Development of an Uncertainty Propagation Equation for Scalar Fields

ABSTRACT

The uncertainty of a scalar field is essential structuring information for any estimation problem. Establishing the uncertainty in a dense gridded product from sparse or random uncertainty-attributed input data is not, however, routine. This manuscript develops an equation that propagates the uncertainty of individual observations, arbitrarily distributed in , to a common estimation location at which they can be used to determine the composite uncertainty of the output field. The equation includes the effect of the distance between the observation and estimation locations, the field and horizontal uncertainty of the observation, and user-parameters to control the expected variability in the field as a function of distance. Two computational versions of the equation, a lower cost conservative approach and a higher cost mean-distance approach, are developed and evaluated for computational cost and resulting accuracy in numerical experiments over simulated bathymetric data. The mean-distance approach is more accurate, but more costly; suitable numerical approximations are proposed to control computational costs. A benefit of the work described is flexibility and enhancement for applications of the model, such as the Combined Uncertainty and Bathymetry Estimator algorithm, which is used as a demonstration of the difference between the two versions of the equation.

KEYWORDS:

• DEM
• mapping
• hydrographic survey
• ocean mapping
• bathymetry
• multibeam sonar

Acknowledgements

We are grateful to one of the anonymous reviewers for bringing to our attention the potential for (11(11) ) as an alternative implementation method.

Funding

This work was supported under National Oceanic and Atmospheric Administration grant NA15NOS4000200, and by the Office of Naval Research under the Naval Research Lab Base Program, both of which are gratefully acknowledged.

Notes

¹ Capture distance 5.0% for depth, IHO S.44 5ed Order 1A survey for maximum observation propagation distance, median pre-filter queue of one sample, estimate offset threshold of 4.0, Bayes factor threshold of 0.135, sequential monitoring threshold of five samples, and prior (most observations assimilated) hypothesis selection.

² Satellite-Based Augmentation System, a system for providing GNSS correctors by satellite transponder, rather than having a local correction system.

³ Global Navigation Satellite System, or colloquially “GPS” (Global Positioning System), most often the U.S.-based NavStar.

⁴ Apple iMac with Intel Core i7 at 4 GHz, 32 GB RAM, running OS X 10.11.6, running unoptimized code from XCode 7.3.1.