Parameter tuning for a multi-fidelity dynamical model of the magnetosphere. (English) Zbl 1283.62243

Summary: Geomagnetic storms play a critical role in space weather physics with the potential for far reaching economic impacts including power grid outages, air traffic rerouting, satellite damage and GPS disruption. The LFM-MIX is a state-of-the-art coupled magnetospheric-ionospheric model capable of simulating geomagnetic storms. Imbedded in this model are physical equations for turning the magnetohydrodynamic state parameters into energy and flux of electrons entering the ionosphere, involving a set of input parameters. The exact values of these input parameters in the model are unknown, and we seek to quantify the uncertainty about these parameters when model outputs are compared to observations. The model is available at different fidelities: a lower fidelity which is faster to run, and a higher fidelity but more computationally intense version. Model output and observational data are large spatiotemporal systems; the traditional design and analysis of computer experiments is unable to cope with such large data sets that involve multiple fidelities of model output.
We develop an approach to this inverse problem for large spatiotemporal data sets that incorporates two different versions of the physical model. After an initial design, we propose a sequential design based on expected improvement. For the LFM-MIX, the additional run suggested by expected improvement diminishes posterior uncertainty by ruling out a posterior mode and shrinking the width of the posterior distribution. We also illustrate our approach using the Lorenz ’96 system of equations for a simplified atmosphere, using known input parameters. For the Lorenz ’96 system, after performing sequential runs based on expected improvement, the posterior mode converges to the true value and the posterior variability is reduced.


62P35 Applications of statistics to physics
86A10 Meteorology and atmospheric physics
62L05 Sequential statistical design
86A32 Geostatistics


Full Text: DOI arXiv Euclid


[1] Banerjee, S., Gelfand, A. E., Finley, A. O. and Sang, H. (2008). Gaussian predictive process models for large spatial data sets. J. R. Stat. Soc. Ser. B Stat. Methodol. 70 825-848. · Zbl 05563371
[2] Bayarri, M. J., Berger, J. O., Paulo, R., Sacks, J., Cafeo, J. A., Cavendish, J., Lin, C.-H. and Tu, J. (2007). A framework for validation of computer models. Technometrics 49 138-154.
[3] Bhat, K. S., Haran, M. and Goes, M. (2010). Computer model calibration with multivariate spatial output: A case study in climate parameter learning. In Frontiers of Statistical Decision Making and Bayesian Analysis (M. H. Chen, P. Müller, D. Sun, K. Ye and D. K. Dey, eds.) 401-408. Springer, New York.
[4] De Cesare, L., Myers, D. E. and Posa, D. (2001). Estimating and modeling space-time correlation structures. Statist. Probab. Lett. 51 9-14. · Zbl 1036.62095
[5] De Iaco, S., Myers, D. E. and Posa, D. (2001). Space-time analysis using a general product-sum model. Statist. Probab. Lett. 52 21-28. · Zbl 1129.62413
[6] Forrester, A. I. J., Sóbester, A. and Keane, A. J. (2007). Multi-fidelity optimization via surrogate modelling. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 463 3251-3269. · Zbl 1142.90489
[7] Fuentes, M. (2006). Testing for separability of spatial-temporal covariance functions. J. Statist. Plann. Inference 136 447-466. · Zbl 1077.62076
[8] Guttorp, P. and Gneiting, T. (2006). Studies in the history of probability and statistics. XLIX. On the Matérn correlation family. Biometrika 93 989-995. · Zbl 1436.62013
[9] Heaton, M. J., Kleiber, W., Sain, S. R. and Wiltberger, M. (2013). Emulating and calibrating the multiple-fidelity Lyon-Fedder-Mobarry magnetosphere-ionosphere coupled computer model. Unpublished manuscript.
[10] Higdon, D., Kennedy, M., Cavendish, J. C., Cafeo, J. A. and Ryne, R. D. (2004). Combining field data and computer simulations for calibration and prediction. SIAM J. Sci. Comput. 26 448-466. · Zbl 1072.62018
[11] Higdon, D., Gattiker, J., Williams, B. and Rightley, M. (2008a). Computer model calibration using high-dimensional output. J. Amer. Statist. Assoc. 103 570-583. · Zbl 1469.62414
[12] Higdon, D., Nakhleh, C., Gattiker, J. and Williams, B. (2008b). A Bayesian calibration approach to the thermal problem. Comput. Methods Appl. Mech. Engrg. 197 2431-2441. · Zbl 1388.80005
[13] Higdon, D., Heitmann, K., Lawrence, E. and Habib, S. (2011). Using the Bayesian framework to combine simulations and physical observations. In Large-Scale Inverse Problems and Quantification of Uncertainty (L. Biegler, G. Biros, O. Ghattas, M. Heinkenschloss, D. Keyes, B. Mallick, L. Tenorio, B. van Bloemen Waanders, K. Willcox and Y. Marzouk, eds.) 87-106. Wiley, Chichester.
[14] Johnson, M. E., Moore, L. M. and Ylvisaker, D. (1990). Minimax and maximin distance designs. J. Statist. Plann. Inference 26 131-148.
[15] Jones, D. R., Schonlau, M. and Welch, W. J. (1998). Efficient global optimization of expensive black-box functions. J. Global Optim. 13 455-492. · Zbl 0917.90270
[16] Kennedy, M. C. and O’Hagan, A. (2000). Predicting the output from a complex computer code when fast approximations are available. Biometrika 87 1-13. · Zbl 0974.62024
[17] Kennedy, M. C. and O’Hagan, A. (2001). Bayesian calibration of computer models. J. R. Stat. Soc. Ser. B Stat. Methodol. 63 425-464. · Zbl 1007.62021
[18] Le Gratiet, L. (2012). Bayesian analysis of hierarchical multi-fidelity codes. Available at [math.ST]. 1112.5389v2
[19] Lorenz, E. N. (1996). Predictability-A problem partly solved, 1-18. Reading, United Kingdom, ECMWF.
[20] Lorenz, E. N. (2005). Designing chaotic models. J. Atmospheric Sci. 62 1574-1587.
[21] Lyon, J. G., Fedder, J. A. and Mobarry, C. M. (2004). The Lyon-Fedder-Mobarry (LFM) global MHD magnetospheric simulation code. Journal of Atmospheric and Solar-Terrestrial Physics 66 1333-1350.
[22] Mitchell, M. W., Genton, M. G. and Gumpertz, M. L. (2005). Testing for separability of space-time covariances. Environmetrics 16 819-831.
[23] National Research Council (2008). Severe space weather events-Understanding societal and economic impacts: A workshop report. National Academies Press, Washington, DC.
[24] Pratola, M. T., Sain, S. R., Bingham, D., Wiltberger, M. and Rigler, J. (2013). Fast sequential computer model calibration of large non-stationary spatial-temporal processes. Technometrics 55 232-242.
[25] Qian, P. Z. G. and Wu, C. F. J. (2008). Bayesian hierarchical modeling for integrating low-accuracy and high-accuracy experiments. Technometrics 50 192-204.
[26] Qian, Z., Seepersad, C. C., Joseph, V. R., Allen, J. K. and Wu, C. F. J. (2006). Building surrogate models based on detailed and approximate simulations. Journal of Mechanical Design 128 668-677.
[27] Rougier, J. (2008). Efficient emulators for multivariate deterministic functions. J. Comput. Graph. Statist. 17 827-843.
[28] Rougier, J., Guillas, S., Maute, A. and Richmond, A. D. (2009). Expert knowledge and multivariate emulation: The thermosphere-ionosphere electrodynamics general circulation model (TIE-GCM). Technometrics 51 414-424.
[29] Sacks, J., Welch, W. J., Mitchell, T. J. and Wynn, H. P. (1989). Design and analysis of computer experiments. Statist. Sci. 4 409-435. · Zbl 0955.62619
[30] Santner, T. J., Williams, B. J. and Notz, W. I. (2003). The Design and Analysis of Computer Experiments . Springer, New York. · Zbl 1041.62068
[31] Tarantola, A. (2005). Inverse Problem Theory and Methods for Model Parameter Estimation . SIAM, Philadelphia, PA. · Zbl 1074.65013
[32] Wikle, C. K. (2010). Low-rank representations for spatial processes. In Handbook of Spatial Statistics 107-118. CRC Press, Boca Raton, FL.
[33] Wilkinson, R. D. (2010). Bayesian calibration of expensive multivariate computer experiments. In Large-Scale Inverse Problems and Quantification of Uncertainty (L. Biegler, G. Biros, O. Ghattas, M. Heinkenschloss, D. Keyes, B. Mallick, L. Tenorio, B. van Bloemen Waanders, K. Willcox and Y. Marzouk, eds.). Wiley, New York.
[34] Wiltberger, M., Wang, W., Burns, A. G., Solomon, S. C., Lyon, J. G. and Goodrich, C. C. (2004). Initial results from the coupled magnetosphere ionosphere thermosphere model: Magnetospheric and ionospheric responses. Journal of Atmospheric and Solar-Terrestrial Physics 66 1411-1423.
[35] Wiltberger, M., Weigel, R. S., Lotko, W. and Fedder, J. A. (2009). Modeling seasonal variations of auroral particle precipitation in a global-scale magnetosphere-ionosphere simulation. Journal of Geophysical Research 114 A01204.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.