Approximating Ocean Acoustic Fields with Finite Basis Function Series for Autonomous Vehicle Applications

Algorithms that enable acoustic sensing vehicles to autonomously map acoustic fields are being developed, but some require an analytical representation of the data collected from the acoustic field with specific continuity properties. One optimal control technique that has these requirements and has been used in autonomous acoustic sensing algorithms is Pontryagin’s Maximum Principle (PMP). Given that most real-world fields are modeled by numerical methods, the need to create continuous analytical representations from data is a hurdle in realistic applications of these autonomous algorithms. In this work, finite series of basis functions are used to solve this problem. The basis functions are selected to meet the criteria of the PMP, and are used to approximate the field from the data collected. While this approach meets the criteria of the algorithm, it can be a computationally expensive approach for complex fields and large datasets. This paper looks at the trade off between accuracy and computational time, for different frequencies and levels of field complexity. A previously published ocean profile developed from data taken off the island of Elba is used in the normal mode software, Kraken, to generate the acoustic fields used to test this method. The candidate basis functions considered are trigonometric functions analogous to finite Fourier series, and Legendre polynomials. The Legendre polynomials are shown to have higher accuracy when used with a dynamically selected exclusion range at low frequencies and a cubic spline interpolation method for generating intermediate data points at higher frequencies. Run times are calculated and while both candidate basis functions are shown to meet the requirements of the PMP algorithm, the Legendre polynomials require the least amount of run time.