The models presented in previous chapters have been deterministic; that is, for a given set of input values, the output is completely determined. In this chapter, descriptions of variability for the input values have been developed. We will now show how variability in input parameters relates to variability in model output. The PDF and its properties have been used to describe the soil characteristics that are model inputs. The next step is to make the model produce a PDF of output values. Once the output PDF is known, the mean, variance and other statistical moments of that property can be found.
In Section 7.4, data were transformed using log and reciprocal functions, and we noted that these transforms produced new PDFs. Other models can be thought of as transforms of the PDF as well. A model changes input variables into output variables and input PDFs into output PDFs. A deterministic model can therefore become a stochastic model by treating input parameters and variables as stochastic processes.
The simplest transforms, from a stochastic model point of view, are linear transform- ations. For these, the output PDF has the same shape as the input PDFs. If the inputs are Gaussian random processes, then the output PDF will be Gaussian. For a Gaussian process, the PDF is described by the mean and variance. Ifyis a function of variablesa, bandc,
y=f(a,b,c) (7.16)
then the expected value ofyis obtained from
y=f(a,b,c) (7.17)
150 Variation in Soil Properties Ifa,bandcare uncorrelated, then
σy2= ∂y
∂a 2
σa2+ ∂y
∂b 2
σb2+ ∂y
∂c 2
σc2 (7.18)
As an example, consider the error in gas-filled porosityφgthat would result from meas- urement error in bulk densityρband water contentw. Combining eqns (2.20) and (2.23) gives
φg= 1 –ρb
1 ρs
– w ρl
(7.19) Assume the ρs and ρl are accurately known. The expected value for φg is the value obtained by calculation from eqn (7.19) using mean values ofρb andw. If errors inρb
andware uncorrelated, then σφ2=
ρb
ρl
2
σl2– 1
ρs
– w ρl
2
σρ2 (7.20)
This equation is evaluated using mean values ofρbandw.
One serious pitfall of this approach is the assumption that the errors are uncorrel- ated. It is likely thatρb andware calculated from the same set of measurements. If, for example, the measured soil mass were too low, that would decrease the calculated bulk density and increase the calculated water content.
Nonlinear transforms, such as those for moisture characteristics or hydraulic con- ductivity functions, often produce log-transformed PDFs. Water content has a Gaussian PDF. Water potential and hydraulic conductivity have log-normal distributions. Equa- tions (7.17) and (7.18) can still be used to calculate means and variances, but these values are more difficult to interpret than for linear transforms.
If the mean and variance of a Gaussian process are known for an input variable of a nonlinear transform, the PDF of the output can be generated using a technique called Monte Carlo simulation. For this technique, input variables are drawn from populations having the specified mean and variance.
7.5.1 Scaling Methods
Because of the heavy computing requirements and the difficulty of visualizing the effects of variability in several input variables at once, it is sometimes useful to use scaled vari- ables. This concept was introduced by Miller and Miller (1956) and is called Miller scaling. The Miller scaling method derives scaling laws between porous media sam- ples that differ in their characteristic length scale only and are geometrically similar, as depicted in Fig. 7.2.
Approaches to Stochastic Modelling 151
Λ*
Λ2
Λ3
Figure 7.2 Similarity for three scales.
The main idea is that the geometry of the pore space is the same, but the media differ in their characteristic length scales, . A reference characteristic length scale (identi- fied by∗ in the figure) must be defined. For practical purposes, a mean diameter of the particle or pore size distribution is often used, but other measures could be used.
To apply the Miller concept of similarity for derivations of hydraulic properties, some assumptions are necessary:
1. The medium is homogeneous, isotropic and permanent, and hence independent of position, orientation and time, when described macroscopically.
2. The liquid phase has uniform surface tension, contact angle, viscosity and density.
3. The medium is assumed to be in theRichardsregime; this means that both the liquid and the gas are connected. Hence there are neither isolated bubbles nor isolated drops.
To find the relation between the scaling factor and the corresponding hydraulic prop- erties, we assume that a given soil, S1, is at a reference state with water content θ∗, matric potentialψ∗, conductivityK∗, characteristic length scale∗and curvature of the water–air interfacerm∗. Another soil,S2, is in a geometrically similar state, defined by a scaling factorκ. Based on the hydraulic property values of soilS1, the corresponding values for soilS2can be obtained from Table 7.2.
While the first relations are quite obvious, it is necessary to discuss the scaling of the matric potential and hydraulic conductivity. The pressure at the air–water interface is proportional to the radius of curvature:
P= 2γ
rm (7.21)
152 Variation in Soil Properties Table 7.2Miller scaling
Soil S1 S2
Scaling factor 1 κ
Characteristic length ∗ =κ∗
Curvature of the water-air interface r∗m rm=κr∗m
Water content θ∗ θ=θ∗
Matric potential ψ∗ ψ=κ–1ψ∗
Conductivity K∗ K =κ2K∗
whereγ is the surface tension [N m–1]; therefore the scaling of the pressure leads to P =2γ
rm = 2γ
κrm∗ =κ–1P∗ (7.22)
Hence the matric potential scales inversely proportional to the scaling factor:
ψ(θ) =κ–1ψ∗(θ) (7.23)
By using a dimensional analysis of Richards’ equation (Richards’ equation will be de- scribed in the following chapters), it is possible to show that the scaling approach can be applied to the hydraulic conductivity:
K(θ) =κ2K∗(θ) (7.24)
The hydraulic properties described by the Miller and Miller (1956) approach are therefore specified by providing a reference point where they are computed and a characteristic length.
The limitation of the Miller assumption is that the soil must have constant poros- ity of the heterogeneous soil, since there is only one scaling factor κ. Some authors have presented different scaling methods, but at the price of increasing the number of parameters.
The Miller characteristic length can be chosen to obtain a spatial distribution of the media that is homogeneous at the macroscopic scale but heterogeneous at the micro- scopic length. This objective can be accomplished by generating the characteristic length as a realization of a stationary random space function, having a distribution function and an autocovariance function. This stochastic approach is useful since it can gener- ate probability distributions of variables such as the soil water retention curve or the hydraulic conductivity.
Roth (1995) investigated fields that have certain statistical properties and are characterized by the scaling parameter κ, where the scaling parameter is used for
Approaches to Stochastic Modelling 153 Miller–Miller-type scaling. It is assumed that the logarithm of the scaling factor,
f = log
∗
(7.25) has the following properties: (a) it is normally distributed, (b) it has variance var(f) =σf2, (c) it has an expectationf= 0 and (d) it has autocovarianceC(h) =E[f(r)f(r+h)].
To generate a realizationf0 of a stationary random spacef with these properties, the Wiener–Khinchintheorem can be used (van Kampen, 1981). This theorem states that the Fourier transform of the autocovariance functionCis equal to the spectrumSoff0, which is defined asS(ν) = |˜f0(ν)|2:
S=F[C] (7.26)
which defines a relation between the autocovariance of a random function and the amp- litude of the different realizations. Since the autocovariance is the covariance of the variable against a time-shifted version of itself, a specific realization can be obtained by assigning fixed values of phasesφ(ν) to the frequency modes. The variableφ(ν) is a fast-changing variable that makes the phases of adjacent frequency intervals mutually independent. A realizationf˜0can be defined by
f˜0(ν) = [F[C]](1/2)exp[iφ(ν)] (7.27) For the numerical simulations, the function is needed only at discrete points. There- fore eqn (7.27) can be modified with the use of a random variable, which simplifies eqn (7.27) and leads to
f˜0(νk) = [F[C]](1/2)exp(i) (7.28) where is a uniformly distributed random variable on the interval [0, 2π]. The fast Fourier transform off˜0finally leads to a discrete realizationf0in position space.
Robinet al.(1993) gave a detailed analysis of this method, where they presented an algorithm, based on Fourier transforms, to generate pairs of random fields. As pointed out by Roth (1995), this method is advantageous for studying the influence of auto- covariance models on phenomena relevant to soil physics, such as flow and transport, because it allows the generation of fields having identical large structure but different autocovariances. Two models for the autocovariance are used: a Gaussian model
C(x) =σ2exp
–π 4
|χ|2 ι2
(7.29) and an exponential model
C(x) =σ2exp
–|χ| ι
(7.30)
154 Variation in Soil Properties where |χ| =
x2+y2andιis the correlation length, which is the characteristic length for the microscopic scale. In the program, these two models are implemented in the functionstatisticalFunction( ). The realizations for the two models are identical, with equal generating sequences ofin eqn (7.28) for both fields. The difference between the two models is that the exponential model is more irregular with respect to the Gaussian model, since the former is not differentiable.