Using PEST to calibrate the G Model

& Inverting it to estimate

Net Delta Outflow

Nicky Sandhu, Yu Zhou

Delta Modeling, BDO, DWR

March 10, 2015

CWEMF 2015

Martinez EC historical 15min-daily

Martinez Salinity Boundary

Delta Outflow

G-Model, Denton 1993

Flow -> G (antecedent history)

Martinez EC Estimate

• From Contra Costa Water District report 1993: Antecedent flow-salinity relations: application to delta planning model

Martinez EC Planning

• Tide effect, Atelievich 2001

• Martinez EC <- Net Delta Outflow (NDO) + Martinez stage (astronomical)

𝑆 𝑡 = 𝑆𝑜 − 𝑆𝑏 ∗ exp(−α𝐺𝑛 𝑡 𝑥(𝑡)) + 𝑆𝑏

From DWR-BDO annual report 2001, Chapter 7: Improving Salinity Estimates at the Martinez Boundary

Historical - Model

EC underestimate up to 5000 us/cm EC upper limit 25000 us/cm

Blue -historical data; red – previous calibration;

Calib 91/8-92/9, valid 93-94; Good fit for main trend, tidal envelope

Historical - Model

Blue – historical data red - previous calibration

Match well at low-mod EC range Underestimate at high EC period High EC is concern

Inverse problem: outputs + inputs -> parameters

Model-Independent, non-linear

Weighted least squares residuals sum:

Jacobian matrix, finite difference, iterative

Parameter estimation with PEST

M Inputs i

Parameters p

Measurements h

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

Objective function: = (hi - oi )2

o - o0 = J (p - p0) Ji , j = oi / pj

14 parameters

5 for G model daily salinity

9 for stage time coefficients

Input is Net Delta Outflow and stage

NDO estimates daily from DAYFLOW, DICU adjusted, DETAW adjusted

Stage 15 min (historical and/or astronomical)

Target is historical 15 min salinity at Martinez

Calibration Setup

original New calib

Calib period 1991/8-1992/9 1991/1-1993/10

phi 8.85E+11 4.43E+11

so 32797 35414

sb 200 2333.1

beta 600 419.7845

npow1 0.75 0.7700082

adel 40000 53662.45

c0 2.76E-03 1.53E-01

c1 -6.07E-05 -6.00E-05

c2 1.52E-04 7.40E-05

c3 -1.05E-05 -1.00E-05

c4 -2.83E-06 -4.00E-05

c5 4.96E-05 2.90E-06

c6 -8.76E-05 -1.00E-04

c7 7.21E-05 4.60E-05

c8 -5.18E-05 -1.00E-04

Parameter Calibrated - original

0

2E+11

4E+11

6E+11

8E+11

1E+12

original new_calib

phi before & after calib

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l par

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ters

ad

just

fir

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Historical - Recalibration

Blue - historical; Red – previous calibration; Green - new calibration;

Historical – Recalibration 2012-2014

Improves in high-EC range Match well in mod-EC range Ignore low-EC range

Blue - historical; Red – previous calibration; Green - new calibration;

PEST makes it easy to try different non-linear additional changes to the model

PEST has parallelism available so performs well with modern multi-core machines

PEST has many options to allow constrained optimizations, regularisation, etc. that are helpful when experimenting with different approaches

PEST outputs correlation information that helped identify highly correlated parameters in the model

Calibration with model changes

Inter-change parameters (Gmodel) with input (NDO)

Use a calibrated G model

Daily NDO becomes the parameters to be estimated match the measured salinity at Martinez

We did a year at a time for ease of analysis

Defining the inverse problem

M

Parameters p

Measurements h

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

Inputs i

Ill-posed problem (365 independent variables)

Non-Uniqueness of solution

NDO daily for a year

Overfitting is a very real possibility.

Fortunately PEST comes to the rescue

Tikhonov Regularisation

Truncated SVD

Inverse problem: NDO Estimation

Prior information provided to limit overfitting NDO

Uses a penalty weight, m

NDO(t)=NDO(t-1)

NDO estimates are overly smooth

NDO(t)-NDO(t-1)=NDO(t-1)-NDO(t-2)

NDO estimates are smooth

Problematic for certain years.

Needs constant tuning to balance overfit vs estimation

Tikhonov Regularisation

1991 historical vs estimated

Blue -hist data; red - calibrated model results;

EC

NDO

A form of regularisation by reducing the dimensionality of the parameter estimation.

Controlled by

MAXSING: maximum number of singular values. Tried with limits but finally allowed all 365 values.

EIGTHRESH: ratio of lowest to highest. Very effective at controlling appropriateness of fit.

Truncated SVD

NDO Estimation – Striking a balance

Red is historical EIGENTHRES Light blue 0.1 Blue 0.01 Dark blue 0.001

Eigen threshold values were the most effective at controlling fit

NDO historical vs estimated EC

NDO

Blue -historical data; red – estimated NDO;

NDO historical vs estimated (1991-1993)

Matches well however concerned with low NDO values

Investigate recent historical information

Estimate NDO another salinity-flow model DSM2:

Consumptive Use, SAC, SJR, Pumping, Gate operations

Real time operations use for forecasting runs

Water cost estimations for planning scenarios

Future Steps

PEST is easy to setup and use for calibration

Can be used for large number of parameters

Has techniques to deal with ill posed problems

Has parallel processing abilities to address performance

Has uses beyond ground water calibration

Calibration of surface water models

Calibration of non-linear regression models

Conclusion