# Sonar Equation Example: Passive Sonar

Acoustic Thermometry of Ocean Climate (ATOC): Signal-to-Noise Ratio

The Acoustic Thermometry of Ocean Climate (ATOC) project used long-range, low-frequency sound transmissions to measure large-scale ocean temperature variability. Acoustic sources on Pioneer Seamount off central California and north of Kauai, Hawaii, transmitted to receivers several thousand kilometers distant. Sound speed increases with increasing temperature, and the measured travel times therefore depend on the average temperatures between the sources and receivers. In the following example the signal-to-noise ratio is computed for a transmission from the Pioneer Seamount source to a receiver located 3000 km away near Hawaii.

The passive sonar equation is:

SNR (decibels) = SL -TL – (NL – AG)

Where SNR is signal-to-noise ratio, SL is the source level, TL is the transmission loss, NL is the noise level, and AG is the array gain. The acoustic sources used for the ATOC project had source levels of:

SL = 195 dB re 1 μPa at 1 m

The array gain is zero for a simple, small hydrophone, as were used on the moored receivers during the ATOC project:

AG = 0 dB

(The SOSUS receivers that were also used during the ATOC project are horizontal arrays and do have substantial array gain.)

The transmission loss TL includes both sound spreading loss and attenuation. At long ranges a combination of spherical and cylindrical spreading provides more accurate estimates of spreading loss than either one alone. Assuming spherical spreading to a range of 5 km, corresponding roughly to the water depth, followed by cylindrical spreading for the remaining distance to a range of 3000 km gives a spreading loss of:

Spherical spreading loss to 5 km = 20 log (5000) = 74 dB

Cylindrical spreading loss from 5 km to 3000 km = 10 log (3000000/5000) = 28 dB

Total spreading loss = 74 + 28 = 102 dB

The attenuation due to sound absorption calculated using the absorption coefficient α (alpha) of about 0.00042 dB/km at a frequency of 75 Hz in the North Pacific Ocean is:

Attenuation (dB) = αR = (0.00042 dB/km x 3000 km) = 1.3 dB
The overall transmission loss is then:

TL = 103 dB

The ambient noise level at 75 Hz is predominantly due to shipping. Near Hawaii the noise level in a 1 Hz wide band at 75 Hz is roughly:

NL = 75 dB re 1 μPa/√Hz

The ATOC receivers operated over frequency bands greater than 1 Hertz wide. The frequency band over which the receiver operates is called the bandwidth BW, given in Hertz. The total noise is:

NLtotal (dB re 1 μPa) = NL + 10 log BW

The relationship between the pulse length and receiver bandwidth for simple signals is roughly:

BW (Hz) = 1 / T, where T is the pulse length in seconds.

The ATOC sources transmitted complex signals that had an effective pulse length of only 0.0267 seconds, giving a bandwidth BW of about 37.5 Hz, even though the ATOC transmissions lasted for 20 minutes. The total noise in the ATOC receiver bandwidth is then:

NLtotal = NL + 10 log BW = 75 + 10 log (37.5) = 91 dB re 1 μPa

Combining all of these values the signal-to-noise ratio at the receiver is:

SNR (decibels) = SL – TL – (NL – AG) = 195 – 103 – (91 – 0) = 1 dB

The SNR is essentially zero, which means that the received signal level is about the same as the background noise level. Sophisticated signal processing methods were used during the ATOC project to increase the SNR by roughly 46 dB, giving useable signal-to-noise ratios, after processing. Without this processing the signal would be undetectable.