Model Fusion and its Use in Earth Sciences R. Romero, O. Ochoa, A. A. Velasco, and V. Kreinovich Joint Annual Meeting NSF Division of Human Resource Development

Need to Combine Data from Different Sources-1 In science and engineering, data are generated by different sources For example in geophysics there are many sources of data for Earth models: –first-arrival passive seismic data –first-arrival active seismic data –gravity data –surface waves Joint Annual Meeting NSF Division of Human Resource Development

Need to Combine Data from Different Sources-2 Datasets from different sources provide complementary information –Different geophysical datasets contain different information on Earth structure Joint Annual Meeting NSF Division of Human Resource Development

Need to Combine Data from Different Sources-3 In general, some datasets provide better accuracy and/or spatial resolution in some areas –gravity measurements have (relatively) low spatial resolution –a seismic data point comes from a narrow trajectory of a seismic signal. Thus, spatial resolution is higher Joint Annual Meeting NSF Division of Human Resource Development

Joint Inversion: An Ideal Solution-1 Currently –Datasets are processed separately –No efficient algorithm to process all datasets simultaneously Ideally –All datasets are used for a single model Joint Annual Meeting NSF Division of Human Resource Development

Joint Inversion: An Ideal Solution-2 Designing such a joint inversion technique presents an important theoretical and practical challenge Joint Annual Meeting NSF Division of Human Resource Development

Proposed Solution: Model Fusion-1 Joint inversion methods still being developed Solution: fuse models from different datasets Simplest case: data fusion, probabilistic uncertainty –several estimates of the same quantity Joint Annual Meeting NSF Division of Human Resource Development

Proposed Solution: Model Fusion-2 –each estimation error is normally distributed with 0 mean and known standard deviation –Least squares: find that minimizes Joint Annual Meeting NSF Division of Human Resource Development solution :

Additional Problem: Different Resolution-1 Different models have different spatial resolution –Seismic data leads to higher spatial resolution estimates of the desnsity at different locations –Gravity data leads to lower spatial resolution estimates of the same densities Joint Annual Meeting NSF Division of Human Resource Development

Additional Problem: Different Resolution-2 Towards precise formulation: –High spatial resolution estimates correspond to small spatial cells –Low spatial resolution estimate is affected by several neighboring spatial cells Joint Annual Meeting NSF Division of Human Resource Development

Towards Formulation of a Problem-1 What is given: –High spatial resolution estimates of the values in several small cells –Low spatial resolution estimates for the weighted averages Joint Annual Meeting NSF Division of Human Resource Development

Towards Formulation of a Problem-2 Objective: based on the estimates and, a more accurate estimate for must be provided Geophysical example: represents the density. Joint Annual Meeting NSF Division of Human Resource Development

Model Fusion: Probabilistic Uncertainty-1 Taking into account several different types of approximate equalities –Each high spatial resolution value is approximately equal to the actual value with a known accuracy : Joint Annual Meeting NSF Division of Human Resource Development

Model Fusion: Probabilistic Uncertainty-2 –Each low spatial resolution value is approximately equal to the weighted average, with a known accuracy : –Prior knowledge of the values is approximately equal to with accuracy : Joint Annual Meeting NSF Division of Human Resource Development

Model Fusion: Probabilistic Uncertainty-3 –Each lower spatial resolution value is approximately equal to the value within each of the smaller cells: Joint Annual Meeting NSF Division of Human Resource Development

Model Fusion: Least Squares Approach Using the Least Squares technique, calculate the desired combined value of by minimizing the corresponding sum of weighted squared differences Joint Annual Meeting NSF Division of Human Resource Development

Example Joint Annual Meeting NSF Division of Human Resource Development +

Conclusion and Future Work A fast practical alternative to joint inversion of multiple datasets was presented Future work planned is to apply this algorithm with a actual gravity and seismic datasets (Summer ‘10) Joint Annual Meeting NSF Division of Human Resource Development

Further Questions Omar Ochoa – Vladik Kreinovich – Aaron Velasco – Rodrigo Romero – Joint Annual Meeting NSF Division of Human Resource Development