Hydro-Mechanical Properties of an Unsaturated Frictional Material

3.2. STEPS OF MODEL BUILDING 71
3.2.2 Selection of the Model Form
Plots of the data, process knowledge and assumptions about the process are used to deter-
mine the form of the model to be fitted to the data. Process modeling is mainly used for
estimation, prediction, calibration and optimization. In the present work the model is used
for prediction of any combination of explanatory (suction ψ) and response variables (volu-
metric water content θ), including also values (within the interval of present data) for which
no measurements are available. In the model it is assumed that the volumetric water content
is subjected to a random error. Hence the model equation should contain at least a random
element with a specified probability distribution. The following general form is supposed for
the process model:
The model contains 3 main parts:
y = f(�x, � β) + ε (3.1)
1. The response variable, that is denoted by y and represents the volumetric water content
θ.
2. The mathematical function f(�x, � β), where x represents the explanatory variable (suction
ψ) and the parameters β (β0, β1...βn) in the model.
3. The random error ε.
Thus in our problem, where the function defines the relationship between the volumetric water
content (water content, saturation) and suction the general form leads to:
θ = f( � ψ, � β) + ε (3.2)
The random error is the difference between the experimental data and the calculated data
and is assumed to follow a particular probability distribution.
Because of its effectiveness and completeness linear least square regression method will
be used for model building and for minimizing the error between observed and predicted
results. Therefor the experimental data of suction and volumetric water content have to be
transformed to linear relationship (see 3.2.3).
3.2.3 Appropriate Data Transformation and Selection of the New Model
As can be seen in Fig. 2.7 for several types of soils the soil-water characteristic curve is a
non-linear relationship. In this study the construction of non-linear regression model for soil
data following the methodology given in Stoimenova et al. (2003b, 2006) is illustrated. It
demonstrates fitting a non-linear model and the use of transformations to deal with violation
of the assumption of constant standard deviations for the residuals. For the construction of the
non-linear model transformation of the data to linear function is used. The transformations
are applied to the experimental data to achieve the following goals: