Library / Sensors & detection / SEN-02
Sensors & detection
Passive sonar equation
Signal excess of a passive sonar: how far the received signal sits above the detection threshold.
- Ref
- SEN-02
- Rev
- v1
- Field
- Sensors & detection
- Result
- SE [dB]
with r in metres in the logarithm and α in dB per metre. The calculator takes r in km and α in dB/km.
Take the level the target radiates, subtract what is lost on the way to you, subtract the background noise, add the gain of your array, then compare with what you need to detect. If what is left over is positive, the target can be detected.
Calculator
- Transmission loss TL81 dB
- SNR at array output14 dB
- Spreading loss share98.77 %
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- Signal excess above zero means the threshold is cleared. To find the detection range, choose Solve for r and set SE to 0 dB.
- With spherical spreading each doubling of range costs 6 dB, so a 6 dB improvement in any term roughly doubles detection range. Absorption makes the real gain a little smaller.
- Source level, noise level, directivity and threshold all trade dB for dB. A quieter target and a bigger array are interchangeable in the equation.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (dB) | Local step | Change in result |
|---|---|---|---|---|---|---|
| SL Source level | 110 to 190 dB re 1 µPa @ 1 m | −36 | 44 | 80 | +1 dB re 1 µPa @ 1 m | +1 dB |
| r Range to target | 0.5 to 100 km | −25 | 30.97 | 55.97 | +10% | −0.928 dB |
| α Absorption coefficient | 0.01 to 5 dB/km | −45 | 4.9 | 49.9 | +10% | −0.1 dB |
| NL Noise level in the band | 55 to 95 dB re 1 µPa | −16 | 24 | 40 | +1 dB re 1 µPa | −1 dB |
| DI Array directivity index | 0 to 35 dB | −16 | 19 | 35 | +1 dB | +1 dB |
| DT Detection threshold | 0 to 20 dB | −6 | 14 | 20 | +1 dB | −1 dB |
Sweep of Range to target
| Range to target (km) | Signal excess (dB) |
|---|---|
| 0.5 | 30.97 |
| 0.9696 | 25.17 |
| 1.88 | 19.33 |
| 3.646 | 13.4 |
| 7.071 | 7.303 |
| 13.71 | 0.8865 |
| 26.59 | −6.154 |
| 51.57 | −14.4 |
| 100 | −25 |
Sweep one variable
Variables and how each one contributes
Signal excess
What the equation gives you. Everything else feeds it.
Source level
Sound level radiated by the target, referred to 1 metre from it. Adds dB for dB. +6 dB of source level roughly doubles detection range under spherical spreading.
Range to target
Distance from target to array. Spreading loss is 20 log10 r: every doubling of range costs about 6 dB, plus absorption that grows linearly with range.
Absorption coefficient
Loss to absorption in seawater, which rises steeply with frequency. Negligible at low frequency and at short range, decisive at high frequency and long range. Roughly 0.07 dB/km near 1 kHz (Thorp).
Noise level in the band
Ambient and self noise at the array in the processing bandwidth. Subtracts dB for dB. Sea state, shipping and the platform’s own noise all feed in.
Array directivity index
Noise rejection gained by the array’s directional response. A larger array adds dB for dB, like a louder target.
Detection threshold
Signal-to-noise ratio needed for the chosen detection and false-alarm probabilities. Raising the required confidence raises DT and eats directly into signal excess.
Worked example with the values above
- Spreading loss: 20 log10(10,000 m) = 80 dB.
- Absorption: α·r = 0.1 × 10 = 1 dB. So TL = 80 + 1 = 81 dB.
- SE = SL − TL − NL + DI − DT = 150 − 81 − 75 + 20 − 10 = 4 dB.
- Signal excess is positive, so the target is detectable at this range.
Assumptions and limits
- Passive sonar: a one-way path from the target to the array.
- Spherical spreading plus linear absorption. In shallow water or ducted conditions spreading is closer to cylindrical (10 log10 r) beyond a transition range, so this overstates loss at long range.
- One frequency band, with a single value each for noise, directivity and absorption.
- Levels in dB re 1 µPa; source level referred to 1 m.
Common mistakes
- Mixing references: source level is dB re 1 µPa at 1 m, noise level is dB re 1 µPa in the band.
- Entering range in km in the logarithm. The spreading term needs metres (the calculator converts for you).
- Using a noise spectrum level without adding 10 log10(bandwidth) to get the level in the band.
- Confusing signal excess with signal-to-noise ratio. SE already has the detection threshold subtracted.
Statistics and uncertainty
The detection threshold is statistical. DT is the signal-to-noise ratio needed for a chosen probability of detection and false-alarm rate, and both ocean noise and propagation fluctuate. At SE = 0 dB the target is detected with that design probability, not with certainty, so operators add margin for variability in transmission loss.
Where it is used
- Estimating the detection range of a passive array against a ship or submarine of known source level.
- Seeing how much a quieter target, a bigger array or a higher threshold changes detection range.
- Trading off array size (DI), processing (DT) and platform noise (NL).
Maths you need
- Decibels: every term is a level, so they add and subtract
- Logarithms: 20 log10 of range gives 6 dB per doubling
- Units and references: µPa reference level; km to m conversion