally,
without considering measurements. Conditional (or posterior) densities are weighted unequally, with the weight for
each sample assigned according its ability to match soil
moisture and chloride measurements.
[76] Long-term recharge probability densities conditioned
on observations from our study sites shifted significantly
away from unconditional densities derived solely from the
model toward the steady state value obtained from a
traditional chloride mass balance (CMB) analysis. The
conditional densities were also narrower than their unconditional counterparts, indicating less uncertainty. The longterm recharge values at the two study sites were about 40–
65 mm/a for the site with coarser soil and about 10– 15 mm/
a for the site with finer soil.
[77] Conditional model input probability densities from
the two study sites indicate that soil moisture and chloride
measurements provide the most information about surface
soil properties and rooting depth. The marginal densities of
most other model parameters did not change significantly
after conditioning. Some of the conditioned parameters
were highly correlated, indicating that unique parameter
values cannot be identified solely from moisture and chloride measurements. However, the parameter correlation
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patterns, as exemplified in the plot of bivariate conditional
probability densities, identify the combinations of parameter
values (e.g., root depth and evaporation parameters) required to explain observed recharge patterns. Overall, the
conditioning process narrows the set of possible input
combinations, giving a better picture of uncertain soil and
vegetation properties at the study sites. This type of uncertainty reduction cannot be obtained in a deterministic model
calibration that focuses only on point estimates of individual
input variables.
[78] Our study uses percolation (moisture flux at 1.5 m
depth, just below the root zone) to analyze recharge mechanisms. Our model-based data assimilation procedure provides conditional percolation probability densities at a fine
time scale, enabling us to examine individual percolation
events that generate recharge. Percolation at the two sites
appears to be highly episodic and variable over the years
and seasons, with 50% of long-term percolation originating
from events occurring only once in about 2.5 years at the
site with coarser soils, and once in about 6 years at the site
with finer soils. In fact, at the finer-textured site, a single
rare intense episodic event probably contributed almost
10% of the net percolation over the 76 year study period.
[79] Conditional probability densities for monthly percolation show that, on average, most recharge occurs in the
early growing season (May – July), when rains are heavy yet
roots have not reached full maturity. This time of year also
has the highest-magnitude percolation events, which typically follow high-intensity rainfall. In contrast, heavy rains
in the middle to late growing season rarely trigger appreciable recharge, especially at the finer-textured L05 – 01 site.
Rain that does fall during the winter dry season tends to
provide weak but sustained recharge. Furthermore, these
winter events often provide the antecedent moisture conditions that facilitate strong recharge peaks in the late
spring, when rains can be intense and crop roots are not
yet well established. Thus wet winters, sometimes brought
by El Niño events, can contribute to both total and highintensity recharge by directly yielding weak winter recharge
and providing moist soil conditions for springtime recharge.
Overall, the recipe for high recharge at our study sites seems
to be a combination of moderate winter rainfall, which
provides antecedent moisture, followed by intense spring
rainfall events that occur at times when root uptake is small.
[80] Data assimilation techniques that rely on both models
and field observations provide the flexibility and temporal
resolution needed to identify the factors responsible for
episodic recharge. Probabilistic approaches such as the
combination of Monte Carlo simulation and importance
sampling used in our study are able to identify the combinations of meteorological conditions and soil and vegetation
properties that facilitate recharge. Episodic percolation
events that give rise to recharge may occur infrequently in
semiarid regions, but they can account for a significant flux of
water to subsurface aquifers. Improved understanding of
such events, derived from models and field observations,
should enable us to make better predictions of the impacts of
land use and climate change on subsurface water supplies.
[81] Acknowledgments. Partial support for this work was provided
by U.S. National Science Foundation grants 0003361, 0121182, 0530851,
and 0540259. We would also like to thank Robert Reedy for assisting with
the soil profile data.
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D. Entekhabi, Department of Civil and Environmental Engineering,
Massachusetts Institute of Technology, Parsons Laboratory, Building
48-216G, Cambridge, MA 02139, USA.
D. McLaughlin, Department of Civil and Environmental Engineering,
Massachusetts Institute of Technology, Parsons Laboratory, Building
48-329, Cambridge, MA 02139, USA.
G.-H. C. Ng, Department of Civil and Environmental Engineering,
Massachusetts Institute of Technology, Parsons Laboratory, Building
48-208, Cambridge, MA 02139, USA. (gng@mit.edu)
B. Scanlon, Bureau of Economic Geology, Jackson School of
Geosciences, University of Texas at Austin, J. J. Pickle Research Campus,
Building 130, 10100 Burnet Road, Austin, TX 78758-4445, USA.
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