Likelihood maximisation techniques for ranging gunfire over grassland

A study into acoustic parameter inversion in the presence of a non-moving, homogeneous atmosphere and grassland impedance ground is carried out using methods of likelihood maximisation. Measured frequency-dependent sound pressure level and power spectra for a blank firing pistol are used to generate simulated data with added Gaussian error to represent variations usually present in real life experiments. Inference is carried out using maximum likelihood estimation (MLE) and maximum a priori (MAP) where model parameters are either given as known or restricted to some uncertain distribution bounded by realistic conditions. The quality of inference is assessed visually and statistically as the error between the true and inferred predictions for a given propagation range. Application of a prior (MAP) greatly improves inference accuracy compared to the sole maximisation of the likelihood function (MLE). It is shown that the use of a single octave band frequency window does not improve the quality of inference, whereas combinations of several low frequency octave bands do. Exact quantification of the true values of the ground and source height are seemingly less important as range increases beyond 500m. Although the techniques presented in this paper are for military/security applications, they are readily applicable to other acoustical problems, e.g. source characterisation in engineering noise control. The methods adopted are likely to benefit from higher-dimensional models, i.e. inhomogeneous atmospheres, complex terrain or urban environments.