Power-budget and echo-integrator equations for fish abundance estimation

For abundance estimation and species identification on fisheries acoustic surveys, control with temperature and finite amplitude effects in the sea demands the calibration factor C of the conventional echo-integrator equation to be fully known in terms of the sonar system’s electrical and acoustical parameters. As no such expression is available from earlier literature, the theoretical basis for the power budget and echo-integrator equations is revisited. The objective is to provide (a) a derivation of these equations for integration in a more complete functional relationship for abundance measurement, including an expression for the calibration factor C; (b) to account for electrical termination; (c) to formulate these expressions in terms of voltage signal echo integration processing; and (d) thereby generalize the Clay-Medwin formulations based on echo integration of “in-water” sound pressure signals, to account for the transducer and electronics components of the echosounder system. Under conditions of small-amplitude (linear) sound propagation, electroacoustic power budget equations are derived for the backscattering cross section in single-target backscattering, used in echosounder calibration; and the volume backscattering coefficient for multiple-target backscattering, used in oceanic surveys. On this basis a more complete echo-integrator equation is derived for two operational cases, “short-ping-and-long-gate”, and “long-ping-and-short-gate”. The results are consistent with and extend previous work in this area