w
m ijk
¼ w
m ¹ 1 ijk
¼ m
]E
m p
]w
m ¹ 1 ijk
þ y w
m ¹ 1 jk
¹ w
m ¹ 2 jk
ÿ m [
, 1
, y [
[ ,
1 ]
ð 7bÞ
where m ¼ training rate and y ¼ momentum factor. Note y ¼ 0 for m ¼ 1. BPA continues updating ¯
w and w until
m ¼ ¯ M, where ¯
M is a user-defined number.
3.3 Genetic algorithm
Holland
20
presented the GA as a heuristic, probabilistic, combinatorial, search-based optimization technique pat-
terned after the biological process of natural evolution. Goldberg
21
discussed the mechanism and robustness of GA in solving nonlinear, optimization problems, and
Montana and Davis
22
applied GA quite robustly in ANN
training. In general, GA determines ¯ w and w in three
steps. First, GA samples an initial population of N random configurations of ¯
w and w from the weight space
defined as ¯
w
ij
[ [
¹ q
, þ
q ]
q . 0 8a
w
ijk
[ [
¹ q
, þ
q ]
q . 0 8b
where q ¼ weight bound. GA encodes each ¯ w and w con-
figuration in binary strings with L
s
bits L
s
¼ string
length and associates each string with a fitness value defined as
F
g s
¼ 1
E
g s
s ¼ 1 ,
2 , …,
N 9a
E
g s
¼ 1
2P X
P p ¼ 1
X
K i ¼ 1
y
g spi
¹ d
pi 2
y
g spi
¼ G ¯
w
g s
,
w
g s
,
x
p
ÿ 9b
where s ¼ string index; F
g s
¼ fitness for sth string after gth
generation; E
g s
¼ mean squared error for sth string after gth
generation; ¯ w
g s
¼ ¯
w in sth string after gth generation; w
g s
¼ w in sth string after gth generation; and y
g spi
¼ y
i
for
¯ w
g s
, w
g s
, and x
p
. As such, GA associates a higher fitness to strings with smaller errors. Second, GA starts updating this
initial population using S
1
for G generations. During an update from g ¹ 1 to g g ¼ generation index, GA per-
forms four operations: scaling, selection, crossover, and mutation: 1 GA scales linearly the fitnesses in the g ¹
1th population within an appropriate range using a scaling coefficient, C, where C is defined as the number of best
strings expected in the scaled population;
21
2 GA updates this population by selecting strings with a higher fitness
with a higher probability; 3 GA perturbs the resulting population by performing crossover with a probability of
p
c
; and 4 GA further perturbs the resulting population by performing mutation with a probability of p
m
. As such, GA evaluates NG þ 1 configurations, and the optimal config-
uration is searched from these configurations.
4 FLOW AND CONTAMINANT TRANSPORT IN THE UNSATURATED ZONE
4.1 Unsaturated flow
In this manuscript, a problem related to flow and transport through a one-dimensional vertical unsaturated soil profile
is selected to evaluate the applicability of ANN and to improve the approach. The steady state unsaturated flow
equation in one-dimension can be written as
d dz
K
z
dh dz
¹ K
z
¼ z [
[ ,
L
c
] 10
where h ¼ pressure head; K
z
¼ hydraulic conductivity; z ¼
depth measured downward; and L
c
¼ column length. The
soil–water retention relationship given by the van Genuch- ten parametric model
23
is used, and this model can be written as
v ¼ v
r
þ v
s
¹ v
r
1 þ lahl
b g
g ¼ 1 ¹ b
¹ 1
11a K
z
¼ K
s
S
1=2 e
1 ¹ 1 ¹ S
1=g e
ÿ
g 2
11b S
e
¼ v ¹ v
r
v
s
¹ v
r
11c where K
s
¼ saturated hydraulic conductivity; v ¼ volu-
metric water content; v
s
¼ saturated water content; v
r
¼ residual water content; a ¼ grain size distribution index;
and b is a curve fitting parameter. The boundary conditions used are a constant steady downward water flux, q, from the
profile top and stipulated water pressure head of h ¼ 0, at the profile base. These boundary conditions can be written
as
¹ K
z
dh dz
þ K
z z ¼ 0
¼ q
12a h
z ¼ L
c
¼ 12b
4.2 Single species reactive contaminant transport
The one-dimensional governing equation for transient trans- port of a single species is written as
] ]t
vRC ¼
] ]z
vD ]C
]z ¹
q ]C
]z ¹
lvC 13
where D ¼ e
q v
14a
R ¼ 1 þ r
b
k
1
C
n ¹ 1
v 14b
ANN and algorithms in flow and transport simulations 149
l ¼ l
1
þ r
b
k
1
C
n ¹ 1
v l
2
14c where C ¼ contaminant concentration; D ¼ dispersion
coefficient neglecting molecular diffusion; e ¼ dispersivity; R ¼ retardation coefficient; l ¼ overall first order decay
coefficient; l
1
¼ dissolved phase, first order, decay coeffi-
cient; l
2
¼ adsorbed phase, first order, decay coefficient;
r
b
¼ soil bulk density; k
1
¼ Freundlich coefficient; and n ¼
Freundlich exponent.
24
The soil profile is initially contami- nant free. The boundary conditions correspond to a steady
inflow of a TCE dissolved water pulse from the profile top at C0,t ¼ C
until t ¼ t followed by fresh water injection.
The profile base is assumed to be under zero dispersive flux condition. These initial and boundary conditions can be
written as C
z ,
t ¼ 0 ¼
0 z L
c
15a C
, t
¼ C
t t 15b
¼ t . t
]C ]z
z ¼ L
¼ 15c
5 RESEARCH APPROACH
In this paper, the focus is to study the six research areas identified in a previous section related to the objectives, and
the study is performed using simulation of a GFCT model related to flow and transport through a one-dimensional
vertical soil profile. Once the base case scenario and related properties are defined, simulations are performed to gener-
ate the breakthrough concentration at the profile base for different scenarios, and these simulations are performed
using the computer code, HYDRUS.
24
Using a selected number of key features of the breakthrough curve BTC,
useful ANN concepts related to architecture, sampling, training, and multiple function approximation are studied,
and the potentials of BPA and GA in ANN training are compared. Finally, a general guideline for simulating
GFCT responses using ANN is suggested.
5.1 Base problem: unsaturated zone flow and single species mass transport
