Neglect of bandwidth of Odontocetes echo location clicks biases propagation loss and single hydrophone population estimates Michael A. Ainsliea) Netherlands Organisation for Applied Scientific Research, P.O. Box 96864, 2509 JG, The Hague, Netherlands
(Received 5 March 2013; revised 14 July 2013; accepted 10 September 2013) Passive acoustic monitoring with a single hydrophone has been suggested as a cost-effective method to monitor population density of echolocating marine mammals, by estimating the distance at which the hydrophone is able to intercept the echolocation clicks and distinguish these from the background. To avoid a bias in the estimated population density, this method relies on an unbiased estimate of the detection range and therefore of the propagation loss (PL). When applying this method, it is common practice to estimate PL at the center frequency of a broadband echolocation click and to assume this narrowband PL applies also to the broadband click. For a typical situation this narrowband approximation overestimates PL, underestimates the detection range and consequently overestimates the population density by an amount that for fixed center frequency increases C 2013 Acoustical Society of America. with increasing pulse bandwidth and sonar figure of merit. V [http://dx.doi.org/10.1121/1.4823804] PACS number(s): 43.30.Sf, 43.30.Cq, 43.80.Ka [JIA]
I. INTRODUCTION
The echolocation clicks of marine mammals can be detected and classified using specially designed passive sonar equipment.1–3 Such detectors and classifiers are proposed as cost-effective tools for estimating population density4 using a single hydrophone5–8 or a hydrophone network.9 For a stationary hydrophone, the rate at which clicks are detected is proportional to the animals’ average vocalization rate, the population density (assumed locally uniform) and the detection area, which is proportional to the square of the detection range. It follows that the population density can be estimated from knowledge of the vocalization probability and detection range. The success of this method relies on obtaining unbiased estimates of both click rate and detection range, the latter of which is the subject of this paper. The usual procedure for estimating detection range is to substitute propagation loss (PL) into the passive sonar equation and solve this equation for the distance at which the signal excess is zero (by definition the detection range).10 Application of this procedure requires an estimate of source level, propagation loss, noise level and detection threshold. A bias in one or more of these terms risks introducing a bias in the population density estimate by either under- or over-estimating the detection probability. Even though typical echolocation clicks are broadband pulses, it is customary to treat them as narrowband pulses for the purpose of estimating PL.6,8 Using the expression for broadband PL derived in Ref. 10, Ref. 11 shows that the narrowband approximation applied by Refs. 6 and 8 results in an error in PL of ca. 17 dB at 5 km for Mesoplodon for the special case of spherical spreading propagation, using a center frequency of 38 kHz and a 10 dB bandwidth of 25 kHz. The present work shows that this error in PL applies to any
a)
Also at: Institute of Sound and Vibration Research, University of Southampton, Highfield, Southampton SO17 1BJ, United Kingdom.
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Pages: 3506–3512
propagation law, suggesting a possible bias in detection range (and hence in the derived population density). The purpose is to explore possible implications of this error for the accuracy of the estimated population density. Similar errors in population estimates of humpback whales from their 150–1800 Hz vocalizations are described in Ref. 12. II. EQUATIONS FOR BROADBAND PROPAGATION LOSS
Propagation loss (PL) can be defined for a narrowband source (indicated by the subscript “NB” for “narrowband”) as the difference between source level (SL) and sound pressure level (SPL), i.e.,10 PLNB SL SPL:
(1)
It is convenient to introduce linear equivalents of these logarithmic quantities. The linear quantity corresponding to SPL is the mean square value of the instantaneous sound pressure p, i.e., hp2 i ¼ 10SPL=10 pref 2 ;
(2)
where pref is the standard reference pressure used to define SPL, usually equal to 1 lPa. Define the source factor S0 and steady state propagation factor F as10 S0 10SL=10 pref 2 rref 2
(3)
FNB 10PLNB =10 rref 2 ;
(4)
and
where rref is the standard reference distance for SL and PL, usually equal to 1 m. Using Eqs. (2), (3), and (4), Eq. (1) can be written in linear form
0001-4966/2013/134(5)/3506/7/$30.00
C 2013 Acoustical Society of America V
FNB ¼
hp2 i : S0
(5)
A narrowband source necessarily has a long duration, not less that the reciprocal of the bandwidth. Concern here is with short transient sounds, which necessarily have a high bandwidth, not less than the reciprocal of their duration. For a broadband transient, Eq. (1) would be replaced by the difference between energy source level10 (SLE) and sound exposure level (SEL), leading to the following alternative PL definition, applicable to transient sounds and indicated by the subscript “BB” for a broadband transient PLBB SLE SEL:
(6)
Linear equivalents to SEL and SLE are the sound exposure E [cf. Eq. (2)], E ¼ 10SEL=10 pref 2 tref ;
(7)
Substitution of Eqs. (11) and (12) in Eq. (10) then results in the equation ð1 ðSE Þf expf2½aðfm Þ aðf Þrgdf ð1 FBB ¼ FNB 0 ; (15) ðSE Þf df 0
where FNB is introduced as shorthand for F(r,fm). Now assume a top-hat source factor spectrum with a uniform spectral density r for frequencies between fm B/2 and fm þ B/2 and zero elsewhere, ðSE Þf fm B=2 fm þ B=2 ¼H 1 H 1 ; r f f (16) where H(x) is the Heaviside step function, such that FBB 1 ¼ FNB B
and energy source factor SE [cf. Eq. (3)], SE 10SLE =10 pref 2 rref 2 tref ;
(8)
where tref is the reference time used for SEL and SLE, usually equal to 1 s. Defining the broadband propagation factor FBB for transients [cf. Eq. (4)], FBB 10PLBB =10 rref 2 ;
(9)
where PLBB is defined by Eq. (6), the propagation factor follows from Eqs. (7) and (8) as FBB ¼
E : SE
(10)
We can write the broadband quantities E and SE as integrals over the respective spectral densities of the exposure (Ef) and energy source factor, (SE)f, such that ð1 Ef ðr; f Þdf (11) EðrÞ ¼
FBB FNB :
0
ðSE Þf df ;
(12)
(13)
and assume that F(r,f) depends on the absorption coefficient a(f), in units of nepers per unit distance, in the following way: Fðr; f Þ ¼ Fðr; fm Þexpf2½aðfm Þ aðf Þrg: J. Acoust. Soc. Am., Vol. 134, No. 5, November 2013
(18)
1 aðf Þ ¼ aðfm Þ þ ðf fm Þaf ðfm Þ þ ðf fm Þ2 af f ðfm Þ þ : 2 (19)
where the arguments r and f denote the range and frequency, respectively, and the subscript f denotes a spectral density. Now consider the following equation for the exposure spectral density [cf. Eq. (10)], Ef ðr; f Þ ¼ ðSE Þf Fðr; f Þ
(17)
Equation (18), valid for a narrowband transient, has been used to model the propagation loss for echolocation pulses of marine mammals such as the killer whale,13 beaked whales,6,8 and finless porpoise.7 Some of these echolocation pulses have a bandwidth comparable to their center frequency. Is this narrowband approximation justified for these broadband echolocation pulses? To answer this question, consider the effect of increasing B from zero in Eq. (17). In general, the absorption a(f) is a function of frequency and needs to be substituted in Eq. (18) before evaluating the integral. This can be achieved by expanding the absorption coefficient as a Taylor series about f ¼ fm, i.e.,
and ð1
expf2½aðfm Þ aðf Þrgdf :
fm B=2
If the bandwidth is sufficiently small, the integrand may be approximated as a constant, equal to its value at the center frequency, in which case
0
SE ¼
ð fm þB=2
(14)
Use of the constant term on its own gives the narrowband approximation, Eq. (18). Including the linear term as well gives FBB sinhx ; FNB x
(20)
where x ¼ r=ra ðfm Þ
(21)
ra ðfm Þ ¼ 1=½af ðfm ÞB
(22)
and
Michael A. Ainslie: Neglecting bandwidth biases propagation loss
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FIG. 1. (Color online) Graph of sinh(x)/x (solid curve); the asymptotic value for large x (dotted curve) is exp(x)/(2x).
is the distance at which the argument of the sinh(x)/x function in Eq. (20) is equal to unity, meaning that at ranges exceeding ra, the broadband propagation factor differs significantly from the narrowband propagation factor. The function sinh(x)/x is plotted in Fig. 1. For large arguments it approximates to exp(x)/(2x), also plotted. Because of this asymptotic behavior, the error incurred by use of Eq. (18) increases exponentially with large x (large product of bandwidth B, distance r, and absorption gradient af). The outcome depends on the frequency dependence of absorption, considered next. III. ABSORPTION SPECTRA FOR ECHOLOCATION PULSES
In this section, some specific echolocation pulses are considered, the properties of which are summarized in Table I. For each pulse, the center frequency and bandwidth are listed. The accuracy of the narrowband approximation
depends on the absorption spectrum in the frequency band relevant to each echolocation pulse, plotted in Fig. 2. Though not listed explicitly in the table, the echolocation click of Blainville’s beaked whale (Mesoplodon densirostris)15 is similar to that of Ziphius cavirostris. The results presented for Ziphius may be considered representative also for Mesoplodon. Two of these echolocation pulses are plotted in Figs. 3(a) and 4(a). These two are selected because of their contrasting nature, one (Phocoena) being a narrowband pulse with multiple cycles of unchanging period of about 8 ls, while the other (Pseudorca) is a pulse comprising a single cycle and twice the bandwidth, despite the lower center frequency. This difference in bandwidth is reflected in the spectra [Figs. 3(b) and 4(b)]. Although bandwidth is not the only consideration [see Eqs. (20)–(22)] it is an important one, and based on these two spectra one might expect the narrowband approximation to work better for Phocoena than for Pseudorca. Figure 2 shows the absorption coefficient10 vs frequency for selected species, in units of nepers per kilometer (Np/ km). The numerical values can be converted to decibels per kilometer (dB/km) by multiplying them by 20log10e 8.686. It can be seen that the absorption changes across the frequency band of each location pulse (from fm Beq/2 to fm þ Beq/2) by up to about a factor 4 (for Pseudorca). With such a large change, the narrowband approach cannot provide an accurate solution unless the absorption is negligible. Nevertheless, none of the absorption curves seem strongly curved, suggesting that the linear approximation, Eq. (20), might prove accurate. Corresponding fractional changes are shown in Fig. 5. The effective bandwidth Beq is calculated from the half power bandwidth B1/2 (a quantity very closely related to the 3 dB bandwidth, with no further distinction being made between the two) as the width of an equivalent energy top-hat spectrum using (assuming a Gaussian spectrum)
TABLE I. Parameters describing broadband echolocation pulses for selected species, in alphabetical order by scientific name: center frequency fm, half power bandwidth B1/2 [from Refs. 13 (Orcinus), 14 (Pseudorca), and Table 10.25 of Ref. 10 (other species)] and equivalent energy bandwidth Beq. fm/kHz
B1/2/kHz
Beq/kHz
Grampus griseus (Risso’s dolphin)
49.0
27.0
28.7
Monodon monoceros (narwhal)
40.0
20.0
21.3
Orcinus orca (killer whale)
45.0
26.7
28.4
Phocoena phocoena (harbor porpoise)
140.0
16.0
17.0
Physeter macrocephalus (sperm whale)
15.0
5.0
5.3
Pseudorca crassidens (false killer whale)
43.0
33.0
35.1
Tursiops truncatus (bottlenose dolphin)
120.0
30.0
31.9
Ziphius cavirostris (Cuvier’s beaked whale)
40.0
12.0
12.8
Species
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Image [# Uko Gorter (narwhal) and Garth Mix (all others)]
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Michael A. Ainslie: Neglecting bandwidth biases propagation loss
FIG. 2. (Color online) Absorption coefficient a(f) vs frequency for frequency range of six of the eight selected species; successive species offset by 0.0 Np/km (Grampus, Phocoena, Physeter), 0.1 Np/km (Pseudorca, Tursiops), and 0.2 Np/km (Ziphius). Monodon and Orcinus (both similar to Grampus) are excluded.
! rffiffiffiffiffiffiffiffiffiffiffi 4loge 2ðf fm Þ2 p B1=2 df ¼ Beq exp 2 loge 2 2 B1=2 1 ð þ1
1:065B1=2 ; (23) such that the integrand is equal to 0.5 at the frequencies f ¼ fm 6 B1/2/2, as illustrated by Fig. 6. Also shown is B1/10, the one tenth power (minus 10 dB) bandwidth, given by
FIG. 4. Pseudorca echolocation pulse (a), pressure time series in arbitrary units and its spectrum (b), spectral density level in decibels relative to the maximum, adapted with permission from Ref. 14. The pulse comprises a single cycle, of duration (the time window containing 97% of the acoustic energy) s ¼ 18 ls. The bandwidth is of the same order of magnitude as the center frequency.
B1=10
sffiffiffiffiffiffiffiffiffiffiffi log10 B1=2 1:823B1=2 : ¼ log2
(24)
Table II shows a, af, aff evaluated at the center frequency of the echolocation pulse [i.e., a(fm), af(fm), and aff(fm)] for the same eight species as Table I. Also listed is the distance ra, the range at which the departure of sinhx/x from unity starts to become noticeable, which takes values between 600 m (for Pseudorca) and 6700 m (Physeter). The error reaches 10 dB (a factor 10 in FBB/FNB) when x ¼ 4.5 (i.e., r ¼ 4.5 ra, which for Pseudorca is about 2.5 km). Most of the factor 11 difference in ra between Pseudorca and Physeter can be explained by the difference in bandwidth between these two species. IV. RESULTS: ERRORS IN BROADBAND PROPAGATION LOSS AND POPULATION ESTIMATE
FIG. 3. Phocoena echolocation pulse (a), pressure time series and its spectrum (b), spectral density level in decibels relative to the maximum, adapted with permission from Ref. 16. The pulse has the appearance of a Gaussian modulated sine wave of duration 8 cycles and period 8 ls. The bandwidth is of order 10% of the center frequency. J. Acoust. Soc. Am., Vol. 134, No. 5, November 2013
A. Error in propagation loss for arbitrary propagation conditions
The error in broadband propagation loss (PL) calculated using Eq. (20) is plotted versus range in Fig. 7 for the eight species. The error at distance 4 km ranges between 0.4 dB Michael A. Ainslie: Neglecting bandwidth biases propagation loss
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FIG. 5. (Color online) Normalized absorption coefficient a(f)/a(fm) vs normalized frequency (f fm)/Beq for the six species of Fig. 2.
FIG. 6. (Color online) Assumed Gaussian spectrum and equivalent energy top-hat spectrum (Grampus).
(Physeter) and 18 dB (Pseudorca). This error is independent of propagation conditions. The only assumption made so far about propagation conditions is that the frequency dependence arises solely or primarily from absorption. However, the effect of the error in PL on the population density estimate depends on the error in detection range, which does depend on propagation conditions. In the following, the consequences for estimation of population density are considered for the special case of spherical spreading.
pulse. The error in propagation loss incurred by approximating FBB by FNB, and its consequences for estimates of detection range and population density, are explored below. The main error is caused by use in Eq. (25) of a decay rate equal to a(fm), which at distances exceeding ra ¼ 1/(afB) overestimates the broadband decay rate implied by Eq. (26), equal to a(fm) (afB)/2, by an amount proportional to bandwidth. Specifically, substituting Eq. (21) in Eq. (20) gives FBB exp½af ðfm ÞBr : FNB 2af ðfm ÞBr
B. Error in propagation loss for spherical spreading conditions
For spherical spreading, the narrowband approximation for the propagation factor is FNB
exp 2aðfm Þr ¼ : r2
(25)
Using the linear approximation, Eq. (20), the broadband equivalent of Eq. (25) is
FBB
exp 2aðfm Þr sinhx ; x r2
(26)
where x is given by Eq. (21). The linear approximation to FBB, Eq. (26), is used in Ref. 10 to calculate propagation loss for an Orcinus pulse and in Ref. 11 for a Mesoplodon
(27)
Further, the extra factor of r in the denominator means that the broadband PL increases with range as 30log10r instead of 20log10r usually associated with spherical spreading. This correction to the power law is smaller than and in the opposite direction to that associated with the difference in exponential decay. C. Error in detection range for spherical spreading conditions
The functions PLNB ¼ 10log10 (FNB1/rref2) (solid lines) and its broadband equivalent PLBB ¼ 10log10 (FBB1/rref2) (dashed lines) are plotted in Fig. 8.10 Interpretation of Fig. 8 is aided by using the concept of a sonar figure of merit (FOM),10 defined as the value of propagation loss for which
TABLE II. Absorption a, absorption gradient af, rate of change of absorption gradient aff, and distance ra beyond which broadband effects are important, evaluated at the center frequency for the echolocation pulses of Table I. The first and second derivatives af and aff are calculated using a first order finite difference approximation. Species Grampus griseus Monodon monoceros Orcinus orca Phocoena phocoena Physeter macrocephalus Pseudorca crassidens Tursiops truncatus Ziphius cavirostris
3510
a(fm)/(Np km1)
af(fm)/(mNp km1 kHz1)
aff(fm)/(mNp km1 kHz2)
ra/(km)
1.76 1.30 1.55 5.10 0.23 1.45 4.57 1.30
50.4 50.1 50.4 24.9 28.2 49.5 28.5 50.5
0.1 0.2 0.1 0.1 1.6 0.1 0.2 0.2
0.7 0.9 0.7 2.4 6.7 0.6 1.1 1.5
J. Acoust. Soc. Am., Vol. 134, No. 5, November 2013
Michael A. Ainslie: Neglecting bandwidth biases propagation loss
FIG. 7. (Color online) Error in PL estimated using Eq. (20) for seven out of the eight species. The y axis is the difference between PLNB and PLBB. Orcinus (similar to Grampus) is omitted.
the detection probability is equal to 50% for a specified false alarm probability. If the detection range (r50) is defined as the distance at which detection probability is 50%, it follows that the range at which PL(r) intersects the FOM value is equal to that detection range. For example, assuming FOM ¼ 120 dB re 1 m2, corresponding to the horizontal line in Fig. 8 results in r50 equal to 1.5 km for Tursiops and 19 km for Physeter. In practice the FOM varies with source level, noise level, and detection threshold10 and would therefore be different for each species. The benefit is that if the FOM is known, the solution to the sonar equation is obtained simply as the range of intersection of this quantity with PL. Calculation of FOM is beyond the present scope. The chosen value of 120 dB re 1 m2 leads to a realistic detection range, but is otherwise arbitrary. The magnitude of the fractional error in detection range increases with increasing range and therefore depends on the figure of merit. For the previously assumed value of
FIG. 9. (Color online) Fractional error in detection range and estimated population density vs figure of merit for six of the eight selected species, assuming spherical spreading. The error for Tursiops and Phocoena, both omitted, is small. The error for Grampus, also omitted, is close to that of Monodon.
FOM ¼ 120 dB re 1 m2, this fractional error, clearly visible in Fig. 8, is 18% (for Grampus) and 34% (Pseudorca). D. Error in population estimate for spherical spreading conditions
If it is assumed that the population density is inversely proportional to the square of the detection range, the error in population density as a result of the error in propagation loss (still for fixed FOM) becomes approximately 50% (Grampus) and 130% (Pseudorca). The values for range error are negative, indicating that the detection range is underestimated, while the population density is overestimated by the amounts stated. In reality the FOM will not take the same value for all pulses (for example, one might expect a larger FOM to apply to Physeter due its high source level17,18), and the above calculation is generalized to arbitrary FOM by plotting the two types of fractional error vs FOM in Fig. 9. The calculation can be further refined by taking into account the possibility that the FOM calculated for a narrowband pulse might be different from that of a broadband pulse of the same center frequency. Parameters that might need adjusting include the in-band noise level (the pulse duration increases while its bandwidth decreases) and the detection threshold (the pulse becomes less recognizable as the bandwidth decreases, increasing the probability of false alarm for a fixed detector threshold19). E. Discussion
FIG. 8. (Color online) Narrowband and broadband propagation loss for four of the eight species, using spherical spreading. Omitted are Orcinus (similar to Grampus), Monodon and Ziphius (also similar to Grampus, but with lower propagation loss due to the lower center frequency), and Phocoena (similar to Tursiops). The horizontal line marks FOM ¼ 120 dB re 1 m2. For Tursiops, the BB result is obscured by the NB curve. J. Acoust. Soc. Am., Vol. 134, No. 5, November 2013
The validity of Fig. 8 and Fig. 9 is limited to conditions of spherical spreading, as described by Eq. (26), which is a reasonable assumption at distances up to a few water depths from the sound source. At greater distances, a region of cylindrical spreading is commonly encountered,20 resulting in lower propagation loss. In such conditions the detection range increases, which increases the influence of absorption and therefore increases both the amount by which PLNB underestimates PLBB and the fraction by which population density is overestimated for fixed FOM. Michael A. Ainslie: Neglecting bandwidth biases propagation loss
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V. CONCLUSIONS
Long range detection of echolocation clicks has been proposed5–8 as a method for population monitoring using a single hydrophone, with the mean distance to the hydrophone position estimated by calculating detection range, requiring propagation loss as an intermediate step. Calculation of propagation loss of a broadband echolocation click as if it were a narrowband pulse at the center frequency of the broadband click (a procedure referred to in this paper as the “narrowband approximation”) is shown to systematically overestimate propagation loss by an amount of order 10–20 dB at distances as short as 5 km. In the absence of any other corrections this would result in a systematic underestimation of the detection range and a biased (overestimated) estimate of the population density. The magnitude of the error depends on the assumed propagation conditions, source and receiver characteristics, and ambient noise level. Assuming spherical spreading and the value FOM ¼ 120 dB re 1 m2, the narrowband approximation overestimates detection range by 30% for the killer whale (Orcinus orca) and 45% for the false killer whale (Pseudorca crassidens), resulting in population density errors exceeding 100% for Orcinus and 200% for Pseudorca. Effects are small for echolocation clicks of the harbor porpoise (Phocoena phocoena) and bottlenose dolphin (Tursiops truncatus). In reality, these estimates are oversimplified, with corrections needed for effects such as differences in FOM between species, effect of broadband corrections to FOM, a more realistic pulse spectrum than a top hat function, and propagation conditions other than spherical spreading. Corrections to the linear approximation might be necessary for very broadband echolocation clicks such as that of Pseudorca. Further research is needed to establish the magnitude of these corrections, but the calculations presented here demonstrate the need to take the broadband nature of the pulse into account for unbiased estimation of population density. ACKNOWLEDGMENTS
The author thanks Alexander M. von Benda-Beckmann (TNO) for reviewing an earlier version of this paper, and Garth Mix and Uko Gorter for their permission to reproduce the images used in Table I. 1
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S. van Parijs, C. W. Clark, R. S. Sousa-Lima, S. E. Parks, S. Rankin, D. Risch, and I. C. van Opzeeland, “Management and research applications of real-time and archival passive acoustic sensors over varying temporal and spatial scales,” Mar. Ecol.: Prog. Ser. 395, 21–36 (2009). 3 W. M. X. Zimmer, Passive Acoustic Monitoring of Cetaceans (Cambridge University Press, Cambridge, UK, 2011), pp. 1–356. 4 L. Thomas and T. Marques, “Passive acoustic monitoring for estimating animal density,” Acoust. Today 8(3), 35–44 (2012). 5 M. A. McDonald and C. G. Fox, “Passive acoustic methods applied to fin whale population density estimation,” J. Acoust. Soc. Am. 105, 2643–2651 (1999). 6 W. M. X. Zimmer, J. Harwood, P. L. Tyack, M. P. Johnson, and P. T. Madsen, “Passive acoustic detection of deep-diving beaked whales,” J. Acoust. Soc. Am. 124, 2823–2832 (2008). 7 S. Kimura, T. Akamutsu, S. Li, S. Dong, L. Dong, K. Wang, D. Wang, and N. Arai, “Density estimation of Yangtze finless porpoises using passive acoustic sensors and automated click train detection,” J. Acoust. Soc. Am. 128, 1435–1445 (2010). 8 E. T. K€ usel, D. K. Mellinger, L. Thomas, T. A. Marques, D. Moretti, and J. Ward, “Cetacean population density estimation from single fixed sensors using passive acoustics,” J. Acoust. Soc. Am. 129, 3610–3622 (2011). 9 T. Marques, L. Thomas, J. Ward, N. DiMarzio, and P. L. Tyack, “Estimating cetacean population density using fixed passive acoustic sensors: An example with Blainville’s beaked whales,” J. Acoust. Soc. Am. 125, 1982–1994 (2009). 10 M. A. Ainslie, Principles of Sonar Performance Modeling (Springer, Berlin, 2010), pp. 29, 30, 59, 60, 61, 63, 85, 96, 147, 543, 616. 11 A. M. von Benda-Beckmann, F. P. A. Lam, D. J. Moretti, K. Fulkerson, M. A. Ainslie, S. P. van IJsselmuide, J. Theriault, and S. P. Beerens, “Detection of Blainville’s beaked whales with towed arrays,” Appl. Acoust. 71, 1027–1035 (2010). 12 T. A. Helble, G. L. D’Spain, J. A. Hildebrand, G. S. Campbell, R. L. Campbell, and K. D. Heaney, “Site specific probability of passive acoustic detection of humpback whale calls from single fixed hydrophones off the California coast,” J. Acoust. Soc. Am. 134, 2556–2570 (2013). 13 W. W. L. Au, J. K. B. Ford, J. K. Horne, and K. A. Newman Allman, “Echolocation signals of free-ranging killer whales (Orcinus orca) and modeling of foraging for chinook salmon (Oncorhynchus tshawytscha),” J. Acoust. Soc. Am. 115, 901–909 (2004). 14 P. T. Madsen, I. Kerr, and R. Payne, “Echolocation clicks of two freeranging, oceanic delphinids with different food preferences: False killer whales Pseudorca crassidens and Risso’s dolphins Grampus griseus,” J. Exp. Biol. 207, 1811–1823 (2004). 15 M. Johnson, P. T. Madsen, W. M. X. Zimmer, N. A. Aguilar de Soto, and P. L. Tyack, “Beaked whales echolocate on prey,” Proc. R. Soc. London, Ser. B 271, S383–S386 (2004). 16 A. Villadsgaard, M. Wahlberg, and J. Tougaard, “Echolocation signals of wild harbour porpoises, Phocoena phocoena,” J. Exp. Biol. 210, 56–64 (2007). 17 B. Møhl, M. Wahlberg, P. T. Madsen, L. A. Miller, and A. Surlykke, “Sperm whale clicks: Directionality and source level revisited,” J. Acoust. Soc. Am. 107, 638–648 (2000). 18 B. Møhl M. Wahlberg, P. T. Madsen, A. Heerfordt, and A. Lund, “The monopulsed nature of sperm whale clicks,” J. Acoust. Soc. Am. 114, 1143–1153 (2003). 19 D. A. Abraham, “Detection-threshold approximation for non-Gaussian backgrounds,” IEEE J. Ocean. Eng. 35(2), 355–365 (2010). 20 D. E. Weston, “Intensity-range relations in oceanographic acoustics,” J. Sound Vib. 18, 271–287 (1971).
Michael A. Ainslie: Neglecting bandwidth biases propagation loss
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(Received 5 March 2013; revised 14 July 2013; accepted 10 September 2013) Passive acoustic monitoring with a single hydrophone has been suggested as a cost-effective method to monitor population density of echolocating marine mammals, by estimating the distance at which the hydrophone is able to intercept the echolocation clicks and distinguish these from the background. To avoid a bias in the estimated population density, this method relies on an unbiased estimate of the detection range and therefore of the propagation loss (PL). When applying this method, it is common practice to estimate PL at the center frequency of a broadband echolocation click and to assume this narrowband PL applies also to the broadband click. For a typical situation this narrowband approximation overestimates PL, underestimates the detection range and consequently overestimates the population density by an amount that for fixed center frequency increases C 2013 Acoustical Society of America. with increasing pulse bandwidth and sonar figure of merit. V [http://dx.doi.org/10.1121/1.4823804] PACS number(s): 43.30.Sf, 43.30.Cq, 43.80.Ka [JIA]
I. INTRODUCTION
The echolocation clicks of marine mammals can be detected and classified using specially designed passive sonar equipment.1–3 Such detectors and classifiers are proposed as cost-effective tools for estimating population density4 using a single hydrophone5–8 or a hydrophone network.9 For a stationary hydrophone, the rate at which clicks are detected is proportional to the animals’ average vocalization rate, the population density (assumed locally uniform) and the detection area, which is proportional to the square of the detection range. It follows that the population density can be estimated from knowledge of the vocalization probability and detection range. The success of this method relies on obtaining unbiased estimates of both click rate and detection range, the latter of which is the subject of this paper. The usual procedure for estimating detection range is to substitute propagation loss (PL) into the passive sonar equation and solve this equation for the distance at which the signal excess is zero (by definition the detection range).10 Application of this procedure requires an estimate of source level, propagation loss, noise level and detection threshold. A bias in one or more of these terms risks introducing a bias in the population density estimate by either under- or over-estimating the detection probability. Even though typical echolocation clicks are broadband pulses, it is customary to treat them as narrowband pulses for the purpose of estimating PL.6,8 Using the expression for broadband PL derived in Ref. 10, Ref. 11 shows that the narrowband approximation applied by Refs. 6 and 8 results in an error in PL of ca. 17 dB at 5 km for Mesoplodon for the special case of spherical spreading propagation, using a center frequency of 38 kHz and a 10 dB bandwidth of 25 kHz. The present work shows that this error in PL applies to any
a)
Also at: Institute of Sound and Vibration Research, University of Southampton, Highfield, Southampton SO17 1BJ, United Kingdom.
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Pages: 3506–3512
propagation law, suggesting a possible bias in detection range (and hence in the derived population density). The purpose is to explore possible implications of this error for the accuracy of the estimated population density. Similar errors in population estimates of humpback whales from their 150–1800 Hz vocalizations are described in Ref. 12. II. EQUATIONS FOR BROADBAND PROPAGATION LOSS
Propagation loss (PL) can be defined for a narrowband source (indicated by the subscript “NB” for “narrowband”) as the difference between source level (SL) and sound pressure level (SPL), i.e.,10 PLNB SL SPL:
(1)
It is convenient to introduce linear equivalents of these logarithmic quantities. The linear quantity corresponding to SPL is the mean square value of the instantaneous sound pressure p, i.e., hp2 i ¼ 10SPL=10 pref 2 ;
(2)
where pref is the standard reference pressure used to define SPL, usually equal to 1 lPa. Define the source factor S0 and steady state propagation factor F as10 S0 10SL=10 pref 2 rref 2
(3)
FNB 10PLNB =10 rref 2 ;
(4)
and
where rref is the standard reference distance for SL and PL, usually equal to 1 m. Using Eqs. (2), (3), and (4), Eq. (1) can be written in linear form
0001-4966/2013/134(5)/3506/7/$30.00
C 2013 Acoustical Society of America V
FNB ¼
hp2 i : S0
(5)
A narrowband source necessarily has a long duration, not less that the reciprocal of the bandwidth. Concern here is with short transient sounds, which necessarily have a high bandwidth, not less than the reciprocal of their duration. For a broadband transient, Eq. (1) would be replaced by the difference between energy source level10 (SLE) and sound exposure level (SEL), leading to the following alternative PL definition, applicable to transient sounds and indicated by the subscript “BB” for a broadband transient PLBB SLE SEL:
(6)
Linear equivalents to SEL and SLE are the sound exposure E [cf. Eq. (2)], E ¼ 10SEL=10 pref 2 tref ;
(7)
Substitution of Eqs. (11) and (12) in Eq. (10) then results in the equation ð1 ðSE Þf expf2½aðfm Þ aðf Þrgdf ð1 FBB ¼ FNB 0 ; (15) ðSE Þf df 0
where FNB is introduced as shorthand for F(r,fm). Now assume a top-hat source factor spectrum with a uniform spectral density r for frequencies between fm B/2 and fm þ B/2 and zero elsewhere, ðSE Þf fm B=2 fm þ B=2 ¼H 1 H 1 ; r f f (16) where H(x) is the Heaviside step function, such that FBB 1 ¼ FNB B
and energy source factor SE [cf. Eq. (3)], SE 10SLE =10 pref 2 rref 2 tref ;
(8)
where tref is the reference time used for SEL and SLE, usually equal to 1 s. Defining the broadband propagation factor FBB for transients [cf. Eq. (4)], FBB 10PLBB =10 rref 2 ;
(9)
where PLBB is defined by Eq. (6), the propagation factor follows from Eqs. (7) and (8) as FBB ¼
E : SE
(10)
We can write the broadband quantities E and SE as integrals over the respective spectral densities of the exposure (Ef) and energy source factor, (SE)f, such that ð1 Ef ðr; f Þdf (11) EðrÞ ¼
FBB FNB :
0
ðSE Þf df ;
(12)
(13)
and assume that F(r,f) depends on the absorption coefficient a(f), in units of nepers per unit distance, in the following way: Fðr; f Þ ¼ Fðr; fm Þexpf2½aðfm Þ aðf Þrg: J. Acoust. Soc. Am., Vol. 134, No. 5, November 2013
(18)
1 aðf Þ ¼ aðfm Þ þ ðf fm Þaf ðfm Þ þ ðf fm Þ2 af f ðfm Þ þ : 2 (19)
where the arguments r and f denote the range and frequency, respectively, and the subscript f denotes a spectral density. Now consider the following equation for the exposure spectral density [cf. Eq. (10)], Ef ðr; f Þ ¼ ðSE Þf Fðr; f Þ
(17)
Equation (18), valid for a narrowband transient, has been used to model the propagation loss for echolocation pulses of marine mammals such as the killer whale,13 beaked whales,6,8 and finless porpoise.7 Some of these echolocation pulses have a bandwidth comparable to their center frequency. Is this narrowband approximation justified for these broadband echolocation pulses? To answer this question, consider the effect of increasing B from zero in Eq. (17). In general, the absorption a(f) is a function of frequency and needs to be substituted in Eq. (18) before evaluating the integral. This can be achieved by expanding the absorption coefficient as a Taylor series about f ¼ fm, i.e.,
and ð1
expf2½aðfm Þ aðf Þrgdf :
fm B=2
If the bandwidth is sufficiently small, the integrand may be approximated as a constant, equal to its value at the center frequency, in which case
0
SE ¼
ð fm þB=2
(14)
Use of the constant term on its own gives the narrowband approximation, Eq. (18). Including the linear term as well gives FBB sinhx ; FNB x
(20)
where x ¼ r=ra ðfm Þ
(21)
ra ðfm Þ ¼ 1=½af ðfm ÞB
(22)
and
Michael A. Ainslie: Neglecting bandwidth biases propagation loss
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FIG. 1. (Color online) Graph of sinh(x)/x (solid curve); the asymptotic value for large x (dotted curve) is exp(x)/(2x).
is the distance at which the argument of the sinh(x)/x function in Eq. (20) is equal to unity, meaning that at ranges exceeding ra, the broadband propagation factor differs significantly from the narrowband propagation factor. The function sinh(x)/x is plotted in Fig. 1. For large arguments it approximates to exp(x)/(2x), also plotted. Because of this asymptotic behavior, the error incurred by use of Eq. (18) increases exponentially with large x (large product of bandwidth B, distance r, and absorption gradient af). The outcome depends on the frequency dependence of absorption, considered next. III. ABSORPTION SPECTRA FOR ECHOLOCATION PULSES
In this section, some specific echolocation pulses are considered, the properties of which are summarized in Table I. For each pulse, the center frequency and bandwidth are listed. The accuracy of the narrowband approximation
depends on the absorption spectrum in the frequency band relevant to each echolocation pulse, plotted in Fig. 2. Though not listed explicitly in the table, the echolocation click of Blainville’s beaked whale (Mesoplodon densirostris)15 is similar to that of Ziphius cavirostris. The results presented for Ziphius may be considered representative also for Mesoplodon. Two of these echolocation pulses are plotted in Figs. 3(a) and 4(a). These two are selected because of their contrasting nature, one (Phocoena) being a narrowband pulse with multiple cycles of unchanging period of about 8 ls, while the other (Pseudorca) is a pulse comprising a single cycle and twice the bandwidth, despite the lower center frequency. This difference in bandwidth is reflected in the spectra [Figs. 3(b) and 4(b)]. Although bandwidth is not the only consideration [see Eqs. (20)–(22)] it is an important one, and based on these two spectra one might expect the narrowband approximation to work better for Phocoena than for Pseudorca. Figure 2 shows the absorption coefficient10 vs frequency for selected species, in units of nepers per kilometer (Np/ km). The numerical values can be converted to decibels per kilometer (dB/km) by multiplying them by 20log10e 8.686. It can be seen that the absorption changes across the frequency band of each location pulse (from fm Beq/2 to fm þ Beq/2) by up to about a factor 4 (for Pseudorca). With such a large change, the narrowband approach cannot provide an accurate solution unless the absorption is negligible. Nevertheless, none of the absorption curves seem strongly curved, suggesting that the linear approximation, Eq. (20), might prove accurate. Corresponding fractional changes are shown in Fig. 5. The effective bandwidth Beq is calculated from the half power bandwidth B1/2 (a quantity very closely related to the 3 dB bandwidth, with no further distinction being made between the two) as the width of an equivalent energy top-hat spectrum using (assuming a Gaussian spectrum)
TABLE I. Parameters describing broadband echolocation pulses for selected species, in alphabetical order by scientific name: center frequency fm, half power bandwidth B1/2 [from Refs. 13 (Orcinus), 14 (Pseudorca), and Table 10.25 of Ref. 10 (other species)] and equivalent energy bandwidth Beq. fm/kHz
B1/2/kHz
Beq/kHz
Grampus griseus (Risso’s dolphin)
49.0
27.0
28.7
Monodon monoceros (narwhal)
40.0
20.0
21.3
Orcinus orca (killer whale)
45.0
26.7
28.4
Phocoena phocoena (harbor porpoise)
140.0
16.0
17.0
Physeter macrocephalus (sperm whale)
15.0
5.0
5.3
Pseudorca crassidens (false killer whale)
43.0
33.0
35.1
Tursiops truncatus (bottlenose dolphin)
120.0
30.0
31.9
Ziphius cavirostris (Cuvier’s beaked whale)
40.0
12.0
12.8
Species
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Image [# Uko Gorter (narwhal) and Garth Mix (all others)]
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Michael A. Ainslie: Neglecting bandwidth biases propagation loss
FIG. 2. (Color online) Absorption coefficient a(f) vs frequency for frequency range of six of the eight selected species; successive species offset by 0.0 Np/km (Grampus, Phocoena, Physeter), 0.1 Np/km (Pseudorca, Tursiops), and 0.2 Np/km (Ziphius). Monodon and Orcinus (both similar to Grampus) are excluded.
! rffiffiffiffiffiffiffiffiffiffiffi 4loge 2ðf fm Þ2 p B1=2 df ¼ Beq exp 2 loge 2 2 B1=2 1 ð þ1
1:065B1=2 ; (23) such that the integrand is equal to 0.5 at the frequencies f ¼ fm 6 B1/2/2, as illustrated by Fig. 6. Also shown is B1/10, the one tenth power (minus 10 dB) bandwidth, given by
FIG. 4. Pseudorca echolocation pulse (a), pressure time series in arbitrary units and its spectrum (b), spectral density level in decibels relative to the maximum, adapted with permission from Ref. 14. The pulse comprises a single cycle, of duration (the time window containing 97% of the acoustic energy) s ¼ 18 ls. The bandwidth is of the same order of magnitude as the center frequency.
B1=10
sffiffiffiffiffiffiffiffiffiffiffi log10 B1=2 1:823B1=2 : ¼ log2
(24)
Table II shows a, af, aff evaluated at the center frequency of the echolocation pulse [i.e., a(fm), af(fm), and aff(fm)] for the same eight species as Table I. Also listed is the distance ra, the range at which the departure of sinhx/x from unity starts to become noticeable, which takes values between 600 m (for Pseudorca) and 6700 m (Physeter). The error reaches 10 dB (a factor 10 in FBB/FNB) when x ¼ 4.5 (i.e., r ¼ 4.5 ra, which for Pseudorca is about 2.5 km). Most of the factor 11 difference in ra between Pseudorca and Physeter can be explained by the difference in bandwidth between these two species. IV. RESULTS: ERRORS IN BROADBAND PROPAGATION LOSS AND POPULATION ESTIMATE
FIG. 3. Phocoena echolocation pulse (a), pressure time series and its spectrum (b), spectral density level in decibels relative to the maximum, adapted with permission from Ref. 16. The pulse has the appearance of a Gaussian modulated sine wave of duration 8 cycles and period 8 ls. The bandwidth is of order 10% of the center frequency. J. Acoust. Soc. Am., Vol. 134, No. 5, November 2013
A. Error in propagation loss for arbitrary propagation conditions
The error in broadband propagation loss (PL) calculated using Eq. (20) is plotted versus range in Fig. 7 for the eight species. The error at distance 4 km ranges between 0.4 dB Michael A. Ainslie: Neglecting bandwidth biases propagation loss
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FIG. 5. (Color online) Normalized absorption coefficient a(f)/a(fm) vs normalized frequency (f fm)/Beq for the six species of Fig. 2.
FIG. 6. (Color online) Assumed Gaussian spectrum and equivalent energy top-hat spectrum (Grampus).
(Physeter) and 18 dB (Pseudorca). This error is independent of propagation conditions. The only assumption made so far about propagation conditions is that the frequency dependence arises solely or primarily from absorption. However, the effect of the error in PL on the population density estimate depends on the error in detection range, which does depend on propagation conditions. In the following, the consequences for estimation of population density are considered for the special case of spherical spreading.
pulse. The error in propagation loss incurred by approximating FBB by FNB, and its consequences for estimates of detection range and population density, are explored below. The main error is caused by use in Eq. (25) of a decay rate equal to a(fm), which at distances exceeding ra ¼ 1/(afB) overestimates the broadband decay rate implied by Eq. (26), equal to a(fm) (afB)/2, by an amount proportional to bandwidth. Specifically, substituting Eq. (21) in Eq. (20) gives FBB exp½af ðfm ÞBr : FNB 2af ðfm ÞBr
B. Error in propagation loss for spherical spreading conditions
For spherical spreading, the narrowband approximation for the propagation factor is FNB
exp 2aðfm Þr ¼ : r2
(25)
Using the linear approximation, Eq. (20), the broadband equivalent of Eq. (25) is
FBB
exp 2aðfm Þr sinhx ; x r2
(26)
where x is given by Eq. (21). The linear approximation to FBB, Eq. (26), is used in Ref. 10 to calculate propagation loss for an Orcinus pulse and in Ref. 11 for a Mesoplodon
(27)
Further, the extra factor of r in the denominator means that the broadband PL increases with range as 30log10r instead of 20log10r usually associated with spherical spreading. This correction to the power law is smaller than and in the opposite direction to that associated with the difference in exponential decay. C. Error in detection range for spherical spreading conditions
The functions PLNB ¼ 10log10 (FNB1/rref2) (solid lines) and its broadband equivalent PLBB ¼ 10log10 (FBB1/rref2) (dashed lines) are plotted in Fig. 8.10 Interpretation of Fig. 8 is aided by using the concept of a sonar figure of merit (FOM),10 defined as the value of propagation loss for which
TABLE II. Absorption a, absorption gradient af, rate of change of absorption gradient aff, and distance ra beyond which broadband effects are important, evaluated at the center frequency for the echolocation pulses of Table I. The first and second derivatives af and aff are calculated using a first order finite difference approximation. Species Grampus griseus Monodon monoceros Orcinus orca Phocoena phocoena Physeter macrocephalus Pseudorca crassidens Tursiops truncatus Ziphius cavirostris
3510
a(fm)/(Np km1)
af(fm)/(mNp km1 kHz1)
aff(fm)/(mNp km1 kHz2)
ra/(km)
1.76 1.30 1.55 5.10 0.23 1.45 4.57 1.30
50.4 50.1 50.4 24.9 28.2 49.5 28.5 50.5
0.1 0.2 0.1 0.1 1.6 0.1 0.2 0.2
0.7 0.9 0.7 2.4 6.7 0.6 1.1 1.5
J. Acoust. Soc. Am., Vol. 134, No. 5, November 2013
Michael A. Ainslie: Neglecting bandwidth biases propagation loss
FIG. 7. (Color online) Error in PL estimated using Eq. (20) for seven out of the eight species. The y axis is the difference between PLNB and PLBB. Orcinus (similar to Grampus) is omitted.
the detection probability is equal to 50% for a specified false alarm probability. If the detection range (r50) is defined as the distance at which detection probability is 50%, it follows that the range at which PL(r) intersects the FOM value is equal to that detection range. For example, assuming FOM ¼ 120 dB re 1 m2, corresponding to the horizontal line in Fig. 8 results in r50 equal to 1.5 km for Tursiops and 19 km for Physeter. In practice the FOM varies with source level, noise level, and detection threshold10 and would therefore be different for each species. The benefit is that if the FOM is known, the solution to the sonar equation is obtained simply as the range of intersection of this quantity with PL. Calculation of FOM is beyond the present scope. The chosen value of 120 dB re 1 m2 leads to a realistic detection range, but is otherwise arbitrary. The magnitude of the fractional error in detection range increases with increasing range and therefore depends on the figure of merit. For the previously assumed value of
FIG. 9. (Color online) Fractional error in detection range and estimated population density vs figure of merit for six of the eight selected species, assuming spherical spreading. The error for Tursiops and Phocoena, both omitted, is small. The error for Grampus, also omitted, is close to that of Monodon.
FOM ¼ 120 dB re 1 m2, this fractional error, clearly visible in Fig. 8, is 18% (for Grampus) and 34% (Pseudorca). D. Error in population estimate for spherical spreading conditions
If it is assumed that the population density is inversely proportional to the square of the detection range, the error in population density as a result of the error in propagation loss (still for fixed FOM) becomes approximately 50% (Grampus) and 130% (Pseudorca). The values for range error are negative, indicating that the detection range is underestimated, while the population density is overestimated by the amounts stated. In reality the FOM will not take the same value for all pulses (for example, one might expect a larger FOM to apply to Physeter due its high source level17,18), and the above calculation is generalized to arbitrary FOM by plotting the two types of fractional error vs FOM in Fig. 9. The calculation can be further refined by taking into account the possibility that the FOM calculated for a narrowband pulse might be different from that of a broadband pulse of the same center frequency. Parameters that might need adjusting include the in-band noise level (the pulse duration increases while its bandwidth decreases) and the detection threshold (the pulse becomes less recognizable as the bandwidth decreases, increasing the probability of false alarm for a fixed detector threshold19). E. Discussion
FIG. 8. (Color online) Narrowband and broadband propagation loss for four of the eight species, using spherical spreading. Omitted are Orcinus (similar to Grampus), Monodon and Ziphius (also similar to Grampus, but with lower propagation loss due to the lower center frequency), and Phocoena (similar to Tursiops). The horizontal line marks FOM ¼ 120 dB re 1 m2. For Tursiops, the BB result is obscured by the NB curve. J. Acoust. Soc. Am., Vol. 134, No. 5, November 2013
The validity of Fig. 8 and Fig. 9 is limited to conditions of spherical spreading, as described by Eq. (26), which is a reasonable assumption at distances up to a few water depths from the sound source. At greater distances, a region of cylindrical spreading is commonly encountered,20 resulting in lower propagation loss. In such conditions the detection range increases, which increases the influence of absorption and therefore increases both the amount by which PLNB underestimates PLBB and the fraction by which population density is overestimated for fixed FOM. Michael A. Ainslie: Neglecting bandwidth biases propagation loss
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V. CONCLUSIONS
Long range detection of echolocation clicks has been proposed5–8 as a method for population monitoring using a single hydrophone, with the mean distance to the hydrophone position estimated by calculating detection range, requiring propagation loss as an intermediate step. Calculation of propagation loss of a broadband echolocation click as if it were a narrowband pulse at the center frequency of the broadband click (a procedure referred to in this paper as the “narrowband approximation”) is shown to systematically overestimate propagation loss by an amount of order 10–20 dB at distances as short as 5 km. In the absence of any other corrections this would result in a systematic underestimation of the detection range and a biased (overestimated) estimate of the population density. The magnitude of the error depends on the assumed propagation conditions, source and receiver characteristics, and ambient noise level. Assuming spherical spreading and the value FOM ¼ 120 dB re 1 m2, the narrowband approximation overestimates detection range by 30% for the killer whale (Orcinus orca) and 45% for the false killer whale (Pseudorca crassidens), resulting in population density errors exceeding 100% for Orcinus and 200% for Pseudorca. Effects are small for echolocation clicks of the harbor porpoise (Phocoena phocoena) and bottlenose dolphin (Tursiops truncatus). In reality, these estimates are oversimplified, with corrections needed for effects such as differences in FOM between species, effect of broadband corrections to FOM, a more realistic pulse spectrum than a top hat function, and propagation conditions other than spherical spreading. Corrections to the linear approximation might be necessary for very broadband echolocation clicks such as that of Pseudorca. Further research is needed to establish the magnitude of these corrections, but the calculations presented here demonstrate the need to take the broadband nature of the pulse into account for unbiased estimation of population density. ACKNOWLEDGMENTS
The author thanks Alexander M. von Benda-Beckmann (TNO) for reviewing an earlier version of this paper, and Garth Mix and Uko Gorter for their permission to reproduce the images used in Table I. 1
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Michael A. Ainslie: Neglecting bandwidth biases propagation loss
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