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Sensors & detection
Active sonar equation
Signal excess in decibels for an active sonar.
- Ref
- SEN-08
- Rev
- v1
- Field
- Sensors & detection
- Result
- SE [dB]
SL is the source level of the sonar, TL the one-way transmission loss, TS the target strength, NL the noise level at the receiver and DI the directivity index of the receiving array, all in decibels. The echo travels out and back, so TL counts twice. SE is the margin by which the echo exceeds the noise at the detection threshold; here the detection threshold is taken as 0 dB, otherwise subtract it as well.
Signal excess in decibels for an active sonar. A positive value means the echo should be detectable.
Calculator
- Echo level at the receiver SL − 2TL + TS115 dB
- One-way loss that can be added before SE reaches zero35 dB
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.
- The defaults leave a 70 dB margin, so the target could be about 35 dB further away in one-way loss, which is many times the range.
- A change of 1 dB in the one-way loss changes the margin by 2 dB, so active sonar range is far more sensitive to the water than a one-way (passive) system.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (dB) | Local step | Change in result |
|---|---|---|---|---|---|---|
| TL One-way transmission loss | 0 to 120 dB | −50 | 190 | 240 | +10% | −12 dB |
| SL Source level | 150 to 260 dB | 0 | 110 | 110 | +10% | +22 dB |
| NL Noise level | 20 to 120 dB | 15 | 115 | 100 | +10% | −6.5 dB |
| TS Target strength | −20 to 50 dB | 35 | 105 | 70 | +1.4 dB | +1.4 dB |
| DI Directivity index | 0 to 40 dB | 50 | 90 | 40 | +10% | +2 dB |
Sweep of One-way transmission loss
| One-way transmission loss (dB) | Signal excess (dB) |
|---|---|
| 0 | 190 |
| 15 | 160 |
| 30 | 130 |
| 45 | 100 |
| 60 | 70 |
| 75 | 40 |
| 90 | 10 |
| 105 | −20 |
| 120 | −50 |
Sweep one variable
Variables and how each one contributes
Signal excess
What the equation gives you. Everything else feeds it.
Source level
The source level: the sound level produced by the transmitter, in dB. One-for-one: each extra decibel of source level adds a decibel to the signal excess.
One-way transmission loss
The one-way transmission loss between sonar and target, in dB. Counts twice: each extra decibel of one-way loss costs two decibels of signal excess, which is why range is so expensive.
Target strength
The target strength: how strongly the target reflects, in dB. One-for-one: a larger or more reflective target adds decibels directly.
Noise level
The background noise level at the receiver, in dB. Minus one-for-one: noisier water, a noisy platform or a high sea state removes decibels.
Directivity index
The directivity index of the receiving array, in dB. One-for-one: a larger array focuses on the target and rejects noise from other directions.
Worked example with the values above
- Out-and-back loss: 2 × TL = 2 × 60 = 120 dB.
- Echo level: SL − 2TL + TS = 220 − 120 + 15 = 115 dB.
- SE = 115 − 65 + 20 = 70 dB, positive, so the echo should be detectable.
Assumptions and limits
- A monostatic sonar: the same path loss in both directions.
- Noise-limited detection (not limited by reverberation), with the detection threshold taken as 0 dB.
Common mistakes
- Counting TL once instead of twice.
- Adding the decibel values of noise and signal as if they were linear powers.
Statistics and uncertainty
All terms are in dB, so independent errors add in quadrature, with TL weighted by 2. With ±3 dB on each of the five terms the margin is uncertain by about ±8.5 dB. Target strength in particular varies by 10 dB or more with the aspect angle, so a single value is an average.
Where it is used
- Deciding whether a sonar will detect a target at a given range.
- Comparing the margin of different sonar settings or platforms.
Maths you need
- Decibels: adding decibels multiplies powers
- Sonar terms: SL, TL, TS, NL, DI