The sonar equation is used in underwater signal processing to relate received signal power to transmitted signal power for one-way or two-way sound propagation. The equation computes the received signal-to-noise ratio (SNR) from the transmitted signal level, taking into account transmission loss, noise level, sensor directivity, and target strength. The sonar equation serves the same purpose in sonar as the radar equation does in radar. The sonar equation has different forms for passive sonar and active sonar.

In a passive sonar system, sound propagates directly from a source to a receiver. The passive sonar equation is

$$SNR=SL-TL-(NL-DI)$$

where SNR is the received signal-to-noise ratio in dB.

The source level (*SL*) is the ratio of the transmitted intensity from the
source to a reference intensity, converted to dB:

$$SL=10\mathrm{log}\frac{{I}_{\text{s}}}{{I}_{\text{ref}}^{}}$$

where *I*_{s} is the
intensity of the transmitted signal measured at 1 m distance from the source.
The reference intensity, *I*_{ref}, is the
intensity of a sound wave having a root mean square (rms) pressure of 1 μPa.
Source level is sometimes written in dB// 1 μPa, but actually is referenced to
the intensity of a 1 μPa signal. The relation between intensity and pressure is

$$I=\frac{{p}_{\text{rms}}^{2}}{\rho c}$$

where *ρ* is the density of seawater,
(approximately 1000 kg/m^{3}), *c* is
the speed of sound (approximately 1500 m/s). 1 μPa is equivalent to an intensity
of *I*_{ref} = 6.667 ✕
10^{-19} W/m^{2}

Sometimes, it is useful to compute the source level from the transmitted power,
*P*. Assuming a nondirectional (isotropic) source, the
intensity at one meter from the source is

$$I=\frac{P}{4\pi}$$

Then, the source level as a function of transmitted power is

$$SL=10{\mathrm{log}}_{10}\frac{I}{{I}_{\text{ref}}}=10{\mathrm{log}}_{10}\frac{P}{4\pi {I}_{\text{ref}}}=10{\mathrm{log}}_{10}P-10{\mathrm{log}}_{10}4\pi {I}_{\text{ref}}=10{\mathrm{log}}_{10}P+170.8$$

When source level is defined at one yard instead of one meter, the final constant in this equation is 171.5.

When the source is directional, the source level becomes

$$SL=10{\mathrm{log}}_{10}\frac{I}{{I}_{\text{ref}}}=10{\mathrm{log}}_{10}P+170.8+D{I}_{\text{src}}$$

where *DI*_{src} is the
directivity of the source. Source directivity is not explicitly included in the
sonar equation.

The sonar equation includes the directivity index of the receiver
(*DI*). Directivity is the ratio of the total noise power
at the array to the noise received by the array along its main response axis.
Directivity improves the signal-to-noise ratio by reducing the total noise. See
Element and Array Radiation and Response Patterns for discussions of directivity.

Transmission loss is the attenuation of sound intensity as the sound propagates through the
underwater channel. Transmission loss (*TL*) is defined as
the ratio of sound intensity at 1 m from a source to the sound intensity at
distance *R*.

$$TL=10\mathrm{log}\frac{{I}_{\text{s}}}{I(R)}$$

There are two major contributions to transmission loss. The larger contribution is geometrical spreading of the sound wavefront. The second contribution is absorption of the sound as it propagates. There are several absorption mechanisms.

In an infinite medium, the wavefront expands spherically with distance, and attenuation
follows a *1/R ^{2}* law, where

$$T{L}_{\text{geom}}=20{\mathrm{log}}_{10}R$$

For ranges greater than the transition depth, the geometric transmission loss is

$$T{L}_{\text{geom}}=10{\mathrm{log}}_{10}R+10{\mathrm{log}}_{10}{R}_{\text{trans}}$$

In Phased Array
System Toolbox, the transition range is one-half the channel depth,
*H/2*.

The absorption loss model has three components: viscous absorption, the boric acid relaxation
process, and the magnesium sulfate relaxation process. All absorption components
are modeled by linear dependence on range, *αR*.

Viscous absorption describes the loss of intensity due to molecular motion being converted to
heat. Viscous absorption applies primarily to higher frequencies. The viscous
absorption coefficient is a function of frequency, *f*,
temperature in Celsius, *T*, and depth, *D*:

$${\alpha}_{\text{vis}}=4.9\times {10}^{-4}{f}^{2}{e}^{-(T/27+D/17)}$$

in dB/km. This is the dominant absorption mechanism above 1 MHz. Viscous absorption increases with temperature and depth.

The second mechanism for absorption is the relaxation process of boric acid. Absorption
depends upon the frequency in kHz, f, the salinity in parts per thousand (ppt),
*S*, and temperature in Celsius,T. The absorption
coefficient (measured in dB/km) is

$$\begin{array}{l}{\alpha}_{\text{B}}=0.106\frac{{f}_{1}{f}^{2}}{{f}_{1}^{2}+{f}^{2}}{e}^{-(pH-8)/0.56}\\ {f}_{1}=0.78\sqrt{S/35}{e}^{T/26}\end{array}$$

in dB/km. *f _{1}* is the
relaxation frequency of boric acid and is about 1.1 kHz at

The third mechanism is the relaxation process of magnesium sulfate. Here, the absorption coefficient is

$$\begin{array}{l}{\alpha}_{\text{M}}=0.52\left(1+\frac{T}{43}\right)\left(\frac{S}{35}\right)\frac{{f}_{2}{f}^{2}}{{f}_{2}^{2}+{f}^{2}}{e}^{-D/6}\\ {f}_{2}=42{e}^{T/17}\end{array}$$

in dB/km. *f _{2}* is the
relaxation frequency of magnesium sulfate and is about 75.6 kHz at T = 10°C and
S = 35 ppt.

The total transmission loss modeled in the toolbox is

$$TL=T{L}_{\text{geom}}(R)+({\alpha}_{\text{vis}}+{\alpha}_{\text{B}}+{\alpha}_{\text{M}})R$$

where *R* is the range in km. In Phased Array System Toolbox, all absorption models parameters
are fixed at *T = 10*, *S = 35*, and
*pH = 8*. The model is implemented in `range2tl`

. Because *TL* is a monotonically
increasing function of *R*, you can use the Newton-Raphson
method to solve for *R* in terms of *TL*. This
calculation is performed in `tl2range`

.

Noise level (*NL*) is the ratio of the noise intensity at the receiver to
the same reference intensity used for source level.

The active sonar equation describes a scenario where sound is transmitted from a source,
reflects off a target, and returns to a receiver. When the receiver is collocated
with the source, this sonar system is called *monostatic*.
Otherwise, it is *bistatic*. Phased Array
System Toolbox models monostatic sonar systems. The active sonar equation is

$$SNR=SL-2TL-(NL-DI)+TS$$

where *2TL* is the two-way transmission loss (in
dB) and *TS* is the target strength (in dB). The transmission loss
is calculated by computing the outbound and inbound transmission losses (in dB) and
adding them. In this toolbox, two-way transmission loss is twice the one-way
transmission loss.

Target strength is the sonar analog of radar cross section. Target strength is the ratio of
the intensity of a reflected signal at 1 m from a target to the incident
intensity, converted to dB. Using the conservation of energy or, equivalently,
power, the incident power on a target equals the reflected power. The incident
power is the incident signal intensity multiplied by an effective
cross-sectional area, *σ*. The reflected power is the reflected
signal intensity multiplied by the area of a sphere of radius
*R* centered on the target. The ratio of the reflected
power to the incident power is

$$\begin{array}{c}{I}_{\text{inc}}\sigma ={I}_{\text{refl}}4\pi {R}^{2}\\ \frac{{I}_{\text{refl}}}{{I}_{\text{inc}}}=\frac{\sigma}{4\pi {R}^{2}}.\end{array}$$

The reflected intensity is evaluated on a sphere of 1 m
radius. The target strength coefficient (*σ*) is referenced to
an area 1 m^{2}.

$$TS=10{\mathrm{log}}_{10}\frac{{I}_{\text{refl}}(\text{1meter})}{{I}_{\text{inc}}}=10{\mathrm{log}}_{10}\frac{\sigma}{4\pi}$$

[1] Ainslie M. A. and J.G. McColm. "A simplified formula for viscous and
chemical absorption in sea water." *Journal of the Acoustical Society of
America*. Vol. 103, Number 3, 1998, pp. 1671--1672.

[2] Urick, Robert J. *Principles of Underwater Sound*, 3rd
ed. Los Altos, CA: Peninsula Publishing, 1983.