Volume_integration
Let A be the set of samples s in the analysis domain for which Volume_integration is being calculated.
Then the volume of any one sample in A is defined as follows (see also About integration algorithms):
for Single beam algorithm 

for Multibeam cruise scanning algorithm 

for V and S mode cruise scanning algorithms 

for H mode cruise scanning algorithm 

for Targetlocked scanning algorithm and V and H mode instrument scanning algorithms 
Where:
Vs = Volume of sample s (m3) = Unit distance along the cruise track = 1 (m) = Unit crosssectional area across the cruise track = 1 (m2) Δxp = Distance along the cruise track for ping p (m) = Vgp·Δtp where Vgp is available, and otherwise Where:
Vgp = Vessel speed over ground at ping p (m/s)
Δtp = Pulse repetition interval for ping p (s)
Φp = Bearing angle of the ping p (radians).
Note: This bearing angle is measured clockwise when viewed from above with 0 bearing in the direction of travel (direction of Vgp). The measure of bearing is however instrument (and mode) specific and translations are performed where required.
Δθb = Angular width of beam b (radians) = θ / (Nb)
Δθbv = Angular width of beam b in the Vmode ping of a targetlocked pair (radians) = θ / (Nb)
Δθbh = Angular width of beam b in the Hmode ping of a targetlocked pair (radians) = θ / (Nb)
Where:
θ = Full sector angle of the respective ping (radians)
Nb = Number of beams in full sector of the respective ping ()
rs = Range of sample s (m) = rminp + (ns+0.5)·Δrs
Where:
rminp = Minimum range for ping p (m)
ns = zero based sample number measured from transducer ()
Δrs = Thickness of sample s in the range dimension (m) = (rmaxp– rminp)/ (Nsp)Where:
rmaxp = Maximum range for ping p (m)
rminp = Minimum range for ping p (m)
Nsp = Number of samples in ping p
Τp = Tilt angle of the ping p (radians), read from the data file for H and S mode and 0 in Vmode.
Τs = The effective Tilt angle of a sample s (radians), defined as follows:
if s is on an Hmode ping then Τs = Τp
if s is on a Vmode ping then Τs is calculated as (b + 0.5) x ΔθbvWhere:
b = the number of the Vmode ping on which sample s lies ()
Δθbv = as defined above (radians)
Note: The Single beam algorithm is used for integration of single beam data, and the remaining algorithms for integrating multibeam data. See About integration algorithms for more information on the various algorithms.
Then the integration volume is determined as:
for Targetlocked scanning algorithm 

for all other algorithms 
Where:
V= the integration volume, Volume_integration (m3) = 0 if sample s is not in the analysis domain 1 otherwise, including below threshold data. Vs= Volume of sample s (m3)  as defined above δ= the set of samples in the analysis domain. Β= the set of beams in the analysis domain and on the Hmode ping of a targetlocked pair. β= the set of samples in the analysis domain and on the beam b of the Hmode ping of a target locked pair. ρ= the set of samples in the analysis domain and on the beam i of the Hmode ping of a target locked pair,
where i is the number of the reference beam,
which is that beam which the Vmode ping of the targetlocked pair intersects. ν= the set of samples in the analysis domain and on the Vmode ping of a targetlocked pair. nβ= the number of samples in the set β nρ= the number of samples in the set ρ
Notes
 Volume_integration is NOT calculated whenever is used. The sample volume, Vs may still be calculated for use in the calculation of Sv_mean but any calculation of Vs by using is not considered useful for the purposes of volume or biomass estimation. If Volume_integration is not calculated for this reason, a special value will be provided instead. The following variables depend upon Volume_integration and cannot be calculated if Volume_integration is not calculated:
See also
Density_number
Density_weight
Fish_number
Fish_weight
N_sigma_bs