Accounting for model errors in iterative ensemble smoothers
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- Θεοφιλά Μπότσαρης
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1 Accounting for model errors in iterative ensemble smoothers Geir Evensen International Research Institute of Stavanger Nansen Environmental and Remote Sensing Center The 3th EnKF Data Assimilation Workshop, Bergen, Ma 8 3, 8 Geir Evensen Slide
2 Eample: ES.E+7 WOPT:B-H (m^3/da) OPR WOPR:B-H (m^3/da) E+7 8E+6 OPT 6E+6 4E+6 E+6 3 Time (das) WGPT:B-H (m^3/da) WGPR:B-H (m^3/da) GPR 3 Time (das) E Time (das) E+7 4 GPT E+9 3 WWPT:B-H (m^3/da) WWPR:B-H (m^3/da) Time (das) 6 Geir Evensen.E+9 E+6 8 WPR Time (das) E+9.E+6 8E+6 6E+6 WPT 4E+6 E+6 3 Time (das) Slide
3 Eample: An iterative ES WOPT:B-H (m^3/da) OPR WOPR:B-H (m^3/da) E+7 8E+6 6E+6 E+6 3 OPT 4E+6 Time (das) WGPT:B-H (m^3/da) WGPR:B-H (m^3/da) GPR 3 Time (das) E Time (das) E+7 4 GPT E+9 3 WWPT:B-H (m^3/da) WWPR:B-H (m^3/da) Time (das) 6 Geir Evensen.E+9 E+6 8 WPR Time (das) E+9.E+6 8E+6 6E+6 WPT 4E+6 E+6 3 Time (das) Slide 3
4 Inverse problem Find R n given a model = g() given measurements d R m of R m d = + e Standard Histor-Matching problem in oil-reservoir models. Geir Evensen Slide 4
5 Data-assimilation problem Find a i+ Rn given a model prediction i+ = g( i ) and measurements d R m of i+ R n d = h( i+ ) + e Standard update step in sequential data assimilation Geir Evensen Slide
6 Baesian update Prior Baes Datum Marginal PDF Geir Evensen Slide 6
7 Ensemble filter update Prior f j = g( f j) Direct update of a j = f j + C f ( f ) ) C + C dd (d j f j Geir Evensen Slide 7
8 EnKF (direct) update Prior Baes Datum ES_D Marginal PDF Geir Evensen Slide 8
9 Ensemble smoother update Prior f j = g( f j) Smoother update of a j = f j + C f ( f ) ) C + C dd (d j f j Indirect update of a j = g( a j) Geir Evensen Slide 9
10 ES (indirect) update Marginal PDF Prior Baes ES Datum ES_D Geir Evensen Slide
11 Indirect update Inverse problems: Measure Update predict = g() Data assimilation: Measure i+ Update i predict i+ = g( i) The indirect update allows for the use of iterative methods. Geir Evensen Slide
12 Nonlinearit Geir Evensen Slide
13 Sensitivit of smoothers to nonlinearit Model = + sin(π) Case : f j N (., C =.) d j N (., C dd =.) Case : f j N (., C =.9) d j N (., C dd =.) Case 3: f j N (., C =.36) d j N (., C dd =.) Cost function p(), C=. p(), C=.9 p(), C= Geir Evensen Slide 3
14 Ensemble vs analtic gradient a j = f j + g ( f j)c ( g ( f j)c g ( f j) + C dd ) ( dj g( f j) ) a j = f j + C (C + C dd ) (d j g( f j)) a j = g( a j) Geir Evensen Slide 4
15 Case : ES.. Prior Joint PDF Baesian Posterior PDF ES_ENS ES_ANA Geir Evensen Slide
16 Case : IES.. Prior Joint PDF Baesian Posterior PDF IES_ENS IES_ANA Geir Evensen Slide 6
17 Case : ESMDA.. Prior Joint PDF Baesian Posterior PDF ES MDA_ENS MDA_ANA Geir Evensen Slide 7
18 Case : ES... Prior Joint PDF Baesian Posterior PDF ES_ENS ES_ANA Geir Evensen Slide 8
19 Case : IES... Prior Joint PDF Baesian Posterior PDF IES_ENS IES_ANA Geir Evensen Slide 9
20 Case : ESMDA... Prior Joint PDF Baesian Posterior PDF ES MDA_ENS MDA_ANA Geir Evensen Slide
21 Case 3: ES... Prior Joint PDF Baesian Posterior PDF ES_ENS ES_ANA Geir Evensen Slide
22 Case 3: IES... Prior Joint PDF Baesian Posterior PDF IES_ENS IES_ANA Geir Evensen Slide
23 Case 3: ESMDA... Prior Joint PDF Baesian Posterior PDF ES MDA_ENS MDA_ANA Geir Evensen Slide 3
24 Model errors in iterative smoothers Geir Evensen Slide 4
25 Including model errors Model Baes Marginal pdf = g(, q) f(,, q d) f(d )δ ( g(, q) ) f()f(q) f(, q d) f ( d g(, q) ) f()f(q) Ensemble of cost functions J ( j, q j ) = ( j f ) TC ( j j f ) j + ( q j q f ) TC ( j qq qj q f j) + ( ) TC ( ) g( j, q j ) d j dd g(j, q j ) d j Geir Evensen Slide
26 ES including model errors Prior Update f j = g( f j, q f j) a j = f j + C f ( f ) ) C + C dd (d j f j q a j = q f j + C f ( f ) ) q C + C dd (d j f j Indirect update a j = g( a j, q a j) or direct update a j = f j + C f ( f ) ) C + C dd (d j f j Geir Evensen Slide 6
27 ESMDA including model errors Cost function for z T j = ( T j, ) qt j at step i J (z i+) = ( z i+ z i ) T ( C i zz ) ( zi+ z i ) where we must have + ( g(z i+) d α ie i ) T ( αic dd ) ( g(zi+) d α ie i ) N mda =. α i i= ) (C i ( + α i C dd i+ = i + C i d + ) α i e i i ) ) q i+ = q i + C i q (C i ( + α i C dd d + ) α i e i i ) i+ = g( i+, q i+ ) Geir Evensen Slide 7
28 IES including model errors Cost function for z T j Iterative solution = ( T j, ) qt j J (z j ) = ( z j z f j ) TC ( zz zj z f j) + ( g(z j ) d j ) TC dd ( g(zj ) d j ) z j,i+ = z j,i + C i zzc zz (z j,i z f j) ) C i z (C i ( + C dd C i zc zz (z j,i z f j) ( ) ) g(z j,i ) d j j,i+ = g(z j,i ) Geir Evensen Slide 8
29 Scalar eample Model = ( + β ) + C qq q Linear case: β =. Nonlinear case: β =.. Prior ensemble for : N(, ). Likelihood for measurement of : N(, ). Geir Evensen Slide 9
30 Linear model without model errors Marginal PDF Prior Baes_. ES_7_. MDA_7_4_. IES_7_. Marginal PDF Prior Baes_. ES_7_. MDA_7_4_. IES_7_. Datum Geir Evensen Slide 3
31 Linear model including model errors Marginal PDF Prior Baes_. ES_7_. MDA_7_4_. IES_7_. Marginal PDF Prior Baes_. ES_7_. MDA_7_4_. IES_7_. Datum Geir Evensen Slide 3
32 Pdfs for model error with linear model Marginal PDF INI_7_. ES_7_. MDA_7_4_. IES_7_... q Geir Evensen Slide 3
33 Joint pdfs linear model Prior Joint PDF Prior Joint PDF Baes Posterior Baes Posterior Geir Evensen Slide 33
34 Joint pdfs linear model Baes Posterior Baes Posterior ES ES Geir Evensen Slide 34
35 Joint pdfs linear model Baes Posterior Baes Posterior ESMDA ESMDA Geir Evensen Slide 3
36 Joint pdfs linear model Baes Posterior Baes Posterior IES IES Geir Evensen Slide 36
37 Nonlinear model without model errors Marginal PDF Prior Baes_. ES_7_. MDA_7_4_. MDA_7_64_. IES_7_. Marginal PDF Prior PDFC_7_. ES_7_. MDA_7_4_. MDA_7_64_. IES_7_. Datum ES_7_.D Geir Evensen Slide 37
38 Nonlinear model including model errors Marginal PDF Prior Baes_. ES_7_. MDA_7_4_. MDA_7_64_. IES_7_. Marginal PDF Prior PDFC_7_. ES_7_. MDA_7_4_. MDA_7_64_. IES_7_. Datum ES_7_.D Geir Evensen Slide 38
39 Pdfs for model error with nonlinear model Marginal PDF INI_7_. ES_7_. MDA_7_4_. IES_7_... q Geir Evensen Slide 39
40 Joint pdfs nonlinear model Prior Joint PDF Prior Joint PDF Baes Posterior Baes Posterior Geir Evensen Slide 4
41 Joint pdfs nonlinear model Baes Posterior Baes Posterior ES ES Geir Evensen Slide 4
42 Joint pdfs nonlinear model Baes Posterior Baes Posterior ESMDA ESMDA Geir Evensen Slide 4
43 Joint pdfs nonlinear model Baes Posterior Baes Posterior IES IES Geir Evensen Slide 43
44 Acknowledgements RCN DIGIRES project with sponsors Equinor, Aker BP, Eni, Petrobras, VNG, and DEA, for funding this work. Y. Chen and D. S. Oliver. Levenberg-Marquardt forms of the iterative ensemble smoother for efficient histor matching and uncertaint quantification. Computat Geosci, 7:689 73, 3. A. A. Emerick and A. C. Renolds. Ensemble smoother with multiple data assimilation. Computers and Geosciences, :3, 3. G. Evensen. Analsis of iterative ensemble smoothers for solving inverse problems. Computat Geosci, (3):88 98, 8a. doi:.7/s URL G. Evensen. Accounting for model errors in iterative ensemble smoothers, 8b. URL P. K. Kitanidis. Quasi-linear geostatistical theror for inversing. Water Resources Research, 3(): 4 49, 99. D. S. Oliver, N. He, and A. C. Renolds. Conditioning permeabilit fields to pressure data. ECMOR th European Conference on the Mathematics of Oil Recover, 996. J.-A. Skjervheim, G. Evensen, J. Hove, and J. Vabø. An ensemble smoother for assisted histor matching. SPE 499,. P. J. van Leeuwen and G. Evensen. Data assimilation and inverse methods in terms of a probabilistic formulation. Mon. Weather Rev., 4:898 93, 996. Geir Evensen Slide 44
45 Summar We noted the dualit of the DA and inverse problem. Iterations improve the estimate for models with modest nonlinearit. Iterative methods allows for inclusion of general model errors. Results published in Evensen (8a,b). Geir Evensen Slide 4
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