four sub-sections the relevant sonar equation is derived, and illustrated by means of a worked example.
3.2 PASSIVE SONAR
3.2 Passive sonar 57 Sec. 3.2]
Figure 3.2. Spectral density level of the radiated power at the source (upper) and intensity at the receiver (lower).
3.2.2 Definition of standard terms (passive sonar)
3.2.2.1 Mean square pressure, sound pressure level, and the decibel
The concept of mean square (acoustic) pressure (abbreviated MSP) is one of key importance to sonar performance modeling and plays a central role in this book. For acoustic pressureqðtÞand averaging timeT, the mean square pressureQis defined as
Q1 T
ðT 0
q2ðtÞdt: ð3:1Þ
The result of Equation (3.1) (MSP) is independent of T for large T if qðtÞ is statistically stationary in time.
Another important parameter is the acoustic intensityI, a vector quantity whose magnitudeI, for a plane-propagating wave, is equal to the MSP (Q) divided by the characteristic impedance of the medium
I jIj ¼Q
c (plane wave). ð3:2Þ
For types of wave other than a plane wave, the relationship between intensity and MSP is a more complicated one, but one can define theequivalent plane wave intensity (EPWI) of any statistically stationary pressure field as the intensity of a plane wave of the same MSP. In other words, denoting this quantityIEPWI,
IEPWIQ
c (any pressure field). ð3:3Þ
In some publications the term ‘‘intensity’’ is used as a synonym for EPWI, and the sonar equation is expressed as a product of ‘‘intensity’’ ratios. In the following, a deliberate choice is made to characterize sound waves by their MSP and not EPWI, as this avoids ambiguities associated with the choice of impedance, and is consistent with the definition of propagation loss in common use (see Appendix B). The distinction becomes an important one if the impedance at the source is different from that at the receiver.
Except in some special situations, neither MSP nor EPWI are proportional to the true acoustic intensity. The sonar equations derived in this chapter and in Chapter 11 are all expressed in terms of MSP ratios and do not rely on any particular relationship between sound pressure and intensity.
Sound pressure levelis the mean square pressure expressed indecibels. The decibel is a logarithmic unit of power or energy (see Appendix B for details). The conversion to decibels involves the following three operations: divide by a standard reference value of the parameter, take the base-10 logarithm, and multiply by 10. Thus, the sound pressure level (SPL) is:
SPL10 log10 Q
p2ref: ð3:4Þ
The reference value of MSP isp2ref and the internationally accepted value ofpref for use in underwater acoustics is one micropascal (1mPa).
Consider a point source at the origin. Regardless of the acoustic environment and source–receiver geometry, the mean square pressureQat an arbitrary locationxcan always be written in the form
QðxÞ ¼p20s20FðxÞ; ð3:5Þ whereFðxÞis the propagation factor, which isdefinedby Equation (3.5); andp0 is the RMS pressure at a small distances0 from the source.7
It is conventional to write Equation (3.5) in logarithmic form 10 log10 Q
p2ref¼10 log p20s20 p2refr2ref
!
þ10 log10 F
r2ref; ð3:6Þ whererref is a constant reference distance, with an internationally accepted value of 1 meter (1 m). The left-hand side of Equation (3.6) is the sound pressure level.
It is conventional to present all sonar equation terms in decibels (dB). Using the above recipe, a power-like quantityxis converted to its dB equivalentLxby dividing by its reference unitxref and applying the relationship
Lx10 log10 x
xref: ð3:7Þ
For this reason the combination 10 log10 appears repeatedly. Often, for notational convenience, the denominator of the argument is omitted, in which case the reference value is quoted instead as a qualifier to the dB unit. For example, the meaning of
X10 log10x dB re xref ð3:8Þ
is identical to that of Equation (3.7). Thus, the power level of a 1-watt source is 10 log10(1 W/1 pW)¼120 dB re pW. Similarly, a sinusoidal pressure wave of ampli- tude 1 Pa has a mean square pressure of 0.5 Pa2, and consequently a sound pressure level of 10 log10[0.5 Pa2/(1mPa)2]¼117.0 dB remPa2. The choice of mPa2 as a refer- ence unit for SPL, adopted here in preference to the more conventionalmPa, follows naturally from Equation (3.8) (or Equation 3.7) with x equal to the mean square pressure.
In general, a level quoted in decibels needs to be accompanied by a statement of the corresponding reference unit, even if an international standard exists for its value.
This is partly because standards can, and do, change with time and circumstances and partly because not all users of decibels adhere to these standards.8 The only safe 3.2 Passive sonar 59 Sec. 3.2]
7Equation (3.5) holds for a point source. The distance s0 from the source at which p0 is measured must be small enough for distortions due to absorption, refraction, reflection, or diffraction to be negligible.
8In water, reference pressures of 20mPa and 1mbar were both used before the modern value of 1mPa became widespread. At the time of writing, a reference pressure of 1mbar (and a reference distance of 1 yd) is still in use by at least one sonar manufacturer. Further, the reference pressure used for sound in air is 20mPa, not 1mPa, making it unclear which value to use in situations involving a mixture of air and water, such as in foam caused by breaking waves.
exception to this general rule occurs with dimensionless quantities, for which it seems reasonable to assume a reference unit of 1.
3.2.2.2 Source level
On the right-hand side of Equation (3.6) there are two terms. The first, a measure of source power, is known as the source level
SL10 log10S0 dB re mPa2m2; ð3:9Þ whereS0 is the product
S0¼p20s20: ð3:10Þ It is remarkable that a parameter of such fundamental importance to sonar asS0does not have a widely accepted name. The term source factor is adopted here.9It is more conventional to define source level as the sound pressure level at a standard reference distance (rref) from the source (ASA, 1994; IEC, www). For a point source in free space, the numerical value is the same provided that rref is the unit distance in whatever units system is used (i.e., 1 meter in the SI system), but the conventional definition leads to difficulties for an extended source such as a ship or an array of sonar projectors (see Chapter 11). Becausep0varies with distance in such a way that p0s0 is constant, the source factor is also a constant, independent of measurement position close to the source, making it a natural physical quantity with which to characterize the source. Although source level is defined in terms of pressurep0, due to the s0 scaling (through Equation 3.10), in practice it is actually a measure of radiated power. Thus, an alternative expression for the source factor is
S0¼cWO; ð3:11Þ
whereWOis the radiant intensity (power per unit solid angle).
For an omni-directional source of power W, the source factor is equal to cW=4. For example, if the source power is 1 watt (W¼1 W), then from Equation (3.9),S0¼0.122 kPa2m2, corresponding to a source level of 170.9 dB remPa2m2.10 3.2.2.3 Propagation loss
The second term on the right-hand side of Equation (3.6) is (minus) thepropagation loss11
PLSLSPL¼ 10 log10FðxÞ dB re m2; ð3:12Þ
9The combinationp0s0, which is the square root of the source factor, is referred to by ASA (1989) as the ‘‘source product’’.
10For the assumed conditions (T¼10C,S¼35, at atmospheric pressure).
11In underwater acoustics a synonymous term to ‘‘propagation loss’’ is ‘‘transmission loss’’.
The term ‘‘propagation loss’’ is used here to avoid possible confusion with alternative definitions of ‘‘transmission loss’’ from other branches of acoustics such as sound transmission through a wall (Morfey, 2001).
or equivalently,
PL¼10 log10S0
Q dB re m2: ð3:13Þ
This term plays the role of transfer function between source and receiver. The ratio S0=Qhas dimensions of area, so the unit of propagation loss is dB re m2.
3.2.2.4 Noise spectrum level and array response
The mean square pressure of background noise within a specified bandwidth (usually the processing bandwidth of the sonar receiver) is denoted QN. Thus, the SPL of background noise in the same bandwidth, known as thenoise level, is
NL10 log10QN dB re mPa2: ð3:14Þ This background noise is often broadband in nature, so it is useful to consider noise spectral density (denotedQNf ) and the corresponding noise spectral density level (or noise spectrum level) NLf, defined as12
NLf 10 log10QNf dB remPa2Hz1: ð3:15Þ The background against which the signal is to be detected is that at the output of the beamformer (i.e., the array response, denoted YN), not at the hydrophone. The spectral density of this quantity (denotedYNf ) can be written as the integral of the noise spectral density over all solid anglesO, weighted by the beam patternBðOÞ:
YNf ¼ ð
QNfOðOÞBðOÞdO: ð3:16Þ For the special case of isotropic noise, meaning that the magnitude of noise spectral density is independent of direction such that
QNfO¼QNf
4 ; ð3:17Þ
it follows that
YNf ¼QNf O
4; ð3:18Þ
whereOis the solid angle footprint of the beam pattern (in steradians), defined by O¼
ð
BðOÞdO: ð3:19Þ
3.2 Passive sonar 61 Sec. 3.2]
12The use of the subscript f for logarithmic quantities (expressed in decibels) indicates a spectrum level (i.e., a spectral density expressed in decibels).
3.2.2.5 Signal-to-noise ratio, array gain, and directivity index
Nowconsider a monochromatic signal13whose MSP per unit solid angle isQSO. By analogy with Equation (3.16), the array response to this signal (denotedYS) is
YS¼ ð
QSOBðOÞdO: ð3:20Þ
Assuming further that this signal is in the form of an incoming plane wave from the directionOS¼ ðS; SÞ, such that
QSO¼QSðOOSÞ; ð3:21Þ whereðxÞis the Dirac delta function (see Appendix A), the array response is then
YS¼ ð
QSðOOSÞBðOÞdO¼QSBðOSÞ: ð3:22Þ The value of the signal-to-noise ratio (SNR) depends on where it is measured in the processing chain. In particular, its value after beamforming (denotedRarr) is different from its value at the hydrophone, before any processing (Rhp). The ratio of these two SNR values, expressed in decibels, is thearray gain; that is,
AG10 log10GA dB; ð3:23Þ
where
GA¼Rarr
Rhp: ð3:24Þ
Here, the hydrophone SNR (Rhp) is defined as the ratio of signal mean square pressure (MSP) to noise MSP at the receiving hydrophone
RhpQS
QN; ð3:25Þ
whereQN is the noise MSP integrated over the sonar processing bandwidth QN¼
ð
QNf df: ð3:26Þ
SimilarlyRarris the SNR at the output of the beamformer, for a ‘‘flat response filter’’
(i.e., one whose response is independent of frequency within a specified passband14 and zero everywhere else)
RarrYS
YN: ð3:27Þ
A related parameter is the directivity index (DI) of an array, which is the array gain for the special case of a plane wave signal and isotropic noise. Unlike AG, DI is a property of the array and the acoustic frequency only. Thus, it is independent of
13The single frequency assumption is relaxed in Section 3.2.4.4.
14Unless otherwise stated, this passband is understood to be the processing bandwidth (denotedf orDf depending on context).
medium and target properties and is usually easier to calculate. For this reason, DI is often used as an approximation to AG in the sonar equation.
3.2.2.6 Signal gain and noise gain
The array gain can be expressed in terms of the signal gain SG:
SG10 log10YS
QS dB ð3:28Þ
and noise gain NG:
NG10 log10YN
QN dB; ð3:29Þ
such that
AG¼SGNG: ð3:30Þ
According to these definitions, SG and NG are both negative. For a well-designed beamformer, the magnitude of NG is usually greater than that of SG, in which case AG is positive.
3.2.2.7 Detection threshold and signal excess
The SNR thresholdR50is the value ofRarr(more generally, that of the SNR after all processing) required to achieve a detection probability of precisely 50 %.15(The value ofR50depends on statistical fluctuations present in both signal and noise, and on the false alarm probability). When expressed in decibels, this quantity is known as the detection threshold
DT¼10 log10R50 dB: ð3:31Þ
Unlike the amplitude threshold AT (in volts, or pascals) introduced in Chapter 2, which is the value of the signalþnoise amplitude in volts, or pascals (or energy thresholdETin V2s or Pa2s) above which a ‘‘signal present’’ decision is triggered, the detection threshold (R50) is a dimensionless parameter. The signal excess is defined as the amount by which the SNR exceeds the detection threshold, in decibels:
SE10 log10RarrDT: ð3:32Þ
It follows from this and from the definition of array gain that
SE¼10 log10RhpþAGDT: ð3:33Þ
Equation (3.33) is thesonar equation. Written like this it looks simple, and in some special circumstances it is. To a large extent, the purpose of this book is to explain howto calculate each of the terms on the right-hand side and corresponding ones for active sonar (see Section 3.3). The examples and special cases considered in the remainder of Chapter 3 are deliberately simplified in order to illustrate the main principles involved.
3.2 Passive sonar 63 Sec. 3.2]
15The symbolX50is used to denote the value of any variableXrequired to achieve a detection probability of 50 %.
3.2.3 Coherent processing: narrowband passive sonar
This section is concerned with the calculation of the probability of detection (pd) for a narrowband passive sonar. The term ‘‘narrowband’’ (abbreviated ‘‘NB’’) implies that the signal may be described, to a first approximation, by sound of a single frequency (see Figure 3.3). The processing considers a very narrowrange of frequencies at a time, thus minimizing the noise in each processing band. The bandwidth is then assumed to be large enough to contain the entire signal, but sufficiently small for the noise power to be directly proportional to the bandwidth. Under these circum- stances the signal-to-noise ratio, and hence also the detection probability, increase with decreasing bandwidth. A narrowband signal is called a tonal because of its resemblance to a single frequency tone in music. The following sub-sections look first at the sonar signal, then the background noise, signal-to-noise ratio, and finally the probabilities of detection and false alarm. A special case involving a horizontal line array is introduced in Section 3.2.3.7, followed by a worked example for this special case (Section 3.2.3.8).
3.2.3.1 Signal (single hydrophone)
For a NB system the sonar signal is the received mean square pressureQSassociated with one of the transmitted tonals (red line in Figure 3.3). With the assumption of an infinite water depth, there are two contributions to the received signal, one from the direct path and one from a surface reflection. We assume that the sea surface is smooth, so that the direct and surface-reflected paths add coherently. If the tonal power isWS it follows from Chapter 2 that
QS¼cWS
4 FNBðr;zarr;ztgtÞ; ð3:34Þ where zarr is array depth;
ztgtis target depth;
ris horizontal separation between array and target; and FNBis the coherent propagation factor.
Assuming that the horizontal separation r is large compared with the product zarrztgt, where is the attenuation coefficient, this can be written:
FNB¼Fcohðr;zarr;ztgtÞ 4
r2e2 rsin2 kzarrztgt r
: ð3:35Þ
The sine-squared behavior in Equation (3.35) is a result of alternate constructive and destructive interference between the two paths, illustrated in Figure 3.416 for a frequency of 300 Hz. The separation between successive peaks (or troughs) is r=kzarr in depth andr2=kzarrztgt in range. This fringe pattern is known as aLloyd mirror interference pattern after an analogous effect from optics. For the example
16This graph and many subsequent ones, as acknowledged in the individual captions, are calculated using the sonar performance model INSIGHT (Ainslieet al., 1996).
3.2 Passive sonar 65 Sec. 3.2]
Figure 3.3. Spectral density level of the transmitter source factor (upper) and mean square pressure at the receiver (lower).
shown (the array depth is 30 m), at a range of 300 m the fringe spacing in depth is approximately 25 m.
3.2.3.2 Noise (single hydrophone)
We consider a broadband noise spectrum illustrated by Figure 3.5. Conceptually, the total noise entering the processing bandwidthf is the area under the curve between the dashed lines; that is,
QNðzÞ ¼fQNf ðzÞ; ð3:36Þ whereQNf is the spectral density of the ambient noise MSP. For the isovelocity case, with infinite water depth, this can be written (see Chapter 2)
QNf ðzÞ 3cE3ð2 zÞWNAf; ð3:37Þ whereWNAf is the power spectral density of the noise source per unit of sea surface area, and E3ðxÞ is a third-order exponential integral (see Appendix B). Here, the processing bandwidth is the analysis bandwidth of the Fourier transform, equal to the reciprocal of the coherent processing time.
Figure 3.4. Coherent propagation loss PL¼ 10 log10ðFcohÞ[dB re m2]vs.rangerand target depthztgt for array depthzarr¼30 m and frequencyf ¼300 Hz (INSIGHT).
3.2.3.3 Signal-to-noise ratio, signal excess, and narrowband passive sonar equation To derive the NB sonar equation we start by calculating the signal-to-noise ratio for the array (rearranging Equation 3.24 for YS=YN and substituting the result in Equation 3.27) as
Rarrðr;zarr;ztgtÞ YS YN¼QS
QNGA: ð3:38Þ Recall that the ratio QS=QN is the SNR measured at a hydrophone, before any processing apart from initial filtering into the frequency bandf (see Figure 3.6).
It follows from Equations (3.34), (3.36), and (3.11) that
Rarrðr;zarr;ztgtÞ ¼S0FNBðr;zarr;ztgtÞ QNf ðzarrÞ GA
f : ð3:39Þ
Recall from Section 3.2.2.7 that the thresholdR50 is the SNR required, after spatial and temporal filtering, to achieve a detection probability of 50 %. The sonar equation is obtained by dividing Equation (3.39) through byR50, and converting to decibels in the usual way. Specifically, the left-hand side becomes the signal excess (SE), defined 3.2 Passive sonar 67 Sec. 3.2]
Figure 3.5. Spectral density level of background noise. The noise termQNis the contribution in the processing bandwidth, markedf.
as the ratio by which the SNR exceedsR50, expressed in decibels SENB10 log10Rarr
R50: ð3:40Þ
Thus,
SENBðr;zarr;ztgtÞ ¼ ½SLPLðr;zarr;ztgtÞ ½NLfðzarrÞ ðAGBWÞ DT; ð3:41Þ where
SL¼10 log10S0 ðsource level: dB re mPa2m2Þ; ð3:42Þ PLðr;zarr;ztgtÞ ¼ 10 log10FNBðr;zarr;ztgtÞ ðpropagation loss: dB re m2Þ; ð3:43Þ NLfðzarrÞ 10 log10QNf ðzarrÞ ðnoise spectrum level: dB remPa2=HzÞ; ð3:44Þ
AG10 log10GA ðarray gain: dBÞ; ð3:45Þ
DT¼10 log10R50 ðdetection threshold: dBÞ; ð3:46Þ and
BW¼10 log10f ðbandwidth: dB re HzÞ: ð3:47Þ Equation (3.41) is the NB passive sonar equation in logarithmic form. The processing Figure 3.6. Spectral density level of signal 10 log10QSf (red) and noise 10 log10QNf (cyan).
bandwidth term BW, originating from the denominator of Equation (3.39), is grouped for convenience with the array gain.17
The figure of merit (FOM) is defined as the propagation loss at which the detection probability is 50 % (i.e., FOMPL50). From this definition it follows that FOMNBðzarrÞ ¼SLþ ðAGBWÞ NLfðzarrÞ DT ð3:48Þ and the sonar equation is then
SENBðr;zarr;ztgtÞ ¼FOMNBðzarrÞ PLðr;zarr;ztgtÞ: ð3:49Þ
3.2.3.4 Array gain and directivity index for a horizontal line array
The array gain for a horizontal line array (neglecting the signal gain) is given by GA¼ QN
ð
QNOBðOÞdO
; ð3:50Þ
where (if the hydrophone spacing is small compared with the acoustic wavelength) BðOÞ ¼sin2u
u2 ð3:51Þ
and, expressing the direction in terms of the spherical co-ordinates(elevation) and (bearing)
u¼uð; Þ ¼kDxcossin
2 ; ð3:52Þ
whereDxis the array length. Given that for a sheet dipole source,QNOis proportional to sin(see Chapter 2), and using dO¼cosdd, it follows that
GA¼
ð=2 0
dsincos ð2
0
d ð=2
0
dsincos ð2
0
d sinuð; Þ uð; Þ
2: ð3:53Þ
The numerator of Equation (3.53) is, and its denominator can be simplified by defining the horizontal beamwidthDðÞas
DðÞ ð2
0
d sinuð; Þ uð; Þ
2
; ð3:54Þ
3.2 Passive sonar 69 Sec. 3.2]
17An alternative convention is to groupf instead with the narrowband detection threshold so that DTNB would be given instead as 10 log10ðfR50Þ, with implied units dB re Hz. The convention of Equation (3.41) is preferred because it makes it possible to standardize on a single definition of DT (as 10 log10R50) for both coherent and incoherent processing, as well as for both passive and active sonar.