Geoderma 134 (2006) 253 – 266
www.elsevier.com/locate/geoderma

Information content and complexity of simulated soil water fluxes
Yakov Pachepsky a,⁎, Andrey Guber b , Diederik Jacques c , Jiri Simunek b ,Marthinus Th. Van Genuchten d , Thomas Nicholson e , Ralph Cady eaUSDA–ARS, Environmental Microbial Safety Laboratory, 173 Powder Mill Road, BARC-EAST, Beltsville, MD, 20705, USAb
Department of Earth and Environmental Sciences, University of California, Riverside, CA, USAc
SCK–CEN, BelgiumdUSDA–ARS, Salinity Laboratory, Riverside, CA, USAe
USNRC, RES, Washington, DC, USA
Available online 19 April 2006

Abstract
The accuracy-based performance measures may not suffice to discriminate among soil water flow models. The objective of thiswork was to attempt using information theory measures to discriminate between different models for the same site. The Richards
equation-based model HYDRUS-1D and a water budget-type model MWBUS were used to simulate one-year long observations ofsoil water contents and infiltration fluxes at various depths in a 1-m deep loamy Eutric Regosol in Bekkevoort, Belgium. We used
the (a) metric entropy and (b) the mean information gain as information content measures, and (c) the effective measure complexityand (d) the fluctuation complexity as complexity measures. To compute the information content and complexity measures, timeseries of fluxes were encoded with the binary alphabet; fluxes greater (less) than the median value were encoded with one (zero).
Fifty Monte Carlo simulation runs were performed with both models using hydraulic properties measured along a trench. The twomodels had the similar accuracy of water flux simulations. Precipitation input data demonstrated a moderate complexity andrelatively high information content. Model outputs showed distinct differences in their relationships between complexity andinformation content. Overall, more complex simulated soil flux time series were obtained with the HYDRUS-1D model that wasperceived to be conceptually more complex than the WMBUS model. An increase in the complexity of water flux time seriesoccurred in parallel with the decrease in the information content. Using both complexity and information content measures allowed
us to discriminate between the soil water models that gave the same accuracy of soil water flux estimates.
Published by Elsevier B.V.

1. Introduction
The search for discriminating between soil water flow
models has intensified as the number of such models has
been steadily growing. Practical problems of soil waterflow are rarely tackled with more than one model.
Therefore, if many models are available, it is desirable tohave a measure of their performance at a specific site tojustify a choice in favor of one of the models.
⁎ Corresponding author. Tel.: +1 301 504 7468.
E-mail address: ypachepsky@anri.barc.usda.gov (Y. Pachepsky).
0016-7061/$ - see front matter. Published by Elsevier B.V.
doi:10.1016/j.geoderma.2006.03.003

Residuals, i.e. differences between simulated andmeasured values, are commonly used to evaluate
performance of models. Most of commonly usedperformance measures such as the root-mean squareerror, the lack-of-fit root mean square error (Whitmore,1991), Williams–Kloot test (Pachepsky et al., 2000),modeling efficiency (Loague and Green, 1991), use thestatistics or probability distribution functions of the setof residuals.
Residual-based performance measures quite often
cannot discriminate models. Residuals quantify themodel accuracy, and one can conclude that the accuracy

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of models is similar if some statistics of residuals are
similar. However, the differences between models
should not be viewed only in terms of the model
accuracy. The accuracy reflects only a specific instanceof model performance with respect to a specific data set
used for testing and/or calibration. Once calibrated, soil
water flow models are usually used in Monte Carlo-typesimulations with multiple scenarios of boundary conditions, initial conditions, and possibly soil and plant
properties. Therefore, other performance measures
should be sought that reveal the differences between
models under scenarios that have not been observed.
The information theory was successfully used for themodel discrimination purpose in cases when the maindifference between models was the difference in thenumber of parameters and parameters were qualitatively
similar, like regression coefficients or hydraulic conductivities of different geological bodies in groundwatertransport modeling. Akaike criterion (Akaike, 1973) andits versions (Bozdogan, 1987; Hurvich and Tsai, 1989)
discriminate models by computing the relative entropy,
or Kullback–Leibler “distance,” which represents theinformation loss when a model is used to approximate
data. Those criteria introduce information theory
measures to gauge a model performance with respect
to a given dataset. But they do not reveal the differences
between model behavior with input scenarios different
from the ones in observations.
Usually the differences between models of flow andtransport in soils are discussed in terms of model
structural complexity and detail. The model complexity
can be evaluated in terms of the number of processes
being considered explicitly, process descriptions, spatial

and temporal discretization/scale, number of parameters,
and speed of computations (Neuman et al., 2003). The
numbers characterizing those model features are hardly
compatible, and their relative significance is unknown.
For example, the total number of parameters is not acomprehensive index of model complexity, because thesensitivity of model outputs to some parameters can be
relatively low, and may depend on the scenario.
Unlike model structures, model outputs are compatible, because soil water flow models under comparison
should generate time series of water fluxes or watercontents for the same depths. Given one cannot discriminate models based on complexity of models structures, isit possible to discriminate models based on the complexity of model output? Some measures of time series complexity are needed to do that. The information theory has
been used to propose several such measures (Wackerbauer et al., 1994). Lange (1999a,b) suggested using
information-related measures to discriminate between
measured time series. He made the distinction between
the information content, or the randomness, of data on
one hand, and their complexity on the other. By the latter
he meant a measure which vanishes or is very small bothfor constant or periodic sequences and for completely
random data, as both types are easy to describe. The
complexity measures should show high values for timeseries not amenable to an easy description involving only
a few parameters. This (admittedly intuitive) distinction
is illustrated in Fig. 1. Complexity and information
content of daily runoff and precipitation are presented inFig. 2 for several catchments in Germany and New
Hampshire (Lange, 1999a,b). Runoff from the natural
catchments operates at high complexities andFig. 1. Complexity and randomness for the sine function, the highly structured Bernoulli process with the probability of repetition 83%, and therandom process with the uniform distribution (Wolf, 1999).

Y. Pachepsky et al. / Geoderma 134 (2006) 253–266

255

probability of the transition from the state “i” to thestate “j” provided state “i” was reached, and pij be
the probability of that the transition from state “i”
to state “j” actually occurs. Those probabilities are
related: pij = pi pi→j.
Knowledge of state and transition probability permits
computing several information variables (Bates andShepard, 1993). Shannon (1948) defined the information Ii specifying the state “i” of a system as 1
Ii ¼ log2
ð1Þ
pi
Fig. 2. Randomness, quantified with the mean information gain, andcomplexity, quantified with the fluctuation complexity, in daily timeseries of precipitation and runoff from ecosystems and urban storm
drain systems (Lange, 1999a).

The average value of the information about the system is
its Shannon's entropy:
H¼

N
X
i¼1

intermediate randomness, urban (channeled) systems
have much more randomness. Precipitation is characterized by extremely high randomness and low complexity.
Data in Fig. 2 show the potential of information theory
measures to discriminate water flux time series generated
in the system of different complexity. Recently Selle andHuwe (2004) proposed to discriminate soil water models
using information theory measures.
The objective of this work was to evaluate theeffectiveness of the information theory measures todiscriminate between different models using simulated
time series of soil water fluxes for the same site. We
wanted to see whether and how the perceived differences between model structures manifest themselvesin differences in information content and complexity ofthe simulated time series of soil water fluxes.

pi Ii ¼ −

N
X

ð2Þ

pi log2 pi

i¼1

It has the maximum value of log(N) when all states areequally probable and pi = 1 / N, i = 1, 2, 3,… N; H ismeasured in bits.
The information gain Gij associated with transition
from the state “i” to state “j” is1
Gij ¼ log2
ð3Þ
piYj
Similarly, the information loss Lij is defined as1
Lij ¼ log2
pjYi

ð4Þ

where the state “j” occurs after the state “i”. The net
information gain is the difference between the information gain and loss:pipj

2. Theory

Cij ¼ Gij −Lij ¼ log ðpjYi =piYj Þ ¼ log2

This section introduces information theory-based
measures of information content and complexity. First,
such measures are defined for systems having a finite
number of states. Then a method is described torepresent a time series with a system with the finite
number of states. Such system appears to be a binary
string. Finally, several information theory variables arelisted that measure both randomness and complexity ofbinary strings.

The average value of Γij over all possible transitions iszero. The mean square deviation σΓ2 of values Γij


N
X
pi 2
r2C ¼
pij log2
ð6Þ
pj
i; j

2.1. Information content and complexity in systems witha finite number of states
Consider a system that has N possible different states.
Let pi be the probability of the ith state, pi→j be theð5Þ

is not zero. This value has been termed fluctuation
complexity (Bates and Shepard, 1993). It captures thefluctuations in the system occurring as it performs
transitions from state to another. The fluctuation
complexity thus measures the spread between information that a system may have in its consecutive states. The
fluctuation complexity is zero in systems with maximum entropy where all states are equiprobable (allpi = 1 / N). The fluctuation complexity approaches zero256

Y. Pachepsky et al. / Geoderma 134 (2006) 253–266

in systems with the entropy approaching zero that have
only one highly probable state.
2.2. Representing a time series with a system with thefinite number of states

The probabilities defined for the binary string are:
pL,i
pL,ij
pL,i→jTo represent a time series with a system with thefinite number of states, the time series is encoded as asymbol string. Lange (1999a) suggested the binary
encoding in which each observation in the time series
is encoded as zero or one. Specifically, all values larger
than the median value of all observations are encoded
as one, and all values less than the median value of allobservations are encoded as zero. Then the word
length L is defined to group consecutive symbols
together. A word of L consecutive symbols (zeros andones) is a state of the system of interest. This system
has the finite number 2L of possible states. An example
of such encoding is shown in Fig. 3. If one considers
three-symbol words (L = 3), then all possible words are000, 001, 010, 011, 100, 101, 110, 111, total of 23 = 8
words.
The transition in the string is the shift by one
symbol. For example, if the string is “00110” andthe word length is three symbols, then the first word
“001” transits to the second word “011”, and thesecond word “011” transits to the third word “110”.
The first and the second word in transitions overlap
by L − 1 symbol which are the last L − 1 symbol of thefirst word and the first L − 1 symbol for the second
word.

probability of the occurrence of the ith word,
probability of the transition from the ith to thejth L-word.
conditional probability of the occurrence ofthe jth word after ith word, i.e. occurrence ofthe jth word provided that the ith word has
occurred.

2.3. Information theory measures for binary strings
representing time series
The information theory offers many variables suitable to measure the information content and complexityof symbolic string sequences (Wackerbauer et al.,1994). The following variables were selected for thiswork.
The information content was measured with themetric entropy and mean information gain. The Shannon
entropy given by Eq. (2) can be computed for words oflength L as
2L
XHðLÞ ¼ −pL;i log2 pL;i
ð7Þi¼1

The Shannon entropy increases linearly with increasing
word length. Normalization with the word length yields
the notion of information independently of the word
length. This measure is called metric entropy Hμ:Hl ¼ HðLÞ=Lð8Þ

The metric entropy vanishes for constant sequences,increases monotonically when the sequence's disorderraises and reaches its maximum at 1 for uniformly
distributed random sequences.
The knowledge of a symbol that follows a word
contributes to the local information. This knowledge
reflects the information gain G as given in Eq. (3). The
average, or mean information gain is
2LX
HG ðLÞ ¼ −pL;ij log2 pL;iYj
ð9Þ
i; j ¼ 1

Fig. 3. Example of the binary encoding of a time series. The line ABshows the median value of the observations in the time series.

The mean information gain HG tells how much additional information will be gained in average for thewhole symbol sequence by knowing the next symbol.
The effect of the knowledge of the next symbol on theinformation can be illustrated by the following example(Wolf, 1999). If one observes the sequence of letters“info” within this text, then the following letter will
be an “r” for sure. The knowledge of this following
letter does not contribute to the information. But, if weY. Pachepsky et al. / Geoderma 134 (2006) 253–266

have observed the sequence “the,” then it is really hardto predict the next letter. Any letter is possible. Herethe knowledge of the next letter is a real gain of
information.
The complexity of symbolic strings was quantified inthis work using the fluctuation complexity value and theeffective complexity values as defined below. The
fluctuation complexity is computed for a string according to Eq. (6) as
r2C ¼

2L
X
i; j



pL;i 2pL;ij log2
pL; j

ð10Þ

Wolf (1999) gives the following example of theinterpretation of the fluctuation complexity. If one
observes the sequence of letters “info” within this text,
then the following letter will be an “r” for sure.
However, if the word ‘nfor’ is observed, then theinformation gain by knowing the “r” is zero within thistext, whereas the information loss of forgetting the “i” ishigher, because “nfor” occurs in the words “unfortunately” and “Information” with a capital “I” (at the begin
of a sentence or in headings). The more these differencesof information gain and loss is fluctuating in theinvestigated string, the more complex is the string inthe sense of the fluctuation complexity.
The effective measure complexity CEM evaluates theminimum total amount of information that has to be
stored at any time for an optimal prediction of the next
symbol (Grassberger, 1986). This measure can approximately be computed as
CEM ¼

2L
X
i; j ¼ 1

pL;ij log2pLL;iYj
pL;i

ð11Þ

The length of words L is arbitrary in theory but is limited
in applications because all measures of information andcomplexity are dependent on probability distributions ofwords and transitions in symbolic strings, and theprobabilities have to be estimated as the relative
frequencies. The availability of data limits the ability
to estimate the probabilities by relative frequencies.
Wolf (1999) suggested to estimate the word length for
the worst-case scenario of uniformly distributed random
sequences. The theoretical value of the measures in thiscase is known. Thus, a condition of tolerance for thedeviation (relative precision) of the expected value
compared to the theoretical value can be found. The
Table 1 presents Wolf's requirements to the length of thebinary string for 3 and 4 symbol words if a 5% relative
error is acceptable.

257

Table 1
Total numbers of symbols in binary strings to provide 5% or better
accuracy in information and complexity measure estimations
Information
theory measureTwo-symbol Three-symbol Four-symbol
words
words
words

Metric entropy
24
Mean information gain 63
Fluctuation complexity 146
Effective measure79
complexity

38
124
262
256

62
246
491
723

All information theory measures were computed inthis work using the SYMDYN software available at theInternet at http://www.dr-frank-wolf.de/diss/dissen.htm.
The length of words permitted by the volume of our data
was two.
3. Experimental data
The experimental setup has been described earlier by
Jacques et al. (2002). In brief, the experimental field
was located at Bekkevoort, Belgium, at the bottom of agentle slope and was covered with a meadow. The soil
was classified as Eutric Regosol (FAO, 1975).
Typically the top 1 m includes three soil horizons: an
Ap horizon between 0 and 25 cm, a C1 horizon
between 25 and 55 cm, and a C2 horizon between 55
and 100 cm. A trench, 1.2 m deep and 8 m long, wasdug at the field site. A dye study revealed theoccurrence of many macropores throughout the soil
profile (Vanderborght et al., 2000). The grass cover was
removed from the experimental area. A plastic sheet
covered the side of the trench where equipment was
installed. Volumetric water content was measured with
TDR. Sixty TDR probes (two rods, 25 cm long, 0.5 cm
rod diameter, 2.5 cm rod spacing) were installed along
the trench at 12 locations with 50 cm spacing at 5
depths (15, 35, 55, 75, and 95 cm deep). The insertion
point of the first TDR-probe was at 10 cm from thetrench reference end. The probe locations on the trenchwall, soil horizons observed on the wall, and soil
texture are shown in Fig. 4. Soil texture was loam at 15,
35, and 55 cm sampling depths, and silty loam at 75
and 95 cm depths.
After all devices were installed, the trench was filled.
A thin layer of gravel (1 to 2 cm) was evenly distributedon the study area: (i) to decrease the erosive effect of therain impact on the bare soil surface, (ii) to minimize theevaporation from the soil surface, and (iii) to decrease
the growth of weed on the experimental plot. Weeds
were regularly removed from the site during thesummer. Passive capillary samplers (PCAPS) were258

Y. Pachepsky et al. / Geoderma 134 (2006) 253–266

Fig. 4. Locations and numbering of TDR-probes (●), tensiometers (○) and temperature probes (□). Patterned rectangles show Ap, C1 and C2
horizons. Dashed lines show the average position of the horizon boundaries. Average values of clay, silt, and sand content are given for the probe
installation depths. The numbers in pentagons show the numbering of locations monitored by five-probe profiling.

installed at two 15 and 55 cm depths at the distance ofabout 5 m from the trench to measure soil water fluxes
once in one to three days. Field measurements started on
March 11, 1998 (day 0) and finished on March 31, 1999
(day 384).
Fig. 5 shows the hourly rainfall rates and thecumulative precipitation during the experiment. A total
rainfall amount of 109 cm was recorded during theperiod of the experiment. The average rainfall in thisregion is around 80 cm. The amount of rainfall was
rather exceptional since it was the largest amount of
rainfall during a 65-year period recorded at an official
nearby weather station (Ukkel, Brussels). In the time
period of the experiment, the pluviograph in Ukkel had
recorded 106 cm.
Soil water fluxes were estimated at the depth of
105 cm using daily soil water balance computations
based on measured water contents, precipitation, andaverage evaporation estimated to be 0.86 mm day− 1.

solves the Richards' equation for saturated-unsaturated
water flow:

4. Soil water flux simulations

are used as the constitutive relationships. In Eqs. (16)and (17), θs is the saturated water content, θr is theresidual water content, Ksat is the saturated hydraulicconductivity, 1α, m, n, and l are empirical shapedefining parameters.
The MWBUS (model of water budget of unsaturated soils) model has been developed to provide a
simple and robust description of water flow in layeredsoils with the daily time step. Soil is divided in layersof thickness Δzi, i = 1, 2,…, M. A downward waterflow from a layer “i” can occur if water content θi

4.1. Soil water flow models
A Richards equation-based model and a water
budget-based model were applied to simulate soil
water fluxes.
The HYDRUS-1D model (Simunek et al., 1998) isa finite element model for simulating the one-dimensional movement of water, heat, and multiple solutes invariably saturated media. The program numerically




Ah
A
Ah
¼
KðhÞ
þ1
At Az
Az

ð12Þ

where θ is the volumetric water content, h is the matricpotential, K is the hydraulic conductivity, z is the verticalaxis directed upward, t is time.The van Genuchten
equation (van Genuchten, 1980):h−hr
1¼
hs −hr ½1 þ ðajhjÞn m

ð13Þ

and the van Genuchten–Mualem equation to computehydraulic conductivity

K ¼ Ksath−hr
hs −hr

l (

"



h−hr
1− 1−
hs −hr1=m #m )2
ð14Þ

Y. Pachepsky et al. / Geoderma 134 (2006) 253–266

259

where τ is the resistance parameter, θr,i is the residual
water content. The daily change in water content is
Dhi ¼ −

hi −hr;is

The value of τ increases with the depth z. Suleiman
and Ritchie (2003) observed that a power function gavea good fit to relationships between τ and z for several
soils and evaporation scenarios:s ¼ s0 þ az b
ð18Þ
where τ0 is the value of resistance parameter at depthz = 0, a and b are parameters of the evaporation model.
The simulated daily evaporation rate E is:E¼
Fig. 5. Cumulative rainfall and daily precipitation during theexperiment.

exceeds the value of water field capacity θFC,i of thislayer. The value of flux qi from the ith soil layer iscalculated as:
qi ¼ minðq1 ; q2 Þ
q1 ¼ qi−1 þ ðhi −hFC;i ÞDzi =Dt
q2 ¼ minðKsat;i ; Ksat;iþ1 Þ

ð15Þ

where qi−1 is the incoming water flux to ith layer, Ksat,iis the saturated hydraulic conductivity of ith layer, Δt
is the time interval set to one day.
The value of flux q0 is calculated from the rainfall
rate R as:
q0 ¼ minðR; Ksat;l Þð16Þ

Fluxes qi, I = 1, 2,…, M are calculated recursively top-tobottom. Fluxes may be corrected beginning from any
internal layer to account for complete saturation of thelayer.
The earlier version of this model was developed
without accounting for evaporation (Guber et al.,1998). For the purposes of this work, the model was
expanded to simulate the decrease of the layer watercontents θi decrease due to evaporation and/or loss tothe upward flow. The exponential equation describes
the decrease in θ with time (Suleiman and Ritchie,2003):
hi ¼ hr;i þ ðhi;initial −hr;i Þexpð−t=sÞ

ð17Þ

n
X

Dhi Dzið19Þ

i¼1

The number of soil layers n subject to evaporation isadjusted to make the value of E equal to the evaporationin the model input. An adjustment over those layers ismade to provide the exact equality.
The set of the MWBUS model parameters includes
the saturation water contents θs,i, the field capacity
values θFC,i, the residual water content values θr,i, andthe saturated hydraulic conductivity values Ksat,i foreach layer, and the parameter of evaporation/upward
loss submodel τ0, a, and b.
4.2. Model parameters
A study of water retention along a 30-m trench in thesame soil at the adjacent site (Mallants et al., 1996) was
performed. Values of van Genuchten parameters were
found from water retention measurements done at 5-cm
long and 5.1-cm diameter cores with sand-box apparatus
for capillary pressures of 1, 5, 10, 50, and 100 cm, andwith pressure cell for capillary pressures of 200, 630,
2500, and 15,000 cm. Statistics of van Genuchten
parameters are in the Table 2 (computed from digitizedgraphs). The variability of the van Genuchten parameters α and n was relatively high in all three horizons.
There was a weak correlation between saturated watercontents θs and log10 α. In all three horizons, and amoderate correlations between log10 α and log10 n. The
same van Genuchten parameters determined in different
horizons did not correlate. The field capacity value for
the MWBUS model was taken as the water retention at
100 cm computed from the van Genuchten Eq. (13).
This value did not have significant correlation with θsand θr (data not shown). The saturated hydraulicconductivity for simulations with both models was

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Y. Pachepsky et al. / Geoderma 134 (2006) 253–266

Table 2
Statistics of van Genuchten parameters of water retention in samples along the 30-m trench

Min
AverageMax
Coefficient of variation (%)

θr,Ap

θr,C1

θr,C2

θs,Ap

θs,C1

0.000
0.040
0.083
57.7

0.000
0.015
0.075
135.1

0.000
0.044
0.101
53.8

0.348
0.411
0.484
7.2

0.330
0.363
0.424
5.2

0.268
–
–
–
1.000

Correlation coefficients significant at 0.05 significance level
θr,Ap
1.000 –––
θr,C1
1.000 ––
θr,C2
1.000 –
1.000
θs,Ap
θs,C1
θs,C2
lg αAplg αC1lg αC2
lg nAplg nC1
lg nC2θs,C2lg α§Aplg αC1lg αC2

lg nAplg nC1

lg nC2

0.379
0.431
0.502
7.7

− 2.966− 2.201− 1.845
11.0− 2.947− 1.951− 1.536
13.2−2.865−2.466−2.113
9.7

0.141
0.236
0.552
41.0

0.103
0.143
0.374
34.8

0.148
0.249
0.424
29.9

–
− 0.383
–
–
–
1.000

− 0.332–− 0.381
0.430
–
–
1.000

0.349
− 0.338
–
–
0.558
–
–
1.000

–
–
−0.684
–
–
0.444
–
–
1.000

0.678

–

–

–
–
–
–
0.376
−0.453
0.282
–
1.000

0.851
–
–
–
−0.285–−0.630
–
–
1.000

0.254
0.834
–
–
− 0.329− 0.313− 0.333− 0.899
–
0.340
1.000

lg — the decimal logarithm.

§

estimated from texture and bulk density as suggested by
Rawls et al. (1998). Values of Ksat were 40.3, 21.9, and42.4 cm d− 1 at depths from 0 to 25 cm, 25 to 65 cm, and55 to 105 cm, respectively. Parameter l in the van
Genuchten–Mualem equations was set to 0.5.
Fifty simulations of soil water flow were done withboth models. Van Genuchten parameters of soil water
retention were randomly sampled from the trench data
for each layer to perform simulations with the
HYDRUS-1D model. To run simulations with the
MWBUS model, values of the saturation water contents
and the residual water content values were randomly
sampled from the trench data, values of the field
capacity were computed from randomly chosen sets of
van Genuchten data using Eq. (13) with h = 100 cm.
Measured daily precipitation and estimated evaporationof 0.86 mm were used to compute the boundary
condition. The free drainage boundary at 150 cm wasused with the HYDRUS-1D model.
5. Results and discussion
Simulated cumulative soil water fluxes are comparedwith measured values in Fig. 6 for three major wetting–drying cycles on days 112–143, 143–232, and 232–275. The average fluxes computed with the two models
were not statistically different (t-test at p b 0.01) at 15
and 55-cm depths for all three wetting–drying cycles.
Only at the depth of 105 cm, average fluxes computedwith the two models were significantly different for thedays 112–143 and 232–275, and the HYDRUS model

had a low accuracy for the days 112–143 whereas the
MWBUS model had a lower accuracy for the days 232–
275 at this depth (Fig. 6).
When the model errors were estimated at other timescales, the two models had similar accuracy. Fig. 7
shows the root-mean-square errors (RMSEs) of theaverage daily flux computed from 4, 8, 16, and 32 daylong time series of measured and simulated cumulative
water fluxes. The time-scale dependencies of the
RMSEs from the two were close (Fig. 7). Values of
RMSE at the 55 cm depth were larger as compared with
RMSE at other depths. This was probably related to thevariability in measured flux values (see error bars in
Fig. 6). Overall, the accuracy of HYDRUS-1D and
MWBUS models was very similar with respect to thesoil water flux values.
Unlike the accuracy, measures of information andcomplexity provided clear distinction between the twomodels. Fig. 8 shows information content and complexity parameters of rainfall and simulated soil water
fluxes at the depth of 105 cm. Values of the metric
entropy, and the mean information gain show thatprecipitation had very high randomness. For example,
the metric entropy of precipitation is very close to one.
For the two-symbol words this means that the
distribution of observed states is close to the uniform
distribution. Complexity of the precipitation time series
is very low, and the precipitation time series are similar
to the random process shown in Fig. 1. These results areanalogous to the earlier results for precipitation in otherhumid regions shown in Fig. 2 (Lange, 1999a,b).

Y. Pachepsky et al. / Geoderma 134 (2006) 253–266

261▪

Fig. 6. Comparison of simulated and measured soil water fluxes over three major wetting–drying cycles on days 112–143 (●, ○), days 143–232,(θ, σ), and days, and 232–275 ( , □). Hollow and filled symbols show results from MWBUS and HYDRUS 1D, respectively.

The time series of simulated water fluxes have smaller
information content as compared with precipitation andestimated soil water fluxes. This can be seen in Fig. 8 bycomparing the metric entropy, and the mean information
gain of simulated fluxes with those values for precipitation and estimated flux. The two flow models seem toserve as the information filters. Simulated redistribution
within soil effectively smoothes the random precipitation
input, and simulated fluxes are much less random than
precipitation. We note that, unfortunately, the complexity and information measures could not be computed forthe measured discharge because the discharge observations were done with varying frequency, and thedischarge was accumulated during one to three days
before being measured.
The HYDRUS-1D model had a tendency to generatesoil water flux time series with less randomness ascompared with the MWBUS model (Table 3). The
average entropy and mean information gain overreplicated simulations were significantly different(p b 0.001 from the t-test). The MWBUS model had
the range of information content measures muchnarrower than the HYDRUS-1D model. A possible

explanation is that the soil water redistribution issimulated in much simpler fashion than in HYDRUS
1D and the signal-filtering, or de-noising, action of theformer model with respect to the precipitation is weaker.
The metric entropy ranges and the mean information
content ranges overlapped for MWBUS and HYDRUS1D and therefore the information content measuresalone could not be used to discriminate between the twomodels.
The effective measure complexity and the fluctuation
complexity show that complexity of simulated fluxes ishigher than the complexity of the precipitation (Fig. 8).
Simulated flux time series have a structure that may berelated to the computational interdependence of soil layers. The larger number of computational layers and thelarger number of the model parameters in HYDRUS-1D
might contribute to the fact that this model generated more
complex output water flux time series as compared with
the MWBUS (Table 3). The MWBUS model had therange of the complexity measure values much narrower
than the HYDRUS-1D model and generated less complex
soil water flux time series. A possible explanation is thatthe larger number of water retention parameters in

Fig. 7. Root-mean-square errors of simulated daily soil water fluxes computed as averages over time intervals shown at the horizontal axis.

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Y. Pachepsky et al. / Geoderma 134 (2006) 253–266

Fig. 8. Complexity and information content of rainfall and simulated soil water flux time series in layered soil profile at the 105-cm depth. Complexity
is measured with the effective complexity measure and the fluctuation complexity, information content is measured with the metric entropy and themean information gain; ● — simulations with HYDRUS-1D, ○ — simulations with MWBUS, ρ — rainfall.

HYDRUS-1D and their effect on the hydraulic conductivity introduces a structure in water fluxes at soil profile
bottom. The metric entropy ranges and the mean
information content ranges overlapped for MWBUS andHYDRUS-1D and therefore the complexity measuresalone could not be used to discriminate between the twomodels.
The good model discrimination was achieved whenboth information content and complexity measures areused to characterize the simulated time series. Using theeffective measure complexity in combination with themean information gain was the most efficient way to discriminate models in this work (Fig. 8). The continuum
mechanics representation of soil water flow in HYDRUS1D resulted in more complex output with less randomness.

The separation of the model outputs in the coordinates‘information-complexity’ depended on the depth where
the soil water fluxes were computed. Mean information
gain, fluctuation complexity, and effective complexity
measure are compared in Fig. 9 at depths of 15, 55, and105 cm for HYDRUS and MWBUS models. Ranges ofthe mean information gain and the measures ofcomplexity depended on the depth. Complexity of theflux simulated with the HYDRUS-1D model at 15 cm wasless at 15 cm than at 55 cm and at 105 cm. The range of themean information gain increased as the depth increased
when the HYDRUS-1D was used. With MWBUS model,
a decrease in the range of both complexity measures andthe information content measure was observed as thedepth increased. No increase in average complexity was

Table 3
Average values of the information theory measures computed for soil water flux time series simulated with HYDRUS-1D and MWBUS models
Information theory measure

Layered soil
HYDRUS-1D

Metric entropy
Mean information gain
Effective complexity measure
Fluctuation complexity

0.73 ± 0.05
0.44 ± 0.10
0.58 ± 0.10
1.52 ± 0.21

The “±” sign separates average and standard deviation.

¶¶

Homogeneous soil profile
MWBUS

HYDRUS-1D

MWBUS

0.81 ± 0.01
0.71 ± 0.01
0.19 ± 0.01
1.07 ± 0.03

0.70 ± 0.05
0.39 ± 0.10
0.61 ± 0.10
1.70 ± 0.15

0.82 ± 0.02
0.72 ± 0.02
0.20 ± 0.01
1.06 ± 0.04

Y. Pachepsky et al. / Geoderma 134 (2006) 253–266

263

Fig. 9. Complexity and information content of simulated soil water flux time series in homogeneous soil profile at the 15, 55 and 105 cm depth.
Complexity is measured with the effective complexity measure and the fluctuation complexity, information content is measured with the mean
information gain; ● — simulations with HYDRUS-1D, ○ — simulations with MWBUS.

computed for precipitation and for the typical model
outputs. All these time series show the persistence inwater fluxes. The probability of today and tomorrow
fluxes to be both less than median, or to be both greaterthan median is larger than probabilities of switchesfrom less than median to the larger than median or viceversa. However, the probability of switch from the ‘lessthan median’ to the ‘greater than median’ state insimulated fluxes is smaller than precipitation. The
continuous sequences of ‘larger than median’ or‘smaller than median’ days are shorter in precipitation
than in simulated soil water fluxes. This may be related
to the fact that large volumes of rainfall accumulate insoil and it takes some time to discharge the water
through the bottom of soil profile. Because theredistribution of soil water in the MWBUS model isinstantaneous, the periods of low (or zero) bottom flux

seen as the depth increased when the MWBUS was used.
A trend of the decrease in average information content
depth can be observed with in Fig. 9 for both models. The
mean information gain at the 15-cm depth is close to theone of the precipitation. As the depth increases, the mean
information gain decreases. The ability of the flow models
to serve as information filters increases with depth. Fig. 9shows that it is not possible to distinguish the two models
with data on information content and complexity at the
15-cm depth (Fig. 9). The differences between models
become substantial at larger depths, probably because
more information filtering occurs as the depth increases.
An insight in the differences in complexity andinformation content can be gained by considering theprobabilities of symbolic words and transitions insymbolic strings representing rainfall and simulated
water fluxes. Table 4 shows those probabilities

Table 4
State and transition probabilities for the symbol strings representing time series of precipitation and simulated soil water fluxes
Precipitation
State “j”¶

State “i”
‘00’
‘01’
‘10’
‘11’

‘00’

‘01’

MWBUS model

HYDRUS-1D model

‘10’

‘11’

‘00’

‘01’

‘10’

‘11’

‘00’

‘01’

‘10’

‘11’

State probabilities pj
0.347
0.151
0.154

0.347

0.576

0.107

0.107

0.209

0.407

0.039

0.042

0.512

Transition probabilities pi→j
0.246
0
0.102
0.099
0
0.052
0
0.039
0
0
0.113
0

0
0
0.113
0.236

0.493
0.081
0
0

0
0
0.034
0.073

0.081
0.026
0
0

0
0
0.073
0.136

0.372
0.034
0
0

0
0
0
0.039

0.037
0.005
0
0

0
0
0.042
0.471

State ‘00’ — less than median value today and tomorrow, state ‘01’ — less than median value today, greater than median value tomorrow, state ‘10’— greater than median value today, less than median value tomorrow, state ‘11’ — greater than median value today and tomorrow.

¶

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Y. Pachepsky et al. / Geoderma 134 (2006) 253–266

are highly probable compared with other time series.
The HYDRUS-1D model has the longest periods ofconsistently larger than median or consistently lower
than median fluxes. This model generates relatively
smooth fluxes because the randomness is filteredduring the water redistributions. Therefore, the corresponding time series have the smallest information
content. On contrary, the precipitation time series iscloser to the uniform distribution than the model
outputs, and thus has the largest information content
expressed as the metric entropy. Similarly, the presenceof relatively large transition probabilities causes
appreciable information gains in precipitation timeseries. Low transition probabilities make transitions
non-informative in model outputs and the mean
information gain is much lower in simulated flux
time series as compared with precipitation. The
complexity depends on the variability among transition
probabilities. High asymmetry of the transition probabilities distribution in the HYDRUS-1D output makes
them the most complex.
An increase in complexity of the conceptual soil
water flow models is often being associated withconsidering a layered flow domain instead of a single
homogeneous layer. For one, this definitely causes anincrease in the number of parameters, as each layer has
its own set of soil hydraulic properties. We ran fiftyMonte Carlo simulations of soil water fluxes for the
single soil homogeneous layer. Hydraulic propertiesof this layer were the same as they were for the topsoil layer in above simulations for the layered soil.
Information content and complexity for the single layer
simulations are shown in Fig. 10, and statistics of theinformation theory measures are shown in Table 3.
Results are very similar to the results for the layeredsoil profile in Fig. 8. The distributions of complexity
measures were slightly tighter for the single layer.
When the graphs in Figs. 8 and 10 were superimposed(not shown), they were mostly indistinguishable. The
information content and complexity of simulated soil
water fluxes appeared to be affected by the type ofwater flow transport mechanism descriptions rather
than by the variation of parameters within the flow
domain.
There seem to exist several interesting avenues ofresearch in assessment of information content and complexity of model output time series. First, the information theory offers a variety of other measures to assess
information content and complexity of systems with afinite number of states For example, Fraser (1989)

Fig. 10. Complexity and information content of rainfall and simulated soil water flux time series in homogeneous soil profile at the 105-cm depth.
Complexity is measured with the effective complexity measure and the fluctuation complexity, information content is measured with the metric
entropy and the mean information gain; ● — simulations with HYDRUS-1D, ○ — simulations with MWBUS, ρ — rainfall.

Y. Pachepsky et al. / Geoderma 134 (2006) 253–266

proposed to use the mean mutual information tomeasure how much information is in the dependency,
or correlation, of two successive words. Rényi (1961)
introduced a generalization of the Shannon entropy —the Rényi entropy of the order α that allows weighing
rare or frequent words differently. High probabilities arestrengthened compared to low probabilities for |α| N 1. If|α| b 1, then the lower probabilities are strengthened.
Wolf (1999) proposed to use differences of Rényi
entropies of conjugated orders as a measure ofcomplexity. Pincus (1995) suggested using the approximate entropy when classical statistics did not show
clear distinction. It is possible that the choice ofmeasures of information content and complexity todiscriminate models may depend on the type of modeland the type of model output.
Another issue of interest to explore is the relation
between the information content and complexity ofsoil water model outputs and the ability of soil watermodels to serve as hydrologic drivers (or vehicles) in
simulating solute transport or energy balance. Outputof two-dimensional flow models can also be analyzed
with information theory measures (Andrienko et al.,2000).
Temporal scale presents yet one more facet of theinformation content and complexity evaluation. The fullreverse-parabola-like complexity–information curve (asshown in Fig. 1) has not been reproduced in this work.
The calculated “mean information gain-fluctuation
complexity” points are mostly on the descending branchof that curve (Figs. 8 and 9). Some part of the dynamics
could be lost in randomness of a relatively coarse
temporal sampling, and finer scales may yield betterreflection of the system dynamics and even better model
discrimination (Wolf, 1999).
Two conceptually different soil water flow models
gave the same accuracy in predicting soil water fluxes inthis work. Such non-uniqueness of model structure
becomes a norm rather than an exception. A comprehensive contemplation on the non-uniqueness of modeling in environmental sciences was recently presented
by Beven (2002, 2003). In the past, the attitude was that
if a model is comprehensive enough, it would bepossible to represent the site uniqueness with a specific
set of model parameters. Recently, there has been amajor increase in detailed studies that, generally, haverevealed a complexity in flow pathways due to heterogeneity at different scales and interactions between
geometry of the flow domain and prevailing hydraulicgradients including both “preferential flow” and “stagnant zones.” Complexity of flow pathways may beeasily perceived but difficult to represent in mathemat-

265

ical terms without making strong simplifying assumptions. This implies that many different model structures
could be consistent with available observations. Beven(2000) argues that no matter how intensively a site isstudied, there is a wide range of models and parameter
sets for each model that will yield acceptable simulations. Such multiplicity of models is simply a summed
expression of the limitations to current models in
representation of flow systems, limitations of measurement techniques with respect to scales of public
interest, and limitations in defining initial and boundary
conditions. One cannot also exclude the possibility that
there still exists a lack of fundamental understanding in
fluid and chemical transport processes in the environment, and this precludes building of widely applicablemodels.
If there is a wide range of models and parameter setsfor each model that will yield acceptable simulations,then what can be an approach to model selection?
Existing approaches to the model selection are basedon the availability of input data, on previous experience, or on physically meaningful model performancewith reasonable scenarios using, i.e., the Generalized
Likelihood Uncertainty Estimation (GLUE) methodology (Beven, 2002). However, an impeccable way toselect “the model” is unknown. Using the information
theory to discriminate models may give additional
insights in their performance and make a site-specificmodel pick.
Overall, more complex simulated soil flux time series
were obtained with the HYDRUS-1D model that wasperceived to be conceptually more complex than the
WMBUS model. An increase in the complexity of water
flux time series occurred in parallel with the decrease in
the information content. Using both complexity andinformation content measures allowed us to discriminate
between the soil water models that gave the sameaccuracy of soil water flux estimates.
Acknowledgements
This work was partially supported through the Interagency Agreement RES-02-008 “Model Abstraction
Techniques for Soil Water Flow and Transport. We are
grateful to Dr. F. Wolf and two anonymous reviewers fortheir valuable comments on the early version of themanuscript.
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