Effect of Scale - Transport and Fate of Chemicals in Soils: Principles and Applications - page 123

comparing linear models of variance pattern with nonlinear ones is equiva-
lent to direct comparison between ADE and FCDE. The comparison between
these two models is achieved by conducting the appropriate F -test (Green
and Caroll, 1978).
The relationship of the measured va ri ance of travel distance to the mean
travel distance or time (i.e.,

σ vs t ) can thus be fitted using
linear or nonlinear models through linear and nonlinear least-squares fit-
ting techniques. Because of the simplicity of the linear least-squares method,
nonlinear models are often converted to linear format by taking logari th m
transformation. Therefore, there are three options to model the

σ versus x or

x x

σ - x or

x

σ - t relationship:

x

1. Linear coordinate: the linear model (model I):

YAX =× (4.40)

2. Linear coordinate: the nonlinear model (model II—nonlinear least-
squares fitting):

YAX B

=× (4.41)

3. Log-log coordinate: the nonlinear model (model III—linear least-
squares fitting):

a

loglog 10 Y = + b log X

(4.42)

σ , X is the mean travel
distance x or time t , and A , B , 10 a , and b are regression coefficients that can
be used to determine the transport parameters. Models II and III are equiva-
lent. For a linear model, the regression coefficient A is twice the longitudinal
dispersivity or dispersion coefficient, depending on whether X refers to the
mean travel distance or time. For nonlinear models, the power B or b can be
used to determine the fractional order a. Model I of Equation 4.37 represents
the CDE equation where α equals 2.
Zhou and Selim (2003) carried out a comparison of the different models
in an attempt to justify use of the FCDE. Specifically, a comparison between
different models is to determine whether an FCDE is necessary and whether
the FCDE is superior to the classical CDE in description of a given data set. In
their investigation, the data from the Cape Cod site are used as an example.
We fitted the above three models with the data f ro m Garabedian et al. (1991).
The functional relationships between

where Y r efers to the variance of travel distance

x

σ versus x as well as σ versus t were

x x

obtained.