An underwater acoustic detection problem is studied in which the ambient noise present at a receiver is calculated, given information describing environmental conditions, including windspeed, and the positions of nearby ships which act as sources of background noise. The signal, whose detection is sought, is narrowband and transmitted from a source that passes by the receiver along a straight track. Cumulative Probability of Detection (CPoD) is calculated along a series of tracks with increasing closestpointofapproach distances to the receiver. Two detection ranges are analyzed, a socalled “defender” detection range and an “intruder” detection range. Both are conservative measures associated with CPoD equaling 0.5 during the transit of the submerged vessel. Predictions of the detection range are compared across independent attempts to solve the same problem with different modelling approaches. The spread of results (i.e., the “reproducibility” of the predictions) is discussed and reasons for differences are highlighted. Environmental conditions that strongly affect detection performance are discussed, as is the use of CPoD as a singlevalued metric to describe detection performance.
I. INTRODUCTION
The performance of underwater detection systems is often characterized in terms of spatial measures of performance. The simplest form of such a measure is a characteristic range at which detection will be achieved. This may be referred to as a “detection range” (Ainslie, 2010; Abraham, 2019).
The detection range is not an easy number to compute. In general, a passivesonar detection range varies as a function of environmental factors, such as bathymetry, sound speed profile, and sediment type (Jensen , 2011). Further, it depends on the object of interest, what its radiated source level is, and whether it is omnidirectional or directional. Finally, it is influenced by the ambient noise, caused, for example, by the wind, rain, or (distant) shipping. These effects are all captured in the (passive) sonar equation (Abraham 2019). By computing the propagation loss from the object of interest to the receiver, computing the ambient noise, and correcting for the type of receiver, a signaltonoise ratio (SNR) can be computed. Using assumptions on the signal and noise distributions, the SNR can then be converted to a probability of detection (Ainslie 2010). Usually, a probability of detection of 0.5 is chosen to determine a detection range. A single detection may not represent a reliable opportunity of unambiguously establishing the presence of a target, and it can be more interesting to look at the probability of detection along a target track. The cumulative probability of detection (CPoD) represents the probability that the target will be detected at least once as function of the location or time along the track (Wagner , 1999; Hodges, 2010). It is an important metric in problems that arise in search and rescue operations, in antisubmarine warfare, mine warfare, in radar applications, or other scenarios where detection along a track is of interest. In this work, the CPoD is used as a tool to compare different modelling implementations for the same detection scenario.
This work is part of the Berlin Ambient Noise Modelling Workshop, held in Berlin on the July 10, 2022. The goal of the Ambient Noise Modelling Workshop was to provide a forum for comparing soundscape modelling for academic, industrial, and governmental researchers (Martin , 2024). The workshop asked acoustic modelling practitioners to model the same described scenarios. In the workshop, two scenarios were considered: a soundscape scenario and a passive sonar detection scenario of a submerged vessel. This work goes into the further details of the detection scenario. Not all groups of the overview paper performed the detection problem scenario calculations; therefore, this work only considers the results of groups that did: groups B, E, and G.
II. TERMINOLOGY
Parameter .  Value .  Notes . 

Vessel heading  300  Degrees clockwise from north 
Vessel speed  3  m/s 
Vessel depth  40  m 
Track length  60 000  m (tracks have CPA at their midpoint) 
Minimum track offset  100  m 
Maximum track offset  10 000  m 
Spacing between adjacent tracks  100  m 
Time period between vessel locations along track  60  s (equivalent to sonar processing timea such that adjacent periods represent statistically independent opportunities) 
Frequencies  63.096, 630.96, 6309.6  Hz 
Source levelb  150 (for 63.096 Hz)  dB re $1\u2009\mu Pa2m2$ 
130 (for 630.96 Hz)  
100 (for 6309.6 Hz)  
Probability of false alarm  5 × 10^{−7}  Gaussian statistics are assumed; results in linear SNR of 20 giving 0.5 probability of detection. Similarly, a detection threshold of 13 dB results in the same probability of detection 
Sonobuoy location  (49.1042, 64.2841)  (latitude, longitude) in degrees [see Fig. 1, or Martin (2024)] 
Sonobuoy depth  10, 70  m (the same as for the civil scenario) 
Start times of tracks  Time of minimum windspeed: September 7, 2019, 3:00:00  
Time of maximum windspeed: September 8, 2019, 09:00:00  
Average soundspeed  1470  m/s (used to convert distance of vessel from sonobuoy to travel time) 
Tonal bandwidth  0.1  Hz (the tonal is a top hat peak, zero outside of this bandwidth) 
Processing bandwidth  1/60  Hz (equal to one over sonar processing time) 
Ambient noise sound pressure level (SPL_{n})  SPL_{n}(t)  dB re $1\u2009\mu Pa2$ (results from civil scenario with 18 kts speed limit; AN must be adjusted for processing bandwidth) 
Vessel depth  40  m 
Water depth  319  m (the propagation loss calculations assumed a constant depth) 
Seabed (compressional) speed of sound  1890  m/s (constant throughout the sediment; it is assumed there is no shear speed of sound) 
Seabed density  2216  kg/m^{3} 
Seabed (compressional) attenuation  0.848  dB/lambda (constant throughout the sediment) 
Parameter .  Value .  Notes . 

Vessel heading  300  Degrees clockwise from north 
Vessel speed  3  m/s 
Vessel depth  40  m 
Track length  60 000  m (tracks have CPA at their midpoint) 
Minimum track offset  100  m 
Maximum track offset  10 000  m 
Spacing between adjacent tracks  100  m 
Time period between vessel locations along track  60  s (equivalent to sonar processing timea such that adjacent periods represent statistically independent opportunities) 
Frequencies  63.096, 630.96, 6309.6  Hz 
Source levelb  150 (for 63.096 Hz)  dB re $1\u2009\mu Pa2m2$ 
130 (for 630.96 Hz)  
100 (for 6309.6 Hz)  
Probability of false alarm  5 × 10^{−7}  Gaussian statistics are assumed; results in linear SNR of 20 giving 0.5 probability of detection. Similarly, a detection threshold of 13 dB results in the same probability of detection 
Sonobuoy location  (49.1042, 64.2841)  (latitude, longitude) in degrees [see Fig. 1, or Martin (2024)] 
Sonobuoy depth  10, 70  m (the same as for the civil scenario) 
Start times of tracks  Time of minimum windspeed: September 7, 2019, 3:00:00  
Time of maximum windspeed: September 8, 2019, 09:00:00  
Average soundspeed  1470  m/s (used to convert distance of vessel from sonobuoy to travel time) 
Tonal bandwidth  0.1  Hz (the tonal is a top hat peak, zero outside of this bandwidth) 
Processing bandwidth  1/60  Hz (equal to one over sonar processing time) 
Ambient noise sound pressure level (SPL_{n})  SPL_{n}(t)  dB re $1\u2009\mu Pa2$ (results from civil scenario with 18 kts speed limit; AN must be adjusted for processing bandwidth) 
Vessel depth  40  m 
Water depth  319  m (the propagation loss calculations assumed a constant depth) 
Seabed (compressional) speed of sound  1890  m/s (constant throughout the sediment; it is assumed there is no shear speed of sound) 
Seabed density  2216  kg/m^{3} 
Seabed (compressional) attenuation  0.848  dB/lambda (constant throughout the sediment) 
The processing time is so large that the processing bandwidth is less than the width of the tonal. This means that the source level should be adjusted to reflect the fact that not all signal energy will be in the processing band.
The source level varies with frequency in such a way that the SNR at each frequency should be in a spread of values corresponding to significant changes in detection probability. Initial calculations with a single value of 130 dB re $1$μPa^{2} m^{2} resulted in zero detection range at the lowest frequency, variable detection range at the middle frequency, and detection for all tracks at the highest frequency.
Halfway along the vessel's trajectory, it is closest to the receiver, where its only distance away from the receiver is its orthogonal distance. This distance is called “closest point of approach” (CPA); and in general, it is expected that, along the track, this would be the point of the highest SNR. At the end of the trajectory of the vessel, when $a=aNmax=60$ km, the CPoD is assumed not to increase anymore if it moved further away from the receiver. This CPoD will be referred to as CPoD_{∞}, which is still a function of the orthogonal distance.
The detection range will be defined as an orthogonal distance for which the CPoD_{∞} crosses a threshold of 0.5 at the end of its track. As will be shown in Sec. IV, the CPoD increases monotonically along a track, but the CPoD does not necessarily monotonically decrease as the orthogonal distance increases. It is therefore possible that the threshold of 0.5 is crossed multiple times as the orthogonal distance increases. Therefore, two detection ranges are defined:

The defender detection range. This is the smallest orthogonal distance for which CPoD_{∞} crosses the threshold of 0.5. It represents a conservative estimate, from the point of view of someone seeking to detect a target passing through the barrier.

The intruder detection range. This is the largest orthogonal distance for which CPoD_{∞} crosses the threshold of 0.5. It represents a conservative estimate, from the point of view of an intruder seeking to pass through the barrier while remaining undetected.
In Sec. V, the effect of these two detection ranges is discussed.
III. PROBLEM DESCRIPTION
The problem is centered around the detection of a submerged vessel in ambient noise. The vessel travels in a straight track from its starting position towards the northwest (300 degrees relative to north), the vessel moves at 3 m/s and is at a constant depth of 40 m. This submerged vessel is perceived as an intruder by a defender (single hydrophone) that can either deploy at 10 or 70 m depth. The general location as well as the location of the single hydrophone and the local soundspeed profile can be seen Fig. 1. The defender's job is to detect the passage of the submerged vessel (the intruder). The track is 60kmlong, starting 30 km before the CPA to the receiver and ending 30 km after it.
The intruder is a passive target with three tonals at 63.096, at 630.96, and at 6309.6 Hz. These tonals each have a bandwidth of 0.1 Hz, and are tophat shaped: constant value inside this bandwidth and zero outside of it. It is assumed that there are no temporal fluctuations in the tonals. Furthermore, it is assumed that Doppler effects can be ignored, as their effect is relatively small (the vessel is moving at 3 m/s) and only complicates computations.
For the propagation loss computations, it is assumed that the environment can be considered rangeindependent (constant bathymetry and constant sound speed profile).
A. Example calculation of inband signal and noise
B. Example of all the computation steps to determine CPoD
To determine the CPoD, the following computation steps have to be performed.

Choose a start time for the submerged vessel.

Determine, along the vessel trajectory and orthogonal distance, the time that the vessel is at this location.

Determine the propagation loss from the submerged vessel to the receiver as function of position.

Determine the received signal as function of position by combining its source level with the propagation loss. Correct the source level with Eq. (7).

Determine the received ambient noise as function of time; this includes both marine traffic noise and wind noise. Correct for the scenario frequency band [Eq. (6)].

Combine the received signal of step 4 with the received ambient noise of step 5, to determine the SNR [Eq. (2)].

Compute the probability of detection [Eq. (3)].

Compute the CPoD as function of vessel position [Eq. (5)].

Compute the detection range. Take the CPoD at the end of the submerged vessel's trajectory, this is CPoD_{∞}, and determine when it crosses 0.5.
An example is shown in Fig. 2, where the starting time is chosen to be the time of the highest windspeed: the 8th of September at 09:00. The vessel moves at 3 m/s (see Table I), so the vessel locations as function of range or time could be determined. The signal in the figures appears as half ellipsoids; this is because the results are not at 11 aspect ratio. In real space, they would be perfect half circles; but for plotting purposes, this is not shown.
IV. RESULTS
A. Variation of CPoD_{∞} with orthogonal distance
The maximum cumulative probability of detection along a track is reached at the end of the submerged vessel track. When analyzing the maximum CPoD, it becomes clear that it does not always decrease monotonically as function of the orthogonal distance away from the receiver. This has to do with both the propagation of signal and noise, as well as the time at which the vessel is at a certain location.
An example of such a result is shown in Fig. 3. In this figure, CPoD_{∞} crosses the 0.5 line five times. The lowest range that it crosses the 0.5 line at is called the defender detection range, and the largest range this happens at is called the intruder detection range.
B. Reproducibility of results across groups
Three groups generated results for the detection scenario. Here, a few example figures are shown for the same case: a receiver depth of 70 m, a frequency of 630.96 Hz. The vessel starts 30 km away from the CPA to the buoy, at the time of the highest windspeed, and therefore highest wind noise. Figure 4 shows the results for group B, for which a ray tracer is used with incoherent addition of the rays. In general, incoherent ray tracers produce smoother propagation results. Figure 5 shows the results for group E, where a parabolic equation model is used. Figure 6 shows the results for group G, who also used a parabolic equation model.
Figures 4–6 show that:

The received signal has a similar shape for all the groups: rings around the receiver. These rings look like ellipsoids, but that is due to an aspect ratio that is not equal to 1:1. These are coming from the interference patterns in the propagation loss applied to the tonal.

The received signal absolute values are close but not the same for the different modelling groups. It is expected that this is due to differences in propagation modelling methods (ray tracer vs parabolic equation modelling).

The noise at the receiver is similar in level but is different over the groups. On the one hand, this is due to differences in propagation modelling, especially near the air–water surface. But it also has a time component. The submerged vessel is moving toward the northwest, at the same time the merchant vessels are moving as well. And finally, the wind changes over time. Interestingly, the noise in Figs. 4 and 6 looks more alike than in Figs. 5 and 6.

The resulting SNR ( $10\u22c5log10(R)$) seems to vary between the groups, but the shape is still similar: rings around the receiver, which decrease in value further away from the receiver.

The resulting CPoD for the groups is somewhat similar in shape but leads to different detection ranges.

It is not easy to map the SNR directly to the CPoD. As can be seen in the figures, it is very beneficial for the CPoD if there is a high SNR region at the start of the track. That leads to a higher local probability of detection, which drastically increases the CPoD. This is especially visible in Fig. 5, where the SNR is higher at the start of the submerged vessel track and has a lower SNR at the end of the submerged vessel track.
C. CPoD_{∞} as function of frequency
The CPoD is a challenging function to compare. A comparison that can be made is by looking at the frequency spread in CPoD_{∞} for a given receiver depth and start time. Figure 7 shows the defender and intruder detection ranges as function of frequency. The results for different groups and models at the same frequencies are spread horizontally, to better compare them. Here, the receiver depth is chosen to be 70 m, and the submerged vessel start time is September 8 at 9:00 in the morning. The different models are shown in different colors.
The following can be seen in the figure:

At 63 Hz, the models all show the maximum detection range of 10 km in the orthogonal direction. The three simulations agree very well; all three were performed with parabolic equation propagation modelling. No other propagation models were considered valid at this low frequency.

At 631 Hz, there are more propagation models. There is a bigger spread in the results, but the detection range lies within a factor two (5 and 10 km), except for one outlier. The outlier in this case is due to a simple mode stripping propagation model, which is both depthindependent and is only a function of range and water depth.

At 631 Hz, there are some models that have defender and intruder detection range that are not equal. In those cases, a phenomenon like that in Fig. 3 occurs.

At 6310 Hz, the fractional spread has become much larger, with detection ranges of 200–700 m.

At 6310 Hz, there are models that still do not have a CPoD_{∞} larger than 0.5 at 100 m in the orthogonal direction, such as the ray tracer with and without scattering of group B. Since 100 m CPA was the minimum, it is unknown what the detection range would be according to those models.
D. CPoD as function of receiver depth
During the simulations, two receiver depths were considered: 10 and 70 m. Varying the receiver depth can give new insights. The frequency and submerged vessel starting time are fixed.
Figure 8 shows such a variation. The figure shows the following:

The detection ranges are higher for the receiver at 70 m depth than for 10 m depth.

The spread in the results is higher for the 10 m receiver depth scenario.

There are many models that report the same defender and intruder detection range.

The detection range of the mode stripping formula is the same (green markers). This is because this model does not take into account depth dependence.
E. CPoD as function of starting times
Like varying receiver depths, varying the submerged vessel starting times can give new insights.
The starting time of September 7 at 3:00 in the morning was the time of the lowest windspeed. September 8 at 9:00 was the starting time with the most wind.
Two subplots are shown in Fig. 9. On the left, the results can be seen for the 631 Hz scenario. Here, the detection ranges are similar. This is because the wind noise is much smaller than the shipping noise at this frequency (see Fig. 10). The variations in detection ranges are due to changing shipping locations as the start time varies.
On the right plot, the frequency is 6310 Hz. There is a big difference with the two starting times. For the 6310 Hz, this difference can mostly be attributed to the wind noise. As can be seen in Fig. 10, at 6310 Hz, the wind noise is similar to the shipping noise. At the maximum wind speed, it is even dominating. For the 631 Hz scenario, this is not the case.
F. CPoD changes when including surface scattering
Changes in windspeed may have strong effects on both signal and noise. Sound propagating from the target and from distant ships may suffer scattering and absorption losses when reflecting from the rough sea surface and when travelling through nearsurface layers of entrained bubbles. This effect will tend to reduce the level of both signal and noise, but it is not obvious which of these will be reduced more and consequently what will be the net effect on SNR of propagation changes. Increased wind speed will certainly increase windgenerated noise, and this will reduce SNR.
While several groups included surface loss in their wind noise modelling, group B also provided ray model results with and without, including surface scattering in the propagation modelling for signal and shipping noise as well. For a given wind speed, a rootmeansquare surface roughness is computed assuming a PiersonMoskowitz (Pierson and Moskowitz, 1964; Chapman, 1983) spectrum for the statistics of the sea surface waves. The ray model then modifies the surface reflection accordingly. For the scenario with 631 Hz and a receiver at 70 m, the resulting effect is worked out in more detail in Fig. 11. As can be seen, and as would be expected, both signal and noise levels are lower with surface scatter included. Surface scattering seems to have a larger influence on the noise than on the signal; hence, the SNR and CPoD are larger when including surface scattering in the propagation modelling.
V. DISCUSSION
Section IV shows how the CPoD over the track, and CPoD_{∞} at the end of the track, can change not just as a function of the model parameters, but as a result of the modelling techniques used. When considering the sensitivity of the detection probability [Eq. (3)] to changes in SNR, this is not surprising. As was shown in the Ambient Noise Modelling workshop overview paper (Martin , 2024), differences in modelling can lead to differences in predicted ambient sound levels on the order of several decibels. A difference of 2 dB in the modelled SNR can lead to a difference as large as ∼0.15 in the detection probability for a single range cell. When these differences in detection probability accrue over a whole track, this can lead to large variations in detection range. Moreover, the detection probability is most sensitive for modest but positive values of the SNR (in the range of 10–20 dB), which are precisely the conditions where detection is marginal.
In interpreting the results, there is still some ambiguity in the choice of intruder versus defender detection range; however, our findings indicate that the two are usually very similar, so in most cases it may be safe to simply use the defender range. The strong frequency dependence is not surprising since both the ambient sound levels and the source levels were explicitly frequencydependent as well. Although the source levels were somewhat arbitrarily chosen for this scenario, in real world scenarios they depend on the specific source and its tonals.
The depth dependence, and especially the shorter detection ranges, for the 10 m hydrophone can probably be understood as a result of the propagation environment and the scenario geometry. Since the sound speed contains a surface layer (Fig. 1), the 10 m hydrophone will receive more energy from the merchant ships, which are simulated as point sources at 6 m source depth, than the 70 m phone, which is at the sound speed minimum. Similar results in this environment show the same behavior (Martin , 2024), where the 63 Hz soundscape levels are found to be higher for the 10 m phone than the 70 m phone. In addition, variations exist across propagation models in the way in which nearsurface sources and receivers are modelled. Fullwave models, such as parabolic equation models, implicitly include a reduction in sound level at distances of a few wavelengths from the (pressurerelease) sea surface. This “surface decoupling” effect leads to high propagation losses, but this phenomenon has to be explicitly taken into account in raybased models.
As can be expected, the starting time of the vessel matters. When there is a lot of environmental noise, it is more difficult to detect the submerged vessel. At times when there is less environmental noise, it is easier to detect the target. This mostly holds true for the higher frequencies, as the lower frequencies are more dominated by the shipping noise. For the lower frequencies, it is therefore more important to know what the locations are of merchant ships in the area; whereas for higher frequencies, the wind and rain conditions are more important.
In Sec. IV, it is shown that including surface reflection as function of windspeed can change the result for the CPoD, as it changes both the signal and noise part of the SNR. Although not all propagation models can handle the change in surface roughness, there were no major outliers when inspecting the CPoD. From a physics point of view, it would be better to include changes in surface roughness in the calculations, if the models allow it. However, from a practical point of view, including surface roughness in the calculations only increases or reduces the CPoD in the same range of uncertainty as using another propagation model.
As the results in this work show, the CPoD can be another tool to understand whether a submerged vessel could pass a passive receiver without being detected. If environmental data, such as windspeed, as well as anthropogenic data, such as shipping, are available, the CPoD can help to determine the placing of a receiver. The operating frequency determines whether environmental data, such as windspeed or precipitation, or anthropogenic data are more prevalent and important for the calculations. Work from the literature (e.g., Wenz, 1962; Ainslie , 2011; Prior , 2019) can help to determine the required level of anthropogenic and environmental details necessary for the considered frequency.
Furthermore, CPoD could be used to determine whether a specific intruder track could be detected by a receiver. Although in this work the tracks of the submerged vessel were straight lines, the method could be used for curved tracks.
VI. EXTENSIONS AND FUTURE WORK
This work could be extended to marine mammal detections and passive monitoring. By replacing the tonals of the submerged vessel with a source model for a vocalizing mammal, one could perform calculations similar to those shown in this work. It is to be expected that marine mammals would not make continuous sound, so it is likely that one must combine this method with a Monte Carlo method. The detection range would now serve as an “effective monitoring range” for a single hydrophone.
Finally, it is worth considering what other metrics could play a similar role to detection range in future research. One possibility is to use the detection currency (Abraham 2023), which is an alternative to detection and false alarm probability based on information theory and relative entropy. Since this scenario successfully highlights the full range of considerations that go into a detector model, it would be interesting to do follow on calculations with other metrics for detector performance.
SUPPLEMENTARY MATERIAL
See the supplementary material for the CPoD∞ data at the end of the vessel track. The models are in random order, such that they cannot be directly linked to the groups. These can be used to reproduce Figs. 3, 7, 8, and 9. Furthermore, they can be used to look into the few cases that have been computed but have not been shown in the figures.
ACKNOWLEDGMENTS
This paper is part of the Berlin workshop. The authors of this work, and the organizers of the workshop (Martin Siderius, Michael Ainslie, Michele Halvorsen, Leila Hatch, Bruce Martin, and Mark Prior), thank the Workshop Scientific Committee of Ross Chapman, Kevin Heaney, and Christ de Jong for insights and support during the development of the workshop. This work was supported by the U.S. Office of Naval Research, Rijkswaterstaat, and U.S. National Oceanographic and Atmospheric Administration for organizing and conducting the workshop.
AUTHOR DECLARATIONS
Conflict of Interest
The authors declare that they have no conflict of interest with respect to the results presented in this work.
DATA AVAILABILITY
The data that support the findings of this study are available from the corresponding author upon reasonable request.