Acoustics basics

Backscattering cross-section and target strength

General definitions

For a given scatterer, the backscattering cross-section (\(\sigma_\text{bs}\), m²) is defined by [MacLennan, 2002]:

\[ \sigma_\text{bs} = \frac{r^2 I_\text{bs}(r) 10^{\alpha r/10}}{I_\text{inc}}, \tag{1.1} \]

where \(I\) is the intensity of the incident (\(\text{inc}\)) and backscattered (\(\text{bs}\)) waves, \(r\) is the range of the target (m), and \(\alpha\) is the acoustic absorption coefficient (dB m^-1). This quantity can be generalized using the isotropic, or spherical scattering, cross-section:

\[ \sigma_\text{sp} = 4 \pi \sigma_\text{bs}, \tag{1.2} \]

where no directional dependence is assumed. For an aggregation comprising \(n\) animals, the mean backscattering cross-section is:

\[ \left< \sigma_\text{bs} \right> = \frac{ \sum\limits_{j=1}^N \sigma_{\text{bs},j} } { n }, \tag{1.3} \]

where \(\sigma_{\text{bs},j}\) is the differential backscattering cross-section of animal \(j\).

This quantity can often be expressed logarithmically via target strength (\(\textit{TS}\), dB re. 1 m²):

\[ TS = 10 \log_{10} \sigma_\text{bs}, \tag{1.4} \]

which is the commonly used representation of individual backscattering cross-sections in the fisheries acoustics literature.

\(\textit{TS}\)-length relationship

Values of \(\sigma_{\text{bs},j}\) often varies as a function of the \(j^\text{th}\) animal’s body length, \(L_j\):

\[ \sigma_{bs,j} = \sigma_{bs,j}(L_j). \tag{1.5} \]

Consequently, one common approach to estimating fish \(\textit{TS}\) uses the empirical relationship between \(\textit{TS}\) and \(L\):

\[ \textit{TS} = m\log_{10}L+b, \tag{1.6} \]

where \(L\) is the total length, \(m\) is the slope, and \(b\) is the \(y\)-intercept. Values of \(m\) and \(b\) are typically species-specific; however, these relationships are often “standardized” by fixing \(m = 20\) since many early experiments yielded \(m\)-values close to 20 [Simmonds and MacLennan, 2005]. This modifies Eq. (1.6) to:

\[ \textit{TS} = 20\log_{10}L+b_{20}, \tag{1.7} \]

where \(b_{20}\) is the \(y\)-intercept associated with the fixed slope. Species-specific values for \(b_{20}\) varies and are frequency-specific (Table 1).

Table 1 Example species-specific \(\textit{TS}\)-\(L\) relationships.

Species	Frequency (kHz)	\(m\)	\(b\)	\(b_{20}\)	Ref.	Note
Clupea harengus	38	8.9	-55.2	-69.5	Foote et al. [1986]	Daytime
C. harengus	38	21.2	-74.2	-72.5	Foote et al. [1986]	Nighttime
Gadus morhua	38			-58.8	Nakken and Olsen [1977]
Gadoids	38	18.0	-66.2	-68.0	Foote [1987]
Merluccius productus	38			-68.0	Traynor [1996]
Pollachius virens	38			-65.8	Foote [1987]
Sprattus sprattus	38	17.2	-60.8		Nakken and Olsen [1977]
S. sprattus	120	21.4	-55.0		Nakken and Olsen [1977]

Converting backscatter into population estimates

Methods for converting (integrated) acoustic backscatter into units of population often requires empirically measuring in situ \(\textit{TS}\) or using models such as the regression coefficients in Table (1). First, the volume backscattering coefficient, \(s_\text{v}\) (m^-1), is computed:

\[ s_\text{v} = \frac{ \sum\limits_{j=1}^N \sigma_{\text{bs},j} } { V }, \tag{1.8} \]

where \(V\) is the integration volume (m³). Similar to the relationship between \(\sigma_\text{bs}\) and \(\textit{TS}\), \(s_\text{v}\) can be expressed logarithmically via the volume backscattering strength:

\[ S_\text{v} = 10 \log_{10} s_\text{v}. \tag{1.9} \]

When \(s_\text{v}\) is averaged over a finite volume of water, \(S_\text{v}\) is often described as being the mean volume backscattering strength (\(\textit{MVBS}\)) [MacLennan, 2002].

Alternatively, \(s_\text{v}\) can also be expressed as a function of \(\left< \sigma_\text{bs} \right>\) and the volumetric animal density (\(\rho_\text{v}\), animals m^-3):

\[ s_\text{v} = \rho_\text{v} \left< \sigma_\text{bs} \right>. \tag{1.10} \]

In fisheries acoustics, backscatter is often integrated over the entire water column to provide estimates per unit area instead of volume. This means that \(s_\text{v}\) can be expressed as the area backscattering coefficient (m²m^-2):

\[ s_\text{a} = \int\limits_{z_1}^{z_2} s_\text{v} dz = s_\text{v} H, \tag{1.11} \]

where \(z\) is a finite depth bounded by depths \(z_1\) and \(z_2\), otherwise known as the integration height \(H\). In practice, this vertical integration occurs over horizontal interval \(x\), which is commonly referred to as an elementary distance sampling unit (EDSU). This effectively breaks up continuous acoustic backscatter measurements into discrete samples (e.g. 1 nmi EDSU) that allows for easier processing and post-survey analyses (e.g. biomass estimation). This changes Eq. (1.11) to:

\[ s_\text{a}(x) = \int\limits_{x_1}^{x_2} \int\limits_{z_1}^{z_2} s_\text{v}(x, z)~ dz~ dx = s_\text{v} H(x), \tag{1.12} \]

where \(x_1\) and \(x_2\) are the along-transect distances at the start and end of the EDSU. With \(s_\text{v}(x)\) integrated over the whole water column, Eq. (1.10) can be modified to compute the areal number density (\(\rho_\text{a}(x)\), animals m²):

\[ \rho_\text{a}(x) = \frac{s_\text{a}(x)}{\left< \sigma_\text{bs} \right>}. \tag{1.13} \]

It is more common to express \(\rho_\text{a}\) relative to nmi² than m² or km². This first requires converting \(s_\text{a}\) into the natucal area scattering coefficient (\(s_\text{A}\), or commonly abbreviated as \(\textit{NASC}\)) [Foote and Knudsen, 1994]:

\[ s_\text{A}(x) = 4 \pi (1852)^2 s_\text{a}(x). \tag{1.14} \]

Once converted, \(\rho_\text{a}\) from Eq. (1.13) can be defined as:

\[ \rho_\text{A} = \frac{s_\text{A}}{4 \pi \left< \sigma_\text{bs} \right>} = \frac{s_\text{A}}{\sigma_\text{sp}} \tag{1.15} \]

where \(\rho_\text{A}(x)\) is in units of animals nmi^-2.