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Calibration
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Manual Model Calibration
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General …..
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Non-linear parameter estimation
infer system inputs or parameters from system outputs part of the mathematics of “inverse methods” referred to as an “ill-posed problem” used in data interpretation and model calibration
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Available Software PEST – Watermark Numerical Computing UCODE – USGS
MODFLOW2000 – USGS SCE(UA) – University of Arizona NLFIT – University of Newcastle
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Theory …..
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What a model does:- M o i o = M (x,p,i) Parameters p Outputs Inputs
x describes system configuration o = M (x,p,i)
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The inverse problem:- M q i p= M-1 (x,i,q) Parameters p Measurements
Inputs i x describes system configuration p= M-1 (x,i,q)
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Model outputs for which there are field or laboratory measurements:
excitation o1 q1 o2 q2 o3 q3 etc p1 M p2 field or laboratory measurements The model
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The inverse problem:- M excitation q1 q2 q3 etc p1 p2 The model
field or laboratory measurements The model
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Model domain and observation bores
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Recharge zones
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Contoured observations
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Recharge zones
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Measured and modelled water levels
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Field or laboratory measurements:-
q1 q2 q3 x
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Field or laboratory measurements:-
value q2 q1 q3 etc distance or time
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Field or laboratory measurements and model output:-
p1 = p11 p2 = p21 Model output value q2 q1 q3 etc distance or time
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Field or laboratory measurements and model output:-
p1 = p12 p2 = p22 Model output value q2 q1 q3 etc distance or time
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Field or laboratory measurements and model output:-
p1 = p13 p2 = p23 Model output value q2 q1 q3 etc distance or time
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Residuals:- p1 = p13 p2 = p23 value q2 q1 q3 etc distance or time
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Residuals:- p1 = p13 p2 = p23 value q2 q1 q3 etc distance or time
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oi ri qi
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Objective Function:- = ri2
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Linear system (two parameters):-
M11 p1 + M12 p2 = o1 M21 p1 + M22 p2 = o2 M31 p1 + M32 p2 = o3 M41 p1 + M42 p2 = o4 etc.
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ie. M p = o
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For a linear system objective function minimized when:-
p = (Mt M)-1 Mt q q1 q2 q3 q4 etc p1 p2 q = p =
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Objective function contours
linear model p2 Objective function minimum p1
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Objective function contours
nonlinear model p2 Objective function minimum p1
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Objective function contours
linear model p2 Objective function minimum p1
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Objective function contours
linear model p2 Objective function minimum p1
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Objective function along section line
model fits data well objective function Objective function minimum parameter values
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Objective function along section line model does not fit data well
minimum parameter values
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Model does not fit data well or data noise is high
Model output value distance or time
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Model does not fit data well or data noise is high
Model output based on parameter set 1 value Model output based on parameter set 2 distance or time
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Objective function along section line model does not fit data well
Objective function is nearly at a minimum at these points Objective function minimum parameter values
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Model fits data well or data noise is low
Model output value distance or time
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Objective function along section line
model fits data well objective function Objective function minimum parameter values
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Objective function contours linear model: high parameter correlation
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Objective function along section line
linear model: high parameter correlation objective function Objective function is nearly at a minimum at these points Objective function minimum parameter values
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Objective function contours linear model: low parameter correlation
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Objective function along section line
linear model: low parameter correlation objective function Objective function minimum parameter values
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Three things which make the objective function minimum less distinct:-
a large amount of data noise poor model suitability a high degree of correlation between parameters
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It seems logical that parameters can be estimated with greater certainty in some settings than in others.
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Stochastic Interpretation …..
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Residuals:- value residuals q2 q1 q3 etc distance or time
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The variance of the residuals is:- 2 = / (m - n)
m = number of observations n = number of parameters
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The variance of the residuals is:- 2 = / (m - n)
m = number of observations n = number of parameters
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The covariance matrix of the estimated parameter set is given by:-
C(p) = 2 (Mt M)-1
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Covariance matrix:- 21,1 21,2 22,1 22,2 C(p1 , p2 ) = 21,2 = E{(p1 - E(p1))(p2 - E(p2))}
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Objective function contours:-
minimum p2 p1
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Contours of equal probability:-
Maximum probability for p1 , p2 p2 p1
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Bivariate probability density function.
Maximum probability Bivariate probability density function.
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Objective function contours
linear model p2 Objective function minimum p1
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Objective function along section line
model fits data well objective function Objective function minimum parameter values
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Objective function along section line
model fits data well maximum probability Parameter probability parameter values
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Objective function along section line model does not fit data well
minimum parameter values
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Objective function along section line model does not fit data well
maximum probability Parameter probability parameter values
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Objective function contours
linear model p2 Objective function minimum p1
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Objective function along section line model does not fit data well
minimum parameter values
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Objective function along section line model does not fit data well
maximum probability Parameter probability parameter values
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Objective function contours
linear model p2 Objective function minimum p1
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Probability contour:-
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Probability contour:-
1 1 p1
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Probability contour:-
2 2 1 1 p1
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Probability contour:-
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Probability contour:-
p1+p2 p1
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Probability contour:-
p1+p2 p1-p2 p1
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high parameter correlation
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insensitivity of at least one parameter
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The covariance matrix of the estimated parameter set is given by:-
C(p) = 2 (Mt M)-1
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The covariance matrix of the estimated parameter set is given by:-
C(p) = 2 (Mt M)-1 Increases with degree of model-to-measurement misfit Increases with degree of parameter insensitivity or correlation
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User’s Input …..
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Observation weights
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Residuals:- value q2 q1 q3 etc distance or time
Assign these points a greater weight value q2 q1 q3 etc distance or time
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Residuals:- value q2 q1 q3 etc distance or time
These points have greater weight value q2 q1 q3 etc distance or time
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Objective Function with weights:-
= (wiri)2
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Model domain and observation bores
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Less data confidence
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High predictive accuracy required here
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High data density
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Heads (m) very different units Conc (mg/l * 10-3)
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Observations have large numerical range
If uniform weighting, peak flows will dominate
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Daily Flow Water quality Monthly volume Exceedence times Φ
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Prior Information
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Information of the type:-
pi = x : weight = w1 a pi + b pj = y : weight = w2
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Prior information:- p1 = a
Contribution to objective function is higher the further is p1 from a p2 a p1
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Before prior information:-
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After prior information:-
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Think of prior information as:-
Additional observations and/or A penalty on the objective function Weights are important.
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Prior information:- removes nonuniqueness
promotes stability in inversion process allows modeller to inject his/her judgement into inversion process
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Nonlinear parameter estimation …..
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Iterative solution improvement:-
Optimal parameters p1
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Iterative solution improvement:-
Optimal parameters Initial parameter estimates p1
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Iterative solution improvement:-
At least n+1 model runs per optimisation iteration (n = no. of adjustable parameters) p1
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Repurcussions of using linearity assumption
on a nonlinear system:- convergence to objective function minimum is an iterative process calculated statistics are only approximately correct
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More on parameter uncertainty …..
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A Confined Aquifer head Fixed Inflow T1 T2 T3 Fixed head
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Objective Function
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A Confined Aquifer head Fixed Inflow T1 T2 T3 Fixed head
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Objective Function
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A Confined Aquifer head Fixed Inflow T1 T2 T3 Fixed head
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A Confined Aquifer head Fixed Inflow T1 T2 T3 Fixed head
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Hillside and Piezometers
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System Properties Transmissivity = 100 m2/day
Creek conductance is very high Recharge = 30 mm/yr
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Groundwater levels
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“Observed” groundwater levels (contoured using SURFER)
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Transmissivity distribution - I
100 m2/day
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Transmissivity distribution - II
12 m2/day 360 m2/day
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Irrigated area
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Groundwater depths after irrigation - transmissivity I
0m to 1m 1m to 2m
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Groundwater depths after irrigation - transmissivity II
0m to 1m 1m to 2m
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SNOW section of the PERLND module of HSPF
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PWATER section of the PERLND module of HSPF
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PWATER section of the PERLND module of HSPF
(continued)
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Daily Flow
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Monthly Volume
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Exceedence fraction
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Parameter values lzsn 0.2732 0.8083 infilt 0.1599 0.0704
agwrc deepfr E E-03 basetp E agwetp E E-03 uzsn intfw irc lzetp
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More parameter values lzsn 0.2732 0.8083 0.0460
infilt agwrc deepfr E E e-2 basetp E e-3 agwetp E E e-3 uzsn intfw irc lzetp
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Bayes Theorem …..
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p(,|yi) p(yi| ,) p(,)
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p(,|yi) p(yi| ,) p(,)
Likelihood function of (,), ie. a measure of the degree of fitness of different parameter combinations given the actual observations, if there were no preferences for any parameter values whatsoever.
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The posterior parameter distribution is proportional to the product of what we first thought the parameters could be, times what the data say they should be.
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Summary …..
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In calibrating a model we try to maximise the fit between model outcomes and field measurements
This will often allow estimation of some parameters with a high degree of certainty, and others with a low degree of certainty The more “noise” that exists in the data (and/or the less the degree to which the model is capable of simulating system fine detail), the higher will be the degree of parameter uncertainty Parameter correlation and insensitivity (which are both the same thing) increase the uncertainty associated with estimation of those parameters The more parameters that require estimation (ie. the more complex is the model) the more likely is this to occur There are quantifiable limits on what can be inferred from a given measurement dataset …..
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The calibration process is the imposition of a set of constraints on parameter values.
That is:- When we make predictions with the model, we should only use those parameters that allow the model to match field measurement over an historical time period. This could still leave us with a wide range of parameter values to use when making model predictions.
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Objective function minimum
p2 Optimal parameters Initial parameter estimates p1
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Objective function minimum
“allowed parameter space” p2 p1
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Objective function minimum
“allowed parameter space” p2 p1
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Objective function minimum
“allowed parameter space” p2 So which parameters do we use to make a prediction? p1
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