Céline Scheidt, Jef Caers and Philippe Renard Uncertainty quantification in inverse problems: Purpose-Focused Inversion (PFI) vs. Model-Based Inversion (MBI) Céline Scheidt, Jef Caers and Philippe Renard 5/9/2013 SCRF 2013

Illustrative Example – Aquifer Analog TI – from Herten 5m Injection of tracer 20m Observation of tracer concentration at 3 depths Observed data 3.5 days 5/9/2013 SCRF 2013

Illustrative Example – Aquifer Analog TI – from Herten 5m Injection of tracer 20m Observation of tracer concentration at 3 depths Prediction of tracer concentration Observed data 3.5 days Prediction data 12 days Uncertainty ? 5/9/2013 SCRF 2013

Model-Based Inversion (MBI) Rejection Sampling: Creation of 30 models matching the data Prediction P10-P50-P90 Posterior (match) Prior (no match) 15,585 forward simulations to estimate uncertainty 5/9/2013 SCRF 2013

Questions Diagnostic tool: does matching the data improve the uncertainty estimation of the prediction? Is there a relationship between the observable response and the prediction response? If the data is only partially informative about the prediction, does it make sense to create Earth models that match the data? Can we avoid time-consuming inverse modeling in certain applications? Can we go further and estimate directly the uncertainty without generating new models matching the data? 5/9/2013 SCRF 2013

Prediction-Focused Inversion (PFI) Observable response (d) Set of prior models Forward modeling Relationship? Prediction response (h) 5/9/2013 SCRF 2013

PFI - Creation of a Joint Space How to analyze the relationship between the observable response and the prediction response? Creation of a (low dimensional) joint space data-prediction (d*,h*) d* and h* are obtained by using any dimensionality reduction technique d* h* Joint space (d*,h*) First component of: observable response  d* prediction response  h* One point = One model 5/9/2013 SCRF 2013

Joint Space (d*,h*) Any dimensionality reduction technique can be used to construct (d*,h*) MDS, PCA and NLPCA all show a relationship data-prediction MDS PCA NLPCA Observable response  d1* Prediction response  h1* High values of d1  High values of h1* Low values of d1*  Low values of h1* 5/9/2013 SCRF 2013

Non-Linear PCA (NLPCA) Input layer Output layer Bottleneck layer h Dimensionality reduction technique h* Train neural network to reproduce the inputs (identity mapping) Middle layer: bottleneck which enforces a reduction of the dimension of the data 5/9/2013 SCRF 2013

PFI: Use of NLPCA to construct (d*,h*) Dimensions: h  h* 100 1 Joint map (d*, h*) NLPCA Dimensions: d  d* 90 1 NLPCA 5/9/2013 SCRF 2013

PFI: Use of NLPCA to construct (d*,h*) Joint map (d*, h*) Prediction response Earlier arrival time Early arrival time Late arrival time Observable response Late arrival time Early arrival time 5/9/2013 SCRF 2013

PFI: Relationship between d* and h* Joint map (d*, h*) PDF for h1* Given the observed data d*obs, what is the uncertainty in h*? posterior for d*obs_2 posterior for d*obs_1 prior Diagnostic Tool d*obs_1 d*obs_2 Case 1: Little reduction of uncertainty in h1* Case 2: Significant reduction of uncertainty in h1* 5/9/2013 SCRF 2013

PFI: Relationship between d* and h* Joint map (d*, h*) PDF for h1* Given the observed data d*obs, what is the uncertainty in h*? posterior for d*obs_2 posterior for d*obs_1 prior Diagnostic Tool d*obs_1 d*obs_2 Comparison of prior/posterior distribution of h* indicates if inverse modeling allows for a better characterization of uncertainty 5/9/2013 SCRF 2013

PFI: Direct Estimation of Uncertainty Can we go beyond this diagnostic process? Can we sample directly the posterior distribution to get new prediction curves? NLPCA -1: h1*  New responses No additional models constructed Creation of new responses directly from sampling 5/9/2013 SCRF 2013

Example – Case 1 Observable response  d1* Prediction response  h1*, h2* NLPCA -1: h1*, h2*  New responses d*obs P10-P50-P90 5/9/2013 SCRF 2013

Example – Case 1 Observable response: New responses created by PFI Poor match of data but accurate uncertainty assessment Prediction P10-P50-P90 200 forward simulations to estimate uncertainty 5/9/2013 SCRF 2013

Example – Case 2 Observable response  d1* Prediction response  h1*, h2* NLPCA -1: h1*, h2*  New prediction P10-P50-P90 5/9/2013 SCRF 2013

Example – Case 2 Observable response: New responses created by PFI Match only at the later time but accurate uncertainty assessment Prediction P10-P50-P90 Rejection Sampling: 24,810 forward simulations to estimate uncertainty 5/9/2013 SCRF 2013

Summary Diagnostic tool indicates if inverse modeling allows for a better characterization of uncertainty It is possible to obtain accurate predictions without creating Earth models that match fully the data PFI approach has similar accuracy as rejection sampling for the tested cases PFI is designed for cases where prediction based on Earth model are needed, not the model itself The prediction responses should be known ahead of time MBI has more appeal when Earth models need to be created for multiple purposes that are not necessarily known a-priori 5/9/2013 SCRF 2013