Regularised Inversion and Model Predictive Uncertainty Analysis

PEST …

Model Input files Output files

Model Input files Output files PEST writes model input files reads model output files

Batch or Script File Input files Output files PEST writes model input files reads model output files

Model calibration conditions Input files PEST Input files Model predictive conditions Output files

Model calibration conditions Input files Model predictive conditions Output files Maximise or minimise key prediction while keeping model calibrated PEST

distance or time q1q1 q2q2 q3q3 etc value Model output Field or laboratory measurements and model output:- calibration datasetprediction

distance or time q1q1 q2q2 q3q3 etc value Model output Field or laboratory measurements and model output:- calibration dataset Lower predictive limit

distance or time q1q1 q2q2 q3q3 etc value Model output Field or laboratory measurements and model output:- calibration dataset Upper predictive limit

distance or time q1q1 q2q2 q3q3 etc value Model output Field or laboratory measurements and model output:- calibration dataset Confidence interval for prediction

distance or time q1q1 q2q2 q3q3 etc value Model output Field or laboratory measurements and model output:- calibration dataset Predictive uncertainty interval

Traditional Parameter Estimation Principal of parsimony Employ no more parameters than can be estimated Calibration complexity dictated by calibration dataset.

Regularised inversion…

Advantages of Regularised Inversion The inversion process is able to put the heterogeneity exactly where it is needed Maximum information content is extracted from the data Predictive error variance is thus minimised Parameterisation complexity determined by prediction Because complexity is retained in the system, we have the ability to realistic assess predictive uncertainty because we do not exclude the detail on which a prediction can depend.

Two Principal Types of Regularisatoin “Tikhonov” – constrained minimisation Subspace methods – principal component analysis

SVD-Assist

Advantages Highly stable numerically. Highly efficient in model run requirements. Can adapt to noise content of data.

Hydraulic conductivity

Specific Yield

Water levels

Snake River Inflow

Local Domain and Air Photo Recovery Well Source area

MTBE concentrations for an elevation of:- –35 ft-msl to –40 ft-msl

Pilot Points and Observations Pilot points – 58 per layer, L1-L7, for HHK, VHK, POR (crosses). Water level observations (circles); MTBE observations (stars) Calibrated ‘mean’ particle. Recovery Well Source area

Example Section Profile Profile across plume at IRM transect

Figure 4 Typical Concentration Profile

Observed MTBE Modelled MTBE -35 to -40 ft msl

Profile - data Source Area FLOW

Source Area FLOW Profile – data and modelled concentrations

Simulated and Observed MTBE at the Recovery Well

Calibrated Horizontal and Vertical Hydraulic Conductivities Ground Water Flow

The cost of uniqueness …..

Model grid Dimensions of model domain 500m by 800m

Boundary H = 0.0 Q = 50 m 3 /day

Particle release point

Reality

True time = days True exit point = easting of

12 head observations

Reality Exit time = 3256 Exit point = 206

Calibration to 12 observations (no noise) Exit time = 7122 [true=3256] Exit point = 241 [true=206]

This model (with its three parameters)…

Calibration to 12 observations Zone-based calibration Exit time = 6364 [true=3256] Exit point = 244 [true=206]

… does not even acknowledge the detail upon which a critical prediction will depend, whereas this model ….

Calibration to 12 observations (no noise) Exit time = 7122 [true=3256] Exit point = 241 [true=206]

Another important point… … does. The former model will grossly under-estimate predictive variance.

Calculation of Model Predictive Error Variance…..

Parameter space Increasing number of parameter combinations

Estimable parameter combinations Unestimable parameter combinations Increasing number of parameter combinations

Error variance calculable from measurement error C(h) Error variance supplied by hydrogeologists C(p) Increasing number of parameter combinations

Error variance calculable from measurement error C(h) Error variance supplied by hydrogeologists C(p) model prediction

σ 2 = y t (I-R) t C(p)(I-R)y + y t GC(h)Gy Therefore total “possible model error” depends on both C(h) and C(p)

Error variance calculable from measurement error C(h) Error variance supplied by hydrogeologists C(p) model prediction

Error variance calculable from measurement error C(h) Error variance supplied by hydrogeologists C(p) model prediction Where do we draw the line on what we try to estimate?

Number of singular values Predictive error variance “Null space” term “Measurement” term Total Predictive error variance vs dimensions of calibrated parameter space

Optimising Data Acquistion…..

Schematic block diagram illustrating model layers and boundary conditions

The prediction

Pumping from layer

Measurements

Observation wells Layer 1 Layer 2 Layer 3

Water levels

Parameters

Hydraulic conductivity – layer 1 Hydraulic conductivity – layer 2 Hydraulic conductivity – layer 3 VCONT – layer 2 VCONT – layer 3 Specific yield – layer 1 Specific yield – layer 2 Primary storage capacity – layer 2 Primary storage capacity – layer 3 Riverbed conductance Recharge Parameters included in analysis

Pre-calibration contribution to predictive error variance

Predictive error variance vs dimensions of calibrated parameter space Minimum = 418 ft 2 at 160 singular values

Contribution to pre- and post-calibration predictive variance by selected parameter types

Optimization of data acquisition:- How can I deepen the minimum in the predictive variance curve?

σ 2 = y t (I-R) t C(p)(I-R)y + y t GC(h)Gy

Reduction in predictive variance if VCONT 2 characterization at each point is reduced from 0.74 to 0.37 (maximum reduction = 112.7ft 2 )

Locations of proposed layer 2-3 differential head measurements (reduction in predictive error variance = 230 ft 2 )

Predictive error variance vs dimensions of calibrated parameter space Previous minimum = 418 ft 2 at 160 singular values New minimum = 188 ft 2 at 190 singular values

Error variance of an existing model…..

IBOUND array

Riverbed K parameters

Log of K (K ranges from 1e-4 to 500)

All lateral Inflow Zones (red cells are fixed head – except for zone 1)

Management zones

Head error variance Number of cells