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## Table 1

Names, symbols, and definitions of group metrics and the various indices, intermediary equations, and scattering quantities used to calculate them.

Symbol Name (and unit) Definition or Algorithm
L Length (m) Maximum distance between all points in ƍ2D
W Width (m) Greatest horizontal distance perpendicular to group length in the ƟL+90° direction
H Height (m) zmax(i) − zmin(i)
R Roundness L/W
V Volume (m3) ƞ3D
Group metrics SA Surface Area (m2) for all k3D
d Density (fish/m3) d(BL) Density (fish/BL3) d/(BL3)
D Depth (m) DB 1Distance above Bottom (m) Bottom depth − D
IHh Horizontal Internal Heterogeneity (m) /L
IHv Vertical Internal Heterogeneity (m) R Range from the ship (m) HD Horizontal distance from the ship (m) n   Total number of measurements within a group
i   Each measurement within a group
Indices k   Number of indices defining ƍ
a   One of 2 measurements in ƍ2D used to define L
b   One of 2 measurements in ƍ2D used to define L
p   Total number of pings
f   Number of triangle facets
v1,v2,v3   The vertices of each triangle facet

x,y Horizontal position (m,m) Universal Transverse Mercator (UTM) coordinate position of the Sv measurement
Positional Z Depth position (m) Depth of the Sv measurement
Intermediaries sx,sy Ship position (m,m) UTM coordinate position of the vessel
Sx, Sy Mean ship position (m,m) Cx, Cy, Cz Density dependent center of gravity (m,m,m) ƍ Convex hull ƍ2D 2D convex hull The smallest 2D convex polygon that contains every one of the n data points (x 1,y 1….xn ,yn ) within the group
Geometrical Intermediaries ƍ3D 3D convex hull The smallest 3D convex volume that contains every one of the n data points (x 1,y 1,z 1….xn ,yn ,zn ) within the group
εi   Convex hull coefficient/ weight such that the coefficients are non-negative and sum to one
ƟL   arctan (|xa – xb | / |ya  – yb |)
f   Number of triangle facets
v1,v2,v3   The vertices of each triangle facet

sv 2Volume backscattering coefficient (m−1) 10 * Svc/10
Scattering quantities FL 3Fish length (m) Mean for k length
BL The number of body lengths per meter (m−1) 1/FL
TS 4Target strength (dB re 1 m2) TS = 20 * log(FL) − 66
σ bs 2Backscattering cross section (m2) 10 * TS/10

1Bottom depth was taken from the nearest (<15 nmi) haul location.

3Based on the population length composition from the nearest (<15 nmi) haul location.

4TS to FLcm model for walleye pollock from Foote and Traynor (1988).

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