# How to incorporate audiogram functions into weighted sound exposure levels based on different hearing groups?

Not all marine mammals have the same hearing abilities, therefore National Marine Fisheries (2020) recommends using audiogram-weighting for sound exposure levels (SELs) based on different marine mammal hearing groups. They also provide weighting functions for different hearing groups. Normally I calculate SEL as an integral in the time domain, so how do I incorporate the weighting function if I want to calculate a weighted SEL?

For a practical implementation, including MatLab code, see Tougaard and Beedholm (2019). This code will provide frequency-specific coefficients for the different weighting curves, which can be either as fractions (between 0 and 1) to be multiplied with the power spectrum density in linear units (µPa-squared) or as dB factors to add to the power spectrum, if in dB re 1µPa.

Tougaard, J. and Beedholm, K. 2019. Practical implementation of auditory time and frequency weighting in marine bioacoustics. Applied Acoustics 145:137-143.

To apply the frequency-specific audiogram weighting, you must calculate SEL in the spectral domain. Parseval's theorem tells us that the sum of a square of a function is equal to the sum of the square of it's transform. That means that the integrated spectrum will be equal to the integrated time series. You can test this by comparing the integral from your time series to the integral of a spectrum; they should be the same.

So you will need to

1. Define appropriate time and frequency bounds for your signal
2. Calculate the power spectral density (Note - applying a window function here may not be appropriate for impulsive signals!)
3. Apply the frequency specific weighting function
4. Integrate the power spectral density. Usually my PSDs are in dB values, but I convert the PSD into linear values, then integrate, then convert back to dB. Make sure you account for FFT size in integral!

Example code for high-frequency weighted Sound Exposure Level (SELwHF): SELwHF<-10*log10(sum(10^(PSDhf/10))/(Nfft * sr))

where: Nfft = number of samples in your signal, sr = sample rate, PSDhf = power spectral density in dB units after weighting function is applied

PAMGuard has a "Filtered Noise Measurement" module which can apply an arbitrary FIR filter to sound data prior to measuring the RMS and SEL outputs of that filter at selected time intervals. If you design a filter which matches the audiogram of the species you're interested in, this module may provide a solution.

• Wow - I didn't know this was a thing. Thanks Doug! Jul 6, 2022 at 18:52