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Model Fusion and its Use in Earth Sciences

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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

- 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

- 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

- 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

- 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

- Designing such a joint inversion technique presents an important theoretical and practical challenge

Joint Annual Meeting NSF Division of Human Resource Development

- 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

- each estimation error is normally distributed with 0 mean and known standard deviation
- Least squares: find that minimizes

solution:

Joint Annual Meeting NSF Division of Human Resource Development

- 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

- 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

- What is given:
- High spatial resolution estimates
of the values in several small cells

- Low spatial resolution estimates for the weighted averages

- High spatial resolution estimates

Joint Annual Meeting NSF Division of Human Resource Development

- 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

- 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

- 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

- 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

- 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

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Joint Annual Meeting NSF Division of Human Resource Development

- 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

- Omar Ochoa – [email protected]
- VladikKreinovich – [email protected]
- Aaron Velasco – [email protected]
- Rodrigo Romero – [email protected]

Joint Annual Meeting NSF Division of Human Resource Development