artificial neural networks for estimating soil hydraulic parameters from dynamic flow experiments

Artificial neural networks for estimating soil hydraulicparameters from dynamic flow experiments

G. H. SCHMITZ, H. PUHLMANN, W. DROGE & F. LENNARTZ

Institute of Hydrology and Meteorology, Dresden University of Technology, 01062 Dresden, Germany

Summary

Inverse methods are often used for estimating soil hydraulic parameters from experiments on flow of

water through soil. We propose here an alternative method using neural networks. We teach a problem-

adapted network of radial basis functions (RBF) the relationship between soil parameters and transient

flow patterns using a numerical flow model. The trained RBF network accurately identifies soil par-

ameters from flow patterns not contained in the training scenarios. A comparison with the inverse

method (Annealing-Simplex) reveals a similarly good prediction by both approaches for randomly

perturbed data and data from the real world. Nonetheless, the inverse method showed dependency on

initial parameter estimates not required by the RBF network. Training demands moderately more

computation and manpower than the inverse technique, but the absolutely stable and simple network

application requires negligible resources. Thus, for individual applications, the network approach is

slightly surpassed by the Annealing-Simplex method. However, the RBF network has to be trained

only once and, subsequently, it can be applied easily and without effort upon any number of laboratory

experiments with standardized experimental setups.

Introduction

The inverse solution of the Richards equation has become

an attractive method for determining hydraulic properties

from dynamic flow experiments (Kool & Parker, 1988; van

Genuchten et al., 1989; Durner et al., 1999). The essential part

of these inverse approaches consists of solving an optimization

problem in which the empirical parameters of the retention

and conductivity models are estimated from a best fit between

the results of a numerical simulation of the Richards equation

and the measured data (e.g. pressure heads, soil water contents

over time). The data are generally obtained from dynamic

flow experiments in the laboratory (Eching & Hopmans,

1993; Wildenschild et al., 1997; Hollenbeck & Jensen, 1998)

or in the field (Eching et al., 1994a; Inoue et al., 1998). The

simulated and observed series of state variables are compared

with each other, and the differences evaluated by an optimiza-

tion algorithm, which feeds the flow model with new soil

parameter estimates for the next simulation (e.g. Finsterle &

Faybishenko, 1999). Several publications show that inverse

optimization routines can provide unique soil parameters

from the outcome of multi-step outflow experiments (Eching

et al., 1994b; Hwang & Powers, 2003). However, many authors

found a great dependence of the identification on the experi-

mental setup, specifically on the type of boundary condition

(Zurmuhl, 1996), the number and size of pressure steps applied

to the soil column (Hollenbeck & Jensen, 1998; Wildenschild

et al., 2001; Hassanizadeh et al., 2002) and the state variables

included in the objective function (Vereecken et al., 1997). The

standardization of the experimental setup is thus a prerequisite

for a meaningful identification of soil hydraulic parameters

from flow experiments.

While most studies of the inverse method confine themselves

to analysing one single soil sample, the question arises as to

how powerful inverse procedures are if asked to identify the

soil parameters for a large number of samples, e.g. to char-

acterize soil heterogeneity statistically at the field scale.

Vereecken et al. (1997) state that the often time-consuming

procedure of locating the global optimum by trying various

different sets of initial parameters was one of the limiting

factors in analysing a large number of measurements and

was often the source of many inconsistencies. Artificial neural

networks (ANN) could be an alternative to the inverse method

of estimating the soil hydraulic parameters from flow experi-

ments. Recent studies in soil hydrology have shown that

neural networks can solve multivariate non-linear problems,

e.g. for establishing pedotransfer functions (Pachepsky et al.,

1996; Schaap & Bouten, 1996; Tamari et al., 1996; SchaapCorrespondence: G. H. Schmitz. E-mail: [email protected]

Received 21 October 2003; revised version accepted 15 December 2003

European Journal of Soil Science, February 2005, 56, 19–30 doi: 10.1111/j.1365-2389.2004.00631.x

# 2004 British Society of Soil Science 19

et al., 1998; Minasny & McBratney, 2002). Parallel develop-

ment is occurring in the field of rainfall-runoff modelling (e.g.

Hu et al., 2001; Rajurkar et al., 2002). Schmitz & Schutze

(1998) took the application of ANN one step further by pro-

posing the use of both ANN and numerical flow models for

optimizing irrigation parameters. Schmitz et al. (2002b) suc-

cessfully used self-organizing maps for evaluating optimal irri-

gation parameters for a desired distribution of soil moisture.

From the soil moisture development resulting from defined

initial and boundary conditions Schmitz et al. (2002a) derived

in an inverse manner the corresponding soil hydraulic para-

meters.

This paper further develops the approach of Schmitz et al.

(2002a) and explores how far ANN are suited for the inverse

solution of the Richards equation and, in particular, for the

estimation of the soil hydraulic parameters from laboratory

flow experiments. The first part explains the basic features of

networks of radial basis functions (RBF) – a special type of

ANN – which we found most suitable for our purpose.

Numerical simulations of multi-step outflow experiments

serve as the basis for training the RBF networks. Three differ-

ent objective functions, including either water contents, pres-

sure heads or fluxes, serve to train the network performances.

In subsequent verification runs, we ask the trained networks to

identify the soil parameters from simulated time-series that

were not contained in the training scenarios. We finally com-

pare the neural network response with the outcome of an

inverse search algorithm. For this purpose, we apply the neural

network approach and the Annealing-Simplex optimization of

Pan & Wu (1998) for identifying the soil parameters from

synthetic data with and without random noise. Moreover,

data obtained from a real laboratory experiment were included

in the overall comparison between both techniques.

Theory

Radial basis function networks

This section summarizes briefly the essentials of networks of

radial basis functions and feedforward selection; a more thor-

ough description can be found in Haykin (1999) or Howlett &

Jain (2001a,b). We outline the specific type of RBF network

we use for one-dimensional network output.

An RBF network consists of an input layer, a hidden layer

and an output layer. The number of nodes in the input layer, n,

corresponds to the number of input variables. The number of

hidden nodes, m, can be chosen arbitrarily. Each of the n

components of the input vector ~xx is connected to m basis

functions, hj. A typical radial basis function is the Gaussian:

hð~xxÞ ¼ exp � ~xx�~cck k2

r2

!; ð1Þ

where ~cc represents the centre and r the radius of the radial

basis function. The output of the hidden-layer neurons is

determined by the distance between ~xx and ~cc. The activating

function of the neurons yields high values only if the input

pattern is located near a supporting node and causes only

minor activations for test patterns outside the trained range.

Their output is combined with the weights fwjgmj¼1 finally

providing the output function f ( x!) (Figure 1). The output

layer of the RBF network is linear and produces a weighted

sum of the outputs of the hidden layer:

f ð~xxÞ ¼Xmj¼1

wjhjð~xxÞ: ð2Þ

The training of an ANN involves a learning algorithm,

which enables the network to adapt its behaviour to a given

relationship between input and output. Training consists of (i)

calculating output sets from given input sets, (ii) comparing

the calculated with the given output sets, and (iii) adjusting the

transfer function, Equation (2), for each neuron in order to

minimize the difference s between the output yi of a given

training set U ¼ (xi!; yi)

u

i¼1 and the network response f (xi!):

s ¼Xui¼1

yi � f ðxi!Þ�� ! min; ð3Þ

with respect to the weights w! as well as to the parameters of

the basis function, c! and r. In accordance with Moody &

Darken (1989), we use for the centres, c!, a subset of the

training data. The values of the radii, r, are derived from the

structure of the training data. Having thus predefined c! and r,

only the weights, w!, need to be adapted during the training. A

more detailed discussion on the selection of c! and r and their

influence on the training process follows in the Methods

section.

In general, the solution of Equation (3) is highly sensitive to

the noise and the choice of sample points within a given

training set. Moreover, the envisaged goal requires a network

that reproduces not only the training data, but also efficiently

w1 wmwj

... ...

... ...

h1(x)

f (x)

... ... ... ...

hj (x) hm (x)

x11 x1i x1nX1 = ... ... )(

xN1 xNi xNnXN = ... ... )(

Figure 1 General structure of RBF networks.

20 G. H. Schmitz et al.

# 2004 British Society of Soil Science, European Journal of Soil Science, 56, 19–30

interpolates between the sample points of the training set. The

addition of a penalty term to Equation (3) penalizes those

nodes which are combined with large weights w! and, by this,

leads to a more stable and general mapping of the shape of the

training set rather than only approximating the discrete sample

points. Equation (3) then becomes

s ¼Xui¼1

yi � f ðxi!Þ�� þ �

Xmj¼1

wj ! min; ð4Þ

where � is the regularization parameter.

Equation (4) can be solved analytically if the complete set of

training pairs is used. However, whilst the network that uses

the complete training set will fit the training data with a

minimum value of s, Equation (4), at the same time it loses

flexibility and will not adequately respond to new data not

included in the training. To minimize the number of neural

nodes whilst preserving the overall reliability of the network,

we use a special type of learning algorithm, the forward selec-

tion with orthogonal least squares (see for example Ahmed

et al., 1995). The forward selection is initiated with a network

that contains one hidden node, which is formed by the training

pair with the largest contribution to s. The network is progres-

sively expanded with the training pair which – out of the

number of remaining training pairs – has the largest influence

on the objective function, Equation (4). The training term-

inates if the residuum, s, falls below a specified tolerance or

is not reduced by adding further basis functions. In so doing,

training data with marginal influence on s are discounted, thus

keeping the number of neural nodes of the RBF network as

small as possible.

Unsaturated water flow

Numerical simulations of multi-step outflow experiments were

made with a model of one-dimensional flow based on the

mixed form of the Richards equation (Celia et al., 1990):

@� hð Þ@t

¼ @

@zK hð Þ @h

@zþ K hð Þ

� �; ð5Þ

where h is the hydraulic head (cm), � is the volumetric

water content (cm3 cm�3), K is the hydraulic conductivity

(cm hour�1), z is the depth taken positive downwards (cm)

and t is time (hours). For describing the unsaturated hydraulic

conductivity and soil water retention characteristics, the

parametric models of Mualem and van Genuchten are used:

� ¼ �� r�s � �r

; ð6Þ

� ð Þ ¼ 1þ � j jnð Þ�m ð7Þ

and

K �ð Þ ¼ Ks� 1� 1��

1m

� �mn o2: ð8Þ

In these equations is the soil matric potential (cm). The

quantities �s, �r and Ks correspond to the volumetric water

content at saturation, the residual water content and the satu-

rated hydraulic conductivity, respectively. In our approach

with artificial neural networks, however, they are merely

fitting parameters, as are � (cm�1), n, m and , the last three

of which are dimensionless.

If y1, y2, . . . , yn are the soil hydraulic parameters within

discrete ranges Yi(yi)¼ {yi1,yi2, . . . ,yin}, then Y¼Y1�Y2� . . .�Yn is the set of the parameter combinations. For

discrete points in time, T¼ t1, t2, . . . , ta, and space, Z¼ z1,

z2, . . . , zb, the numerical flow simulation F: Y ! X maps a

parameter combination y!2 Y in the result x!2 X with

x! ¼ F y!� �

¼ � y!� �

Z � Tð Þ; y!� �

Z � Tð Þ; q y!� �

Z � Tð Þ �

:

The quantities �, and q are the resulting moisture contents,

soil matric potentials and water fluxes, respectively. For the

sake of clarity in the following development, we denote the

flow patterns �( y!) (Z � T), ( y!) (Z � T) and q( y!) (Z � T),

respectively, by �( y!), ( y!) and q( y!).

Methods

Numerical and laboratory experiments

We first made numerical experiments for generating the data-

base for training the RBF networks and, second, for verifying

the predictive accuracy of the trained networks on independent

test data.

The first series of numerical experiments consisted of simu-

lations of the transient water flow in a vertical soil column

(height¼ 22 cm, diameter¼ 20 cm) according to common prac-

tice for multi-step outflow experiments (Figure 2). From

hydrostatic initial conditions, a Neumann condition with

q(0,t)¼ 0 was applied at the upper boundary of the soil col-

umn. As the lower boundary a Dirichlet condition was taken:

(L,t)¼ 0, �26, �55, �100, �150, �200 cm at t¼ 0, 0.25,

26.25, 75, 141.5, 218.5 hours, respectively. A constant time

step of �t¼ 10 minutes was used for an overall simulation

time of t¼ 280 hours (1680 time steps). The flow domain was

discretized with eight computational nodes at depths of 0, 4, 7,

10, 13, 16, 19 and 22 cm. Each numerical simulation resulted in

a flow pattern described in time and space by the values of the

three state variables �(zk,tl), (zk,tl), q(zk,tl), k¼ 1, 2, . . . , 8 and

l¼ 1, 2, . . . , 1680. Observations of water contents and pressure

heads were considered at 4, 7, 9, 10, 13, 16 and 19 cm and

the cumulative outflow from the lower boundary of the soil

column was recorded (Figure 2).

The second series of numerical experiments aims at validat-

ing the network performance when dealing with real-world

data and orientates itself on a laboratory multi-step outflow

experiment. The experiment was made on a sample of fine

sand (10 cm-tall sample above a 0.64 cm-thick ceramic plate)

Artificial neural networks for estimating soil parameters 21


and was started at hydrostatic equilibrium with a pressure

head of �25 cm at the top of the sample. Six pressure steps

were applied successively at the bottom of the soil column:

(L,t)¼ 0, �19.3333, �24.3333, �29.3333, �34.3333,�49.3333, �69.3333 cm at t¼ 0, 1501, 3502, 5503, 8504,

11 505 s, respectively. For the numerical simulations of this

experiment, a constant time step of �t¼ 1 s was used for an

overall simulation time of t¼ 14 587 s. The flow domain was

discretized equidistantly with �z¼ 0.1 cm. The recorded

cumulative outflow from the soil sample was used in the

objective function for the training of an adequate RBF

network.

Generating the training database

Successful training of the RBF network requires an adequate

representation of the relationship between the soil hydraulic

parameters and the corresponding flow pattern. To avoid any

detrimental impact of measurement errors, we formed these

required training data pairs by numerical simulations. The

dimension of the training database as well as the computa-

tional effort for the network training increases disproportion-

ately with the number of parameters to be identified by the

network. We fixed the initial and boundary conditions and

dimensions of the soil column, thus keeping the degrees of

freedom for training the network as few as possible.

Table 1 shows the soil parameter domains chosen for our

analyses, which were defined from the UNSODA database

(Nemes et al., 2001).

The applications to synthetic data involve the parameter

domain of typical loamy sands and were based on the vari-

ation of three of theMualem and vanGenuchten parameters, �,

n and Ks. The parameter m was taken as 1� 1n, was fixed at

0.5 and the water content at saturation and the residual water

content were assumed to be experimentally predetermined, as

�s¼ 0.41 cm3 cm�3 and �r¼ 0.14 cm3 cm�3. A further analysis

focused on how RBF networks perform when five soil hydrau-

lic parameters are determined from synthetic flow data, i.e. �sand �r are also treated as variable soil parameters.

The application to a real laboratory flow experiment uses

the parameter domain of sands. Again, m was taken as 1� 1n

and was fixed at 0.5, and the value of �r was fixed at 0.

We discretized the defined parameter domains by equally

spaced grids for each of the variable parameters (Table 1). The

scenario databases for teaching the ANN were then generated

by running the specified experimental setups for every single

combination of the soil hydraulic parameters resulting from

the parameter grids. The resulting pairs of soil parameters and

corresponding flow patterns then formed the ‘raw’ data for

training the RBF networks and provided comprehensive infor-

mation for mapping the relationship between the soil hydraulic

parameters and the corresponding numerical solution of the

Richards equation.

The dimensions of the input part of the ‘raw’ training sets

are determined by the number of elements in the discrete time-

series of �, and q and by the number of possible parameter

combinations arising from the permutation of the soil par-

ameters. This large dimension of the ‘raw’ training sets prevents

an efficient training of the network. A further obstacle for a

successful training arises from the non-unique mapping of the

Upper boundary: no flow

4 cm

Soil

sample

TD

R p

robe

s

7 cm

10 cm

13 cm

16 cm

19 cm

4 cm

7 cm

10 cm

13 cm

16 cm

Tensiom

eters

19 cm

Outflow

Lower boundary: pressure steps

Figure 2 Setup for numerical multi-step outflow experiments.

Table 1 Range of soil parameters

Synthetic experiments Laboratory experiment

Soil parameter Minimum value Increment Maximum value Minimum value Increment Maximum value

� /cm�1 0.01 0.001 0.03 0.005 0.005 0.15

n 1.2 0.05 2.2 1.5 0.5 10

Ks /cm day�1 1 0.5 11 �7a 0.5a �1a

�s /cm3 cm�3 0.38 0.01 0.45 0.25 0.01 0.45

�r /cm3 cm�3 0.08 0.01 0.15 – Fixed –

aIn order to cover a broader parameter range, Ks was transformed to log10(Ks).



soil hydraulic parameters in the flow patterns, which also

represents one of the hazards of the inverse method (van

Dam et al., 1992; Mous, 1993; Gribb, 1996).

Forming clusters of those sets of the soil parameters which

yield similar flow patterns proved to be an efficient means for

reducing the dimension of the training set and, at the same time,

avoiding non-unique relations between soil parameters and flow

patterns. We group two input and output pairs (xi!; yi

!), (xj!; yj

!)

in the same cluster G ¼ ( x!; y!) �

if the distance, dist, between

their flow patterns is less than a tolerance limit, ":

xi!; yi

!� �; xj!; yj

!� �2 G ! dist yi

!; yj!� �

< ": ð9Þ

The distance, dist, between the flow patterns – either water con-

tents �, pressure heads or fluxes q in Equation (9) – is given by

dist� yi!; yj

!� �¼ 1

ab� yi!� �

� � yj!� �� ; ð10Þ

dist yi!; yj

!� �¼ 1

ab yi

!� �� yj

!� �� ; ð11Þ

and

distq yi!; yj

!� �¼ 1

abq yi!� �

� q yj!� �� ; ð12Þ

where all soil parameters and flow patterns are normalized in

the interval [0,1].

A further necessary condition for grouping two combin-

ations of parameters in the same cluster is that each parameter

combination, yi!, yj

!, must be adjacent to at least one element

of the cluster. The centre of the cluster is then taken as the

centre of gravity, i.e. a single pair of soil parameter combin-

ation and corresponding flow pattern represents the complete

cluster.

Defining the objective function for the network training

In our application, training of the ANN involves clusters Gi

with centres (xi!; yi

!), where yi! is a combination of the soil

hydraulic parameters and xi! is the corresponding (numerically

simulated) flow pattern. In contrast to the problem considered

in the Theory section, the output of the RBF network (the soil

parameters) is now multidimensional. Following Park & Sand-

berg (1993), we developed the network architecture for dealing

with this task. For adapting Equation (4), the distance

yi � f (xi!)

�� is replaced and Equation (4) becomes

s ¼Xui¼1

Xnj¼1

yij � f xi!� �

j

n o2dij þ �

Xmj¼1

wj ; ð13Þ

where di!

is given by

di!¼ max

xi!

;yi!� �

2Gyk!� �

� minxi!

;yi!� �

2Gyk!� �

: ð14Þ

Adapting the network architecture

We modified the general structure of the RBF network and

adapted it to the purpose of our investigation.

. For the centres c! in Equation (1), we use a subset of the

training data, i.e. the flow patterns of the cluster centres

chosen from the complete set of cluster centres by the for-

ward selection.

. The radii of the basis function are derived from the structure

of the training data and are a function of the dimensions of

the cluster G. The radius, r, in Equation (1) is replaced by

r ¼ �rG!; ð15Þ

in which

rG!¼ max

xi!

;yi!� �

2Gxi!� �

� minxi!

;yi!� �

2Gxi!� �

; ð16Þ

and � is a scaling factor, which effects an overlapping of the

different radial basis functions in the input space.

. One can derive the values of �, Equation (15), and �, Equ-

ation (13), by minimizing Equation (13) with respect not only

to the connecting weights w!, but also to � and �. This task

represents a problem of highly non-linear optimization. For

this reason, these network parameters are chosen by trial-

and-error. For specified c! and rG!, which derive directly

from the structure of the training data, different networks

are trained for different values of � and �. The network

chosen is that which yields the smallest predictive error

during both the training and the subsequent verification.

Defining the objective function for the inverse method

The outcome of the network methodology is compared with

the results obtained from the inverse Annealing-Simplex algo-

rithm. The inverse method employs objective functions which

include either water contents, pressure heads or fluxes and

which are similar to those used for the clustering, Equations

(10), (11) and (12). The values of � (yj!), (yj

!) and q (yj!) in

Equations (10)–(12) are then replaced by the observed water

contents, pressure heads and water fluxes, �$, $ and q$.

Results

Data processing

We first focused on simultaneously identifying three soil par-

ameters within the loamy sand range. We did so by using

numerically generated flow patterns for the range of the par-

ameters �, n andKs with 21 discrete values (Table 1). Hence, the

training set contained 213¼ 9261 pairs of parameter combin-

ations and related flow patterns. Analogously, the training set

contained 592 704 training pairs for the case that five par-

ameters are to be estimated and 147 420 training pairs for the

sand example (Table 1). These numbers could be significantly



reduced – with a negligible loss of information – by the for-

mation of clusters. Table 2 shows the number of clusters and

their typical sizes for the loamy sand example with three vari-

able soil parameters. The number of developing clusters

depends mainly on the chosen tolerance limit, ", and the state

variables considered. A relatively homogeneous distribution of

the cluster centres with a similar cluster size over the entire

parameter space is a precondition for reliable interpolation

between the cluster centres. Large clusters which cover too

extensive a range of the parameter space have a detrimental

effect on the network performance. Clusters formed on the

basis of moisture contents and water fluxes, rather than only

water potentials, are more evenly distributed and, hence, will

provide a more favourable database for training an RBF net-

work.

Network training and verification

The progress during the training very much depends on the

structure of the training data, i.e. on the type of state variables

included in the objective function, Equations (10)–(12), and

the tolerance, ", Equation (9), used for the clustering. From the

clustered training set, the centres c! and the radii rG! of the

basis functions are derived according to the rules outlined in

the Methods section. Successful training then relies on the

right choice of � and �. To obtain a first impression of the

influence of � and � on the network training, both parameters

were varied systematically and different RBF networks were

trained, each using a unique combination of � and �. Figure

3(a) shows ‘contour’ plots of the residual training error, s, as a

function of the parameters � and �. All networks included in

the contour plot were trained on moisture contents and use the

same tolerance for forming clusters ("¼ 0.01). Obviously, the

sensitivity of s with respect to � is very pronounced whereas a

broad range of � values with similar residuals, s, is present.

From theoretical considerations, it is clear that the fit to the

training data is best for very small values of �. To ensure that

the trained network will also fit data not contained in the

training set, the trained networks were verified on 100 numeri-

cally generated flow patterns which were not included in the

previous training process. The validation criterion is expressed as

svalid ¼ 1

100

X100i¼1

yi!� f

!xi!� �� ; ð17Þ

where y! is the original parameter combination and f!(xi!) is

the identified parameter combination (network response). Fig-

ure 3(b) shows the values of svalid as a function of � and �. By

comparing the surfaces of s and svalid, those ranges of � and �

can be indicated where both training and verification yield

satisfying results. For the case shown in Figure 3, the optimal

parameters were found to be �¼ 2.9 and �¼ 10�4.

Table 2 Number of clusters and cluster sizes depending on the toler-

ance limit, ", Equation (9), and the state variables considered, Equ-

ations (10)–(12), for the loamy sand example with three variable

parameters (�, n and Ks)

State variable Number of

Parameter combinations per cluster

considered " clusters Mean number Maximum number

0.01 1876 5 31

� 0.02 806 11 87

0.03 454 20 209

0.01 1955 4 151

0.02 975 9 437

0.03 732 13 803

0.01 1354 7 27

q 0.02 610 15 149

0.03 429 22 390

κ

(a) Training data

0.010.010.010.010.010.01

0.010.01

0.030.030.030.050.050.05

1.3 1.7 2.1 2.5 2.9 3.3

10–5

10–3

10–1

κ

λ

(b) Test data

0.02

0.040.04

0.04

0.06

0.060.06

0.02

0.02

0.08

1.3 1.7 2.1 2.5 2.9 3.3

10–7

10–5

10–3

10–1

λ

10–7

Figure 3 Contour plots in the �� plane of (a) the residual training

error, s, and (b) the validation criterion, svalid.



Table 3 demonstrates the reliability of three different net-

works for predicting – within the range of loamy sand

considered – all three soil hydraulic parameters from 100

arbitrary flow patterns not included in the training scenarios.

Along with the network parameters �, Equation (15), and �,

Equation (13), the type of state variable taken to describe the

flow pattern (either � (z,t), (z,t) or q (z,t)) has a significant

impact on the prediction (Table 3).

Figure 4 shows the development of the residual, s, Equation

(13), in the course of the training of the three networks in

Table 3. The residuum decreases continuously as new basis

functions are added to the network. While networks using flux

or water content reduce the residuum most significantly, the

training of the network considering only pressure head term-

inates with a comparatively large number of neural nodes with

a relatively large residual error. This implies that the predictive

accuracy of networks that employ pressure heads exclusively

will remain rather unsatisfactory.

Comparison of the RBF networks with an inverse method

The following comparative analysis of the network method-

ology and the inverse optimization focuses on the reliability,

robustness and efficiency of the two approaches, when applied

to determination of the soil hydraulic parameters from the

outcome of multi-step outflow experiments with standardized

initial and boundary conditions, according to common labora-

tory practice. The first part of this investigation uses synthetic

data with and without noise. It aims to cover the more general

aspects such as the impact of the choice of the reference vari-

ables in the objective function or the initial guess for starting

the parameter identification procedure. The second part refers

to an application of both approaches for identifying the soil

hydraulic parameters from the outcome of a laboratory multi-

step outflow experiment.

Application to synthetic data without noise

The first applications of the trained networks were based on

numerically simulated flow patterns. The simulations also indi-

cated the soil parameters originally used for generating these

patterns along with the exact location of the optimal parameter

set where the corresponding objective function is zero.

One hundred randomly chosen different parameter sets

within the loamy sand domain, together with the correspond-

ing flow patterns for each of the three objective functions,

Equations (10)–(12), served as the basis for a comparative

application of both an inverse method – the Annealing-

Simplex algorithm (Pan & Wu, 1998) – and the neural network

approach. The 100 initial parameter estimates for starting the

inverse optimization routine were also chosen randomly within

the confined domain of loamy sand (Table 1).

Figure 5 displays the reference values together with the

corresponding parameter estimations made by the trained

RBF networks and the Annealing-Simplex optimization for

each of the three objective functions. Both the network

approach and the Annealing-Simplex method generally iden-

tify the soil parameters reliably. However, some deterioration

in the quality of the parameter identification by both

approaches can be seen when soil water pressure is used exclu-

sively in the objective function. This can be attributed to the

comparatively low sensitivities of the simulated pressure heads

to changes in the soil hydraulic parameters. In some cases, the

Annealing-Simplex identifies combinations of parameters far

from the optimum value. These we regard as outliers which

arise from ‘wrong’ (random) initial guesses of the soil par-

ameters. In contrast to the inverse optimization, the network

approach does not rely on user-specified initial values and is

comparatively safe from outliers in the soil parameters identi-

fied.

We then extended the application to the case where five

Mualem and van Genuchten parameters (�, Ks, n, �r and �s)

are to be evaluated from synthetic flow patterns. This task

necessarily resulted in more training which, because of present

restrictions on computer resources, could not be done for

ultimate accuracy. Nonetheless, presenting the network with

100 different flow patterns in the form of the transient devel-

opment of the water content also produced very satisfactory

results. Figure 6 shows the frequency distribution of the rela-

tive deviations of the predicted parameter values of �r, �s, �, n

and Ks from the ones used in the numerical experiments which

had initially produced the flow patterns and which had not

been part of the training. The result demonstrates high reliabil-

ity and robustness even for an RBF network generated with a

reduced training effort.

Application to synthetic data with noise

Laboratory experiments often produce data with noise. We

therefore extended our analyses with synthetic data and

Table 3 Network performance for verification as a consequence of the state variables considered and the network parameters � and �; " was taken to

be 0.01

State variable considered � � Neural nodes Training residual, s Validation error, svalid

� 2.9 10�4 692 0.0076 0.0125

1.7 10�6 1036 0.0168 0.0487

q 2.1 10�7 910 0.0020 0.0275



added random noise to the (normalized) synthetic flow pat-

terns used in the previous paragraph:

�0 z; tð Þ ¼ � z; tð Þ þ ; ð18Þ

0 z; tð Þ ¼ z; tð Þ þ ; ð19Þ

and

q0 z; tð Þ ¼ q z; tð Þ þ ; ð20Þ

in which is normally distributed with ¼N(0,�2). To

simulate various degrees of measurement errors we considered

standard deviations for ranging from 0.01% to 1%. Both the

Annealing-Simplex and the RBF networks were asked to

re-identify the soil parameters from the noisy flow patterns.

For this we developed two distinct RBF networks. The first

network was trained on smooth, synthetic data without noise

as described in the paragraph above. The second network was

trained on noisy training data, i.e. cluster centres with added

random noise.

Both the RBF networks and the Annealing-Simplex optim-

ization proved to be fairly immune to noise given the variances

arising from measurement errors in the real world. In this

0 200 400 600 800 10000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Count of neural nodes

Res

idua

l err

or, s

Figure 4 Residual error, s, for using either � (solid line, with �¼ 2.9,

�¼ 10�4), (dash-dotted line, with �¼ 1.7, �¼ 10�6), or q (dotted

line, with �¼ 2.1, �¼ 10�7).

2 4 6 8 10

2

4

6

8

10

Ks,reference

Ks,

iden

tifie

d

(a) Soil moisture

0.01 0.015 0.02 0.025 0.030.01

0.015

0.02

0.025

0.03

αreference

α ide

ntifi

ed

1.2 1.4 1.6 1.8 2.0 2.21.2

1.4

1.6

1.8

2.0

2.2

nreference nreference nreference

n ide

ntifi

ed

2 4 6 8 10

2

4

6

8

10

Ks,reference

Ks,

iden

tifie

d

(b) Pressure head

0.01 0.015 0.02 0.025 0.030.01

0.015

0.02

0.025

0.03

αreference

α ide

ntifi

ed

1.2 1.4 1.6 1.8 2.0 2.21.2

1.4

1.6

1.8

2.0

2.2

n ide

ntifi

ed

2 4 6 8 10

2

4

6

8

10

Ks,reference

Ks,

iden

tifie

d(c) Flux

0.01 0.015 0.02 0.025 0.030.01

0.015

0.02

0.025

0.03

αreference

1.2 1.4 1.6 1.8 2.0 2.21.2

1.4

1.6

1.8

2.0

2.2

n ide

ntifi

edα i

dent

ified

Figure 5 Reference parameters and identified parameters when using either �, or q. Triangles indicate results of the Annealing-Simplex

optimization, circles indicate the solution of the network approach.



context, note that no significant improvement in the network

prediction was achieved when noisy data were used in the

network training. Figure 7 shows the parameters identified

by the first network and the inverse routine together with the

true (reference) values for the case that only soil moisture is

included in the objective functions and ¼N(0,0.25).

Application to laboratory data

The subsequent comparison of the network approach and the

Annealing-Simplex optimization involves data obtained from

a real laboratory multi-step outflow experiment. We fixed �r at

0 and at 0.5. The parameter range was confined by lower and

upper bounds for each of the four variable Mualem and van

Genuchten parameters (Table 1).

For the specified parameter domain of fine sands (Table 1),

we trained an appropriate RBF network based upon the water

fluxes, Equation (12). We subsequently applied the trained

network to the observed time-series of the cumulative outflow.

The soil parameters provided by the RBF network (Table 4)

were then used for computing the corresponding cumulative

outflow. The simulated outflow agrees almost perfectly with

the observed cumulative outflow from the soil column (Figure 8).

The remarkable accuracy of the network prediction con-

firmed that the new methodology, despite still being in the

early stages of its development, can predict soil hydraulic

parameters from the outcome of real laboratory experiments

with convincing reliability.

The soil parameters identified by the RBF network were

confirmed by the Annealing-Simplex optimization, which

yielded almost identical results (Table 4 and Figure 8).

Conclusions and visions for future development

We have presented an alternative approach to the inverse

method for estimating soil hydraulic parameters from dynamic

flow experiments. It employs a numerical solution of the

Richards equation for teaching an artificial neural network

(ANN) the relationship between the van Genuchten par-

ameters and transient flow patterns. To achieve this goal, we

developed an adequate ANN for the problem using the theory

of radial basis functions (RBF) networks. The first part of our

contribution explains the basic features of this type of ANN,

which we found most suitable for the application envisaged.

Soil hydrological practice recommended basing the training of

the RBF networks on numerical multi-step outflow experi-

ments, which allowed generation of not only the training

databank, but also a separate validation databank for an

unbiased and accurate examination of the predictive accuracy

of the trained networks. The experiments were made for a

standardized experimental setup with initial and boundary

2 4 6 8 10

2

4

6

8

10

Ks,

iden

tifie

d

Ks,reference

0.01 0.015 0.02 0.025 0.030.01

0.015

0.02

0.025

0.03

αreference

α iden

tifie

d

1.2 1.4 1.6 1.8 2.0 2.21.2

1.4

1.6

1.8

2.0

2.2

nreference

n iden

tifie

d

Figure 7 Reference parameters and identified parameters for noisy data for the Annealing-Simplex optimization (triangles) and the network

approach (circles).

0 0.05 0.10 0.15 0.20 0.250

0.2

0.4

0.6

0.8

1.0

Fre

quen

cy

Relative deviation

Ks

αnθr

θs

Figure 6 Cumulative frequencies of relative deviations between

reference parameters and identified parameters for training data with

five variable soil parameters.



conditions chosen according to common laboratory practice.

Varying the soil hydraulic parameters systematically within the

ranges of loamy sands as well as fine sands as specified in the

UNSODA databank provided the raw data, i.e. the simulated

time-series of water contents, pressure heads and fluxes (input

vector) together with corresponding sets of soil hydraulic par-

ameters (output vector). We then condensed the raw data by

grouping similar flow patterns in clusters which finally formed

the input–output scenario databank used for the training. For

a comprehensive analysis of the experimental setup and a

sound overall comparison with the inverse method, we built

different RBF networks based exclusively on either water

contents, pressure heads or fluxes, respectively.

After training, these RBF networks were applied to identi-

fying soil parameters from generated flow patterns not con-

tained in the training scenarios. This subsequently revealed a

surprisingly significant impact of the type of measurement on

the predictive capability. The RBF networks trained on the

basis of water content or flux were found to perform better

than those trained on pressure heads only. Nonetheless, the

soil hydraulic parameters were predicted by each of the three

RBF networks with excellent accuracy. Prediction was some-

what less accurate with synthetic time-series flow patterns

obtained by randomly perturbed data from numerical multi-

step outflow experiments. In a comparative application deal-

ing with the same task, the inverse method with Annealing-

Simplex optimization was equally accurate. We then applied

the two strategies for identifying the parameters to data from

the real world deriving from a laboratory multi-step outflow

experiment. Again, the soil hydraulic parameters were pre-

dicted by both approaches with convincing accuracy. None-

theless, the inverse method showed some dependency on the

initial estimate of the optimum parameter set, which one need

not specify when using the RBF networks.

Although the absolutely stable and straightforward applica-

tion of the trained network is shown to require small

resources, the network training procedure requires more com-

putation and manpower than the inverse technique for a spe-

cific parameter identification task. With respect to individual

predictions on the basis of a small number of soil samples, the

network approach is therefore slightly surpassed by the

Annealing-Simplex one. However, the proposed network

methodology also features advantages over the common

inverse methods. After the network has been trained, which

has to be done only once, it can be routinely applied without

effort to any number of laboratory experiments. In this con-

text, we should emphasize that the network approach does not

rely on user-specified initial estimates for the location of the

optimum parameter vector. The actual application of the

trained network on laboratory experiments requires no exper-

tise in working with numerical methods because no numerical

flow models are involved. All problems with respect to numer-

ical errors, sensitivities to time and space discretization and

numerical instabilities are handled in the training procedure

and do not appear in any application mode. For these reasons,

the application of the trained networks is not only very fast,

but is also characterized by its stability and its objectiveness,

which may also be of particular interest to investigators

exploiting data from evaporation experiments or other types

of experimental flow scenarios.

Further development of the new methodology targets the

extension of this latest technology to cope with the identifica-

tion of soil hydraulic parameters for a wider range of soil

parameters corresponding to a larger variety of soil types.

This will include the parallel application of separately trained

sub-networks within an overall network structure and finally

should lead to the development of a global parameter identi-

fication tool that covers the whole spectrum of natural soils.

Hand-in-hand with this development, standardized multi-step

outflow experiments are required. These may be done on each

soil type of interest, and networks can be optimized for the

experimental setup with respect to efficiency, information from

the measurements, labour and cost.

Acknowledgements

The funding of this research project by the Deutsche

Forschungsgemeinschaft (DFG) under Grant Schm 1183/6-1

0 1 2 3 40

0.5

1.0

1.5

2.0

2.5

3.0

Time /hours

Cum

ulat

ive

outfl

ow /c

m

–70

–60

–50

–40

–30

–20

–10

Pressure head /cm

Figure 8 Observed cumulative outflow (solid line) and simulated

cumulative outflow using the soil parameters identified by the RBF

network (dashed line) and by the Annealing-Simplex method (dash-

dotted line). The dotted line shows the applied pressure steps.

Table 4 Soil hydraulic parameters identified by the RBF network and

the Annealing-Simplex optimization

�s /cm3 cm�3 � /cm�1 n Ks /cm s�1

RBF network 0.310 0.036 6.70 0.0158

Annealing-Simplex 0.305 0.036 6.72 0.0156



is gratefully acknowledged. We thank Mark J.L. Orr who

submitted his neural network routines to the Internet. The

algorithms were adapted for the purposes of this paper. The

finite difference code for solving the Richards equation was

provided by Rudolf Liedl. The laboratory experiment data

were kindly provided by Hans-Jorg Vogel (University of

Heidelberg).

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