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Nonlinear Regression
When all K and S parameters are log-transformed, the regression for the transient problem will converge, and
optimal estimates of the nine model parameters will be obtained.
EXERCISE 9.7: Estimate parameters for the transient system by nonlinear regression.
Evaluate Model Fit
Now, we will perform the same analysis of the regression results for the transient problem that was performed for the steady-state problem.
EXERCISE 9.8: Evaluate measures of model fit
Statistical measures of overall model fit, S, s2, and s, are shown in Figure 9.13, p. 246.
Evaluate Model Fit
EXERCISE 9.9: Use Graphs for Analyzing Model Fit and Evaluate Related Statistics
EXERCISE 9.9a: Evaluate graphs of weighted residuals and weighted and unweighted simulated and observed values.
See Figure 9.14, p. 247 of Hill and Tiedeman and
statistic R in Figure 9.13.
Which graphs are most useful to understanding
model fit? Is R helpful?
Weighed Residuals vs. Simulated Values
Figure 9.14a of Hill and Tiedeman (page 247)
-3
-2
-1
0
1
2
-100 -50 0 50 100 150 200
Simulated value
Wei
ghte
d re
sidu
al
Heads
Drawdowns
Flows
Weighted Observed Values vs.Weighted Simulated Values
Figure 9.14b of Hill and Tiedeman (page 247)
-800
-600
-400
-200
0
200
-800 -600 -400 -200 0 200
Weighted simulated value
Wei
ghte
d ob
serv
ed v
alue
Evaluate Model Fit
EXERCISE 9.9b. Evaluate graphs of weighted residuals
against independent variables and the runs statistic.
The runs statistic is given in Figure 9.16, p. 249.
EXERCISE 9.9c: Assess independence and normality
of the weighted residuals.
The normal probability graph and the RN2 statistic are
shown in Figure 9.17, p. 250.
Normal Probability Graph
Figure 9.17 of Hill and Tiedeman (page 250)
-3
-2
-1
0
1
2
3
-3 -2 -1 0 1 2 3
Weighted residual
Stan
dard
nor
mal
sta
tist
icHeads
Drawdowns
Flows
Evaluate Parameter Estimates
EXERCISE 9.10: Evaluate Estimated Parameters
EXERCISE 9.10a. Composite scaled sensitivities.
EXERCISE 9.10b: Parameter estimates and confidence
intervals.
EXERCISE 9.10c: Reasonable parameter ranges.
EXERCISE 9.10d: Parameter correlation coefficients.
CompositeScaled Sens.
258.3
44.3
158.9
0.6 5.3 7.6
53.4
15.7 14.4
0
50
100
150
200
250
300
Q_ 1&2 SS_ 1 HK_ 1 K_ RB VK_ CB SS_ 2 HK_ 2 RCH_ 1 RCH_ 2
PA RA M ETER
CO
MP
OS
ITE
SC
AL
ED
SE
NS
ITIV
ITY
Figure 9.18 of Hill and TiedemanFinal Composite Scaled Sensitivities
(page 251)
197.2
18.2
142.9
0.7 3.2 1.2
41.1
6.317.0
0
50
100
150
200
250
Q_1&2 SS_1 HK_1 K_RB VK_CB SS_2 HK_2 RCH_1 RCH_2
PARAMETER
CO
MPO
SIT
E S
CA
LE
D S
EN
SIT
IVIT
Y
Figure 9.11 of Hill and TiedemanInitial Composite Scaled Sensitivities
(page 243)
ConfidenceIntervals
Figure 9.19 of Hill and Tiedeman:Confidence Intervals for Transient Regression (page 252)Figure 7.7 of Hill and Tiedeman:
Confidence Intervals for Steady State Regression (page 153)
0
100
200
300
400
500
600
700
800
HK_1 K_RB VK_CB HK_2 RCH_1 RCH_2 Q_1&2 SS_1 SS_2
Parameter
Perc
ent o
f est
imat
ed v
alue
Reasonable Range
True value
Starting value
-400
-300
-200
-100
0
100
200
300
400
500
600
HK_1 K_RB VK_CB HK_2 RCH_1 RCH_2
Per
cent
of
estim
ated
val
ue
Reasonable Range
True value
Starting value
Final Parameter Correlation Coefficients
Q_1&2 SS_1 HK_1 K_RB VK_CB SS_2 HK_2 RCH_1 RCH_2
Q_1&2 1.00 -0.75 -0.99 -0.089 -0.50 -0.056 -0.95 -0.17 -0.91
SS_1 1.00 0.74 -0.19 0.82 -0.60 0.70 0.12 0.68
HK_1 1.00 0.0003 0.51 0.057 0.91 0.18 0.90
K_RB 1.00 -0.38 0.42 0.28 0.005 0.095
VK_CB 1.00 -0.70 0.43 0.090 0.44
SS_2 symmetric 1.00 0.078 0.021 0.065
HK_2 1.00 0.14 0.88
RCH_1 1.00 -0.23
RCH_2 1.00
Table 9.7 of Hill and Tiedeman (page 253)
Model Linearity
EXERCISE 9.11: Test for linearity.
See Figure 9.20, p. 253.
The modified Beale’s measure is 84.
The model is effectively linear if this measure is less than 0.04, and
the model is nonlinear if this measure is greater than 0.44.
Update: Ground-Water Management Issues
Results from the recalibrated model can now be used to update the advective transport predictions.
Many of landfill developer’s concerns have been addressed:
Model has been calibrated with head and flow data collected under same stress conditions that will exist during operation of the landfill, and under which the advective transport will be predicted.
Uncertainty of most flow model parameters has been reduced, compared to their uncertainty in steady-state model.
Advective travel will be analyzed under steady-state pumping conditions, because these are the conditions under which the landfill will operate.
PredictingAdvectiveTransport
Figure 9.21 of Hill and Tiedeman (page 255)
Exercise 9.12a: Plot predicted path
ADVECTIVE-TRANSPORT OBSERVATION NUMBER 1 PARTICLE TRACKING LOCATIONS AND TIMES: LAYER ROW COL X-POSITION Y-POSITION Z-POSITION TIME -------------------------------------------------------------------------------- 1 2 16 15500. 1500.0 100.00 0.0000 ................................................................................ OBS # 1- 3 OBS NAME: AD10 1 2 16 15178. 1575.8 85.940 0.31500E+09 ................................................................................ 1 2 15 15000. 1615.4 79.690 0.47394E+09 1 2 14 14000. 1875.5 56.849 0.12269E+10 1 3 14 13600. 2000.0 51.405 0.14794E+10 2 3 14 13469. 2037.2 50.000 0.15518E+10 PARTICLE ENTERING CONFINING UNIT ................................................................................ OBS # 4- 6 OBS NAME: AD50 2 3 14 13469. 2037.2 48.862 0.15700E+10 ................................................................................ 2 3 14 13469. 2037.2 40.000 0.17114E+10 PARTICLE EXITING CONFINING UNIT 2 3 13 13000. 2167.8 34.419 0.20230E+10 2 3 12 12000. 2539.7 25.685 0.26478E+10 ................................................................................ OBS # 7- 9 OBS NAME: A100 2 3 12 11165. 2909.6 20.380 0.31500E+10 ................................................................................ 2 3 11 11000. 2988.7 19.436 0.32485E+10 2 4 11 10980. 3000.0 19.336 0.32603E+10 2 4 10 10000. 3609.3 14.987 0.38208E+10 2 5 10 9464.0 4000.0 13.057 0.41490E+10 2 5 9 9000.0 4426.0 11.385 0.44536E+10 2 6 9 8497.7 5000.0 10.083 0.48233E+10 2 7 9 8046.1 6000.0 8.1157 0.53184E+10 ................................................................................ OBS # 10- 12 OBS NAME: A175 2 7 9 8018.8 6524.4 6.9647 0.55200E+10 ................................................................................ 2 7 8 8000.0 6988.7 6.1411 0.56728E+10 2 8 8 7999.0 7000.0 6.1113 0.56810E+10 2 8 9 8000.0 7001.1 6.1068 0.56817E+10 2 9 9 8384.8 8000.0 3.0823 0.59752E+10 2 9 10 9000.0 8186.7 1.6827 0.60413E+10
Predicting Advective Transport
Riv
er
Well
Path in original steady-state modelTrue pathPath in updated steady-state model
Landfill
Figure 9.22 of Hill and Tiedeman (page 256)
0
1
2
3
HK_1 K_RB VK_CB HK_2 RCH_1 RCH_2 POR_1&2
Parameter Name
0
40
80
120
160 AD10x
AD10y
AD10z
AD50x
AD50y
AD50z
A100x
A100y
A100z
A175x
A175y
A175z
css
Abs
olut
e va
lue
of p
redi
ctio
n sc
aled
sen
siti
vity
(pss)
Com
posi
te s
cale
d se
nsit
ivit
y (css)
Parameters Important to Advective Paths
EXERCISE 9.12b: Evaluate the model’s ability to simulate predictions using composite and prediction scaled sensitivities, and parameter correlation coefficients.
Q_1&2 SS_1 HK_1 K_RB VK_CB SS_2 HK_2 RCH_1 RCH_2
Q_1&2 1.00 -0.75 -0.99 -0.089 -0.50 -0.056 -0.95 -0.17 -0.91
SS_1 1.00 0.74 -0.19 0.82 -0.60 0.70 0.12 0.68
HK_1 1.00 0.0003 0.51 0.057 0.91 0.18 0.90
K_RB 1.00 -0.38 0.42 0.28 0.005 0.095
VK_CB 1.00 -0.70 0.43 0.090 0.44
SS_2 symmetric 1.00 0.078 0.021 0.065
HK_2 1.00 0.14 0.88
RCH_1 1.00 -0.23
RCH_2 1.00
Q_1&2 SS_1 HK_1 K_RB VK_CB SS_2 HK_2 RCH_1 RCH_2
Q_1&2 1.00 -0.65 -0.99 -0.066 -0.40 -0.035 -0.92 -0.37 -0.84
SS_1 1.00 0.63 -0.26 0.80 -0.71 0.58 0.22 0.53
HK_1 1.00 -0.050 0.42 0.036 0.84 0.38 0.82
K_RB 1.00 -0.43 0.42 0.32 0.016 0.076
VK_CB 1.00 -0.75 0.30 0.15 0.32
SS_2 symmetric 1.00 0.063 0.028 0.047
HK_2 1.00 0.31 0.79
RCH_1 1.00 -0.17
RCH_2 1.00
Table 9.7 of Hill and Tiedeman: without predictions
Table 9.8 of Hill and Tiedeman: with predictions
Prediction Uncertainty:Linear Simultaneous Confidence Intervals
10 yrs
50 yrs
100 yrs
175 yrs
Riv
er
Well
True particleposition at:
Predicted pathConfidence intervalTrue path
50 yr100 yr
10 yr
175 yr
Fig 8.15b, p. 210
From calibration with
transient data
From calibration with steady-state data
50 yr
Riv
er
Well
100 yr
10 yr
Fig 9.23a, p. 258
EXERCISE 9.12c: Evaluate prediction uncertainty using inferential statistics.
Riv
er
Well
50 yr100 yr
10 yr
175 yr
Fig 8.15d, p. 210 Fig 9.23d, p. 258
From calibration with
transient data
From calibration with steady-state data
Prediction Uncertainty:Nonlinear Simultaneous Confidence Intervals
50 yr
Riv
er
Well
100 yr
10 yr