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Because a primary task of multi-sensor systems is to make estimates based on the data collected and shared throughout the sensor system, it is important to design data compression methods that reduce the volume of data to be shared, while causing only a minimal degradation of the quality of these estimates. An important aspect, not specifically considered previously in compression research for sensor systems, is that sensor systems generally have to make multiple estimates from the data. Furthermore, it is unrecognized in the literature that these multiple estimates generally have conflicting compression requirements and that finding the right way to balance these conflicts is crucial. The key tools we bring to bear on this area are: 1) the use of a Fisher-information-based distortion measure that is designed specifically for multiple estimates, and 2) the use of numerical optimization methods to achieve desired compression trade-offs among the multiple estimates. We first develop results that support using the trace of the Fisher information matrix (FIM) as a distortion measure for the simultaneous multiple-parameter estimation problem. We then apply these results to the problem of time-difference-of-arrival (TDOA) and-frequency-difference-of-arrival- (FDOA) based location estimation of an RF emitter. We show that within this problem there is a fundamental trade-off between the impact of compression on TDOA estimation and the impact of compression on FDOA estimation; furthermore, we show that this trade-off can be addressed by adapting the compression scheme to the sensor-emitter geometry using a geometry-adaptive compression scheme.