1. 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

2. 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

3. 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

4. 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

5. 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

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

7. 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

8. 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

9. 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

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

11. 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

12. 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

13. 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

14. 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

15. 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

16. 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

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

18. 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

19. 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