Introduction
Echosounders provide data on 1) the time delay between transmission and reception of the echo and 2) the intensity of the returning echo (echo level). The time delay indicates the range or depth to the target. The echo level is the measure that is translated to abundance of fish in the water column.
The echo level depends on the intensity of the transmitted sound wave (the source level), the loss in intensity as the sound wave spreads in the water and is absorbed by water (transmission loss), the reflectivity of the target (TS), the position of the target in the beam, and various losses in the instrument associated with converting sound pressure to an electric voltage. These processes are combined in the SONAR (SOund NAvigation and Ranging) equation. It is necessary to know the different terms in this equation to correctly relate the measured echo level to fish density.
General Equation
The SONAR equation relates returned energy to transmitted energy through the generalized form (in logarithmic units):


[1] 
The full SONAR equation has energy transmission, loss, and reflection components, but also accounts for the position of the insonified target relative to the acoustic beam (B(θ)). The equation is simple, as most of the complex underwater physics are hidden in the TS and B(θ) terms, and takes the form:


[2] 
Which, by combining terms, is equivalent to:


[3] 
where:
EL is the returning echo intensity (dB)
SL is the transmitted sound intensity (dB)
TS is the TS of the fish (dB)
B(θ) is the beam directivity (intensity of the sound at angle θ )
TL is the transmission loss (dB)
The TL depends on spreading and absorption. TL in one direction is:


[4] 
and in two directions is twice that, or:


[5] 
where:
R is the range to the target (m)
a is the absorption loss (dB•m^{1})
Substituting these into Equation (3), the logarithmic version of the SONAR equation becomes:


[6] 
This equation can also be written as:


[7] 
where:
I_{EL} is the intensity of the returning sound
I_{SL} is the intensity of the transmitted sound
σ_{bs} is the backtransformation of TS (m^{2})
b is the directivity of the transducer in the direction (θ) of the target on a linear scale
is the backtransformation of the TL.
Beam width or 3dB angle
The beam width or 3 dB angle are equivalent terms that refer to the angle between the half intensity direction on either side of the main lobe, measured in degrees. If the transducer beam is elliptical, both minor (athwart) and major (along) values are different. Values for the 3 dB angle are usually provided by the manufacturer and may be obtained during calibration. Some authors use halfbeam angle, which is the angle from the center of the beam to the 3 dB intensity direction. The halfbeam angle is half of the full beam angle.
The equivalent beam angle
The equivalent beam angle (ψ in steradians, EBA in dB) is also known as the reverberation angle. This value represents the angle at the apex of the ideal transducer beam (a transducer with beam directivity of 1 within and 0 outside the beam) that gives the same S_{v} values as the actual transducer, including side lobes (Simmonds and MacLennan 2005). Equivalent beam angle is defined as:


[8] 
where:
θ and Φ are spherical polar coordinates used to determine the direction of a point (P) relative to the origin (O) of the transducer.
θ is the angle of OP from the acoustic axis
Φ is the azimuthal angle of OP projected onto the plane of the transducer face, and;
b is the beam pattern, defined in terms of intensity.
The entire beam patterns is used in the integration, from θ=0 to π and from Φ=0 to 2π.
Values for ψ are supplied by some manufacturers and can be modified in postprocessing software. Some manufacturers report equivalent beam angle in steradians while others use the dB format. The conversion between the two forms is:


[9] 
Values of ψ or EBA can be calculated for a circular transducer with Equation 15.
Volume backscattering
The sampling volume is related to the volume of a shell in the shape of a half sphere in front of the transducer with a thickness of (cτ/2). All fish within this volume contribute to the measured echo level. The contribution of fish is not uniform in all directions; therefore we integrate the beam directivity over the whole halfsphere (Equation 8) to obtain the equivalent beam angle (ψ or EBA), a two way process with the unit steradians (sr) or dB. This defines a sampling volume of the ideal beam. The sampling volume (also called the reverberation volume) increases with range squared (R^{2}):


[10] 
The number of fish contributing to the volume backscattering at any one instant in time is equal to fish density multiplied by the sampling volume [ρ V]. The total volume backscattering is this number of fish multiplied by their average backscattering cross section (σ_{bs}).
Therefore, the volume backscattering coefficient (s_{v}) is defined as fish density (ρ) multiplied with the average backscattering cross section (σ_{bs}).


[11] 
and the total echo intensity I_{EL} is proportional to [s_{v} V], which is [s_{v} ψ R2 (cτ/2)].
Equation 6 and 11 can be combined to calculate the echo level from an ensemble of fish (after dB transformations):
Combining terms yields:


[12] 
In these equations, S_{v} is s_{v} in dB units (S_{v} = 10 log_{10}(s_{v})). The same equation may be written in backtransformed units for the volume backscattering coefficient (s_{v}):


[13] 
All terms in these equations are known from manufacturersupplied transducer specifications or calibrations except s_{v} (S_{v}), which can therefore be calculated. These values are what we are interested in to translate acoustic data into fish density. Note that an addition or subtraction using logarithmic units is equivalent to a multiplication or division in untransformed units.