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Predicting outdoor sound propagation in uncertain conditions remains a challenge. This increases the complexity of the inverse problem, e.g. parameter recovery in the presence of a particular sound source such as like gunfire. This paper investigates the use of maximum likelihood methods, both frequentist and Bayesian, in inverting true parameters from measured and simulated data. A simple source-receiver acoustic model is used which assumes; a homogeneous atmosphere, soft impedance ground and some medium range sound propagation to predict the deviation in sound pressure at the receiver. A blank firing pistol, Bruni Mod 92, is used to record a realistic sound source spectrum in an anechoic chamber. Gaussian noise is added to model predictions for this type of source to mimic uncertainty of real-life observations. Error analysis is performed by repeatedly generating observations and then evaluating the errors between the true range and recovered range estimate. This analysis is performed in broadband and octave frequency bands. It was found that the frequentist method greatly underestimates the range while the Bayesian method, even with a particularly flat prior, greatly reduces both over-and underestimations, significantly improving the range estimate to within ±5 of the true value in the majority of cases. The inclusion of octave band filters in the infrasonic frequency showed these bands were mostly responsible for the accurate range estimates. This paper paves the way for applications of this class of statistical models to real-life acoustic data for source parameter recovery.

Predictions of outdoor sound propagation in uncertain conditions is a challenging task. Evidence suggest that using more than one receiver can reduce this effect of uncertainties. This paper studies via numerical simulations the effects of uncertainty in the source/receiver geometry and impedance ground condition on the sound pressure ratio recorded using the two-microphone method. A Monte Carlo method is employed study the effect of uncertainties in the range and ground parameters. The range and frequency are found to be key parameters which control the resultant probability density function for the absolute sound pressure ratio and phase difference. The introduction of small uncertainty only matters if the uncertainty is present in the distance between the source and receiver. Uncertainties in the impedance ground are found to have a negligible effect. The sound pressure ratio is affected by the uncertainty more strongly at a shorter range. These findings pave the way to the development of more robust methods for outdoor acoustic source localisation and identification from two-microphone data.

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 500 . 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.