IMPLEMENTATION OF PEST TO IMPLEMENTATION OF PEST TO THE PHASE 5 SEDIMENT AND THE PHASE 5 SEDIMENT AND

WATER-QUALITY WATER-QUALITY CALIBRATIONCALIBRATIONHSPF – CBWM HSPF – CBWM

Modeling SubcommitteeModeling Subcommittee

Quarterly ReviewQuarterly Review

January 25, 2006January 25, 2006

o = M (x,p,i)

What a model does:-

MInputs

iOutputs

o

Parameters p

x describes system configuration

MInputs

iMeasurements

q

Parameters p

x describes system configuration

p= M-1 (x,i,q)

The inverse problem:-

PESTPEST Parameter estimation: iterative Parameter estimation: iterative

processprocess Pest minimizes the weighted sum of Pest minimizes the weighted sum of

squared differences between model squared differences between model predictions and measured data.predictions and measured data.

Gauss-Marquardt-Lavenberg Gauss-Marquardt-Lavenberg algorithmalgorithm

Singular Value DecompositionSingular Value Decomposition

qi

oi

ri

m

1i

2ii )rw(

Hea

ds (

m)

Con

c (m

g/l *

10-3

)

very different units

The weight pertaining to each observation should be proportional to the inverse of the standard deviation associated with each observation.

Theoretically:-

Nonlinear parameter estimation …..

Initial parameter estimate:-

o0 = M( p0 )

Initial estimates

Model outcomes Nonlinear model

Using Taylor’s theorem

o=o0 + J(p - p0 )

Parameter vector

Model outcomes Jacobian matrix

Jacobian matrix:-

o1 / p1 o1 / p2 o1 / p3 o1 / p4

o2 / p1 o2 / p2 o2 / p3 o2 / p4

o3 / p1 o3 / p2 o3 / p3 o3 / p4

o4 / p1 o4 / p2 o4 / p3 o4 / p4

o5 / p1 o5 / p2 o5 / p3 o5 / p4

o6 / p1 o6 / p2 o6 / p3 o6 / p4

o7 / p1 o7 / p2 o7 / p3 o7 / p4

o8 / p1 o8 / p2 o8 / p3 o8 / p4

etc

One model run (at least) per adjustable parameter to calculate derivatives using finite differences.

p1

p2

Iterative solution improvement:-

At least n+1 model runs per optimization iteration (n = no. of adjustable parameters)

Regularization and SimultaneousCalibration…..

Simultaneous calibrationL

ZS

N

Watershed #1 #2 #3 #4 #5

ad

jus

tab

le

Simultaneous calibration with regularisationL

ZS

N

Watershed #1 #2 #3 #4 #5

ad

jus

tab

le

minimum deviation from estimated average

TSPROC

Input files

Output files

PEST

writes model input files

reads model output files

Modelcalibration conditions

Input files Input files

Output files Output files

PEST Defaultcondition

pi36 1.0 * log(irc1) - 1.0 * log(irc12) = 0.0 1.00 regulirpi37 1.0 * log(irc12) - 1.0 * log(irc23) = 0.0 1.00 regulirpi38 1.0 * log(irc23) - 1.0 * log(irc34) = 0.0 1.00 regulirpi39 1.0 * log(irc34) - 1.0 * log(irc45) = 0.0 1.00 regulirpi40 1.0 * log(irc45) - 1.0 * log(irc1) = 0.0 1.00 regulir

For the IRC parameter

pi26 1.0 * log(uzsn1) - 1.0 * log(uzsn12) = 0.0 1.00 reguluzpi27 1.0 * log(uzsn12) - 1.0 * log(uzsn23) = 0.0 1.00 reguluzpi28 1.0 * log(uzsn23) - 1.0 * log(uzsn34) = 0.0 1.00 reguluzpi29 1.0 * log(uzsn34) - 1.0 * log(uzsn45) = 0.0 1.00 reguluzpi30 1.0 * log(uzsn45) - 1.0 * log(uzsn1) = 0.0 1.00 reguluz

For the UZSN parameter

pi55 1.0 * log(lzsn1) = 0.90308998699 1.00 regullz2pi56 1.0 * log(lzsn12) = 0.90308998699 1.00 regullz2pi57 1.0 * log(lzsn23) = 0.90308998699 1.00 regullz2pi58 1.0 * log(lzsn34) = 0.90308998699 1.00 regullz2pi59 1.0 * log(lzsn45) = 0.90308998699 1.00 regullz2

We may also have…

Application to the Monocacy……

Future Work……

Evaluate the accuracy of the model Evaluate the accuracy of the model predictions with regularizationpredictions with regularization

Determine an optimum period of Determine an optimum period of calibration/verificationcalibration/verification

Try additional criteria for the Objective Try additional criteria for the Objective function (i.e., rating curve for sediment function (i.e., rating curve for sediment calibration)calibration)

Evaluate/Eliminate insensitive Evaluate/Eliminate insensitive parametersparameters

Extend application to entire watershedExtend application to entire watershed