Questions

1. Does an array of sonar transducers separated by hundreds of metres or kilometres give sensitivity advantages over conventional towed arrays for detecting surface ships and submarines?
2. How do vertically suspended arrays compare with towed arrays?
3. If each distributed node is itself a vertical array, does that offer desirable traits?

Approach. Published measurements and systems (notebook 01) combined with simulations built for this study: array gain with ocean coherence limits and directional noise, sonar-equation coverage maps, focused-beamforming and time-of-arrival localisation, normal-mode depth discrimination, and finite-difference time-domain (FDTD) wave simulations across a thermocline (notebooks 02–05). Node positions and clocks are assumed known, as the brief allows; the conclusions concern physics, not engineering or cost.

Answers in brief

Q1 · Widely spaced nodes versus a towed array

Not for sensitivity: spacing itself buys no gain. Array gain against noise comes from the number of sensors whose signals add coherently. Ocean signals stay coherent over only ~20–40 wavelengths in shallow water (≈100–200 m at 300 Hz), so nodes hundreds of metres to kilometres apart cannot be added coherently across most of the passive band. Their outputs can only be fused incoherently, worth about 5 log₁₀ M dB for a steady signal. In the coverage model (150 m shelf, 300 Hz), nine single-hydrophone nodes need a target ~14 dB louder than a 1.5 km towed array needs to cover the same 20 × 20 km box, despite being closer to it.

Wide spacing wins on localisation: a position fix from one event (when 3–4 nodes hear it), with no left/right ambiguity. It also brings freedom from tow-platform noise, diversity against fading and target aspect, and the choice of where to put sensors. A coherent sum across 5 km-spaced nodes beats plain incoherent fusion only below ~8–30 Hz, depending on the coherence length.

Q2 · Vertical versus towed arrays

A vertical array trades bearing for depth; whether it has more gain depends on the water. For equal element counts (32 elements, 300 Hz):

On a shelf the target's signal is spread over many modes and wind noise fills the deep-water notch, so the horizontal array has more gain. The vertical array's advantages are information: it spans the thermocline (a fixed-depth array gains or loses 1–2 dB relative to the column average depending on the side of the layer the target is), and it can tell a surface ship from a submarine by estimating source depth, which a broadside towed array cannot. It gives no bearing on its own, and it does not move.

Q3 · A network of vertical arrays

Yes: roughly towed-array sensitivity, plus 3D position and classification.

• Sensitivity: on a shelf each node brings ~6–14 dB of array gain and the network adds proximity. 9 VLA nodes with matched-field processing cover the box against the same targets as the 1.5 km towed array (~101 dB for a surface ship) and ~3 dB quieter ones when the target is below the layer. That is an upper bound (known environment and range, no vertical-coherence loss); with conventional beams they need targets ~2–8 dB louder. Real tow self-noise (3–6 dB) moves the balance towards the network.
• Localisation: horizontal position from the network geometry, and depth and range from each node.
• Classification: 92–100% correct surface/submerged calls from about −10 dB SNR per hydrophone in the model.

The ocean still forbids coherent gain between nodes at normal passive frequencies. The case for the concept rests on what it adds at equal sensitivity; the open questions are engineering ones: moorings, power, communications and suspension noise.

What drives the answers (notebook 06)

• Coherence caps array gain at about 10 log₁₀(4 L_c/λ): ~21 dB for L_c = 30 λ, ~26 dB for 100 λ. Longer arrays buy resolution, not sensitivity.
• Frequency: small (hull) apertures are pushed up in frequency until absorption and bottom loss stop them (a 5 m bow array: a few kHz); long apertures are capped by coherence and want low frequency.
• The thermocline is large in deep water at mid frequency: a submarine below a 60 m layer is ~9–12 dB (1 kHz) to ~26–34 dB (3.5 kHz) harder for a hull sonar in the layer to hear than a surface source. It is transparent at 100 Hz and only ~5–7 dB on a 150 m shelf, where the seabed refills the column.

A 10-USV formation at 100 Hz (notebook 07)

Ten USVs with one hydrophone each at λ/2 make a 70 m array: ~11 dB, about 10 dB short of a 1.5 km towed array. Giving each USV a dipped 20-hydrophone vertical line (200 hydrophones) brings it within 3–4 dB with matched-field processing, and ahead by 3–5 dB only if the vehicles and lines are much quieter than the tow. The requirements are hydrophone count, quietness, an environment model, and ~1 m knowledge of every hydrophone including line tilt.

Key figures

Array gain is set by coherent element count, not aperture (notebook 02)

Coherent gain across nodes 5 km apart vanishes above a few tens of Hz (notebook 02)

Coverage against quiet targets: towed array, single-hydrophone nodes and VLA nodes (notebooks 02 and 04)

Horizontal versus vertical array gain: deep-water picture, then a shallow shelf (notebook 03)

Where the energy arrives: surface ship versus submarine below the layer (notebook 03, from FDTD)

Only the vertical array (or an endfire towed array) sees source depth (notebook 03)

Pre-rendered animations

Every case is a set of named parameters in sonarsim/cases.py; notebook 05 shows how to change them and render your own.

Assumptions and limits

• Coherence is modelled as exponential decorrelation with length $L_c$ (20–100 λ swept; 30 λ default, from Carey's shallow-water measurements). Real coherence varies with internal waves, range and path.
• Propagation for coverage is range-only: spherical to the water depth, cylindrical beyond, with absorption and 0.5 dB/km bottom loss. Depth structure (the layer) is applied from two FDTD cases at 100 Hz, not a general law.
• Noise is modelled, not measured: a free-field directional model (wind from above, shipping near horizontal) for the deep-water picture, and Kuperman–Ingenito normal-mode noise in a Pekeris waveguide for the shelf.
• Vertical-array gain on the shelf assumes perfect vertical coherence of the modal field; matched-field figures assume the environment is known. Real mismatch lowers both.
• Detection uses a steady signal, $P_d \approx 0.6$, $P_{fa} = 10^{-4}$, no fading, no tow self-noise, and conventional (not adaptive) beamforming. All coverage is on a 150 m shelf at 300 Hz; deep-water, low-frequency surveillance is outside the model.
• FDTD is 2D: sources are line sources and spreading is cylindrical, so absolute levels are not 3D levels. Depth structure, refraction and multipath are physical.
• Matched-field depth estimation uses a known Pekeris waveguide; real environmental uncertainty widens the peaks.
• Target source levels are swept (90–135 dB re 1 µPa²/Hz at 1 m), because real submarine levels are not public.
• Not modelled: cost, deployment, communications, power, platform motion and strumming, active sonar in detail.

Using this report

• Hosted: open the notebooks in the browser (JupyterLite). The code runs in your browser; heavy FDTD cases have a quick=True preview mode.
• Locally: pip install numpy scipy matplotlib pillow jupyterlab, then jupyter lab in this folder. ffmpeg enables MP4 output.
• Regenerate everything: python scripts/render_cases.py && python scripts/build_notebooks.py.

Sources

Carey (1998), The determination of signal coherence length based on signal coherence and gain measurements in deep and shallow water, JASA 104, 831 (link) and the Carey number review in Acoustics Today (pdf) · TB-16 (GlobalSecurity) · SURTASS (Wikipedia; GlobalSecurity) · SOSUS (DOSITS) · DARPA DASH / TRAPS (DARPA) · Noise directionality (Internoise 2014; Kuperman & Ingenito 1980) · VLA depth discrimination (JASA EL 2016); HLA mode-subspace depth discrimination (JASA 2013) · Reliable acoustic path (JASA 2009) · Random arrays (Lo 1964) · Towed-array left/right ambiguity (Kaouri) · Multistatic sonar (Fewell & Ozols; multistatic sonobuoy fields) · Textbooks: Urick, Principles of Underwater Sound (1983); Jensen, Kuperman, Porter & Schmidt, Computational Ocean Acoustics (2011); Albersheim (1981), IEEE Trans. AES 17, 131.

01 · Background: arrays, coherence and the sonar equation

This notebook sets out the physics and the published facts the rest of the study relies on. The later notebooks (02–04) answer the three questions with simulations; the summary is in 00_summary.

Scope. Passive detection of surface ships and submarines by the sound they radiate, mainly at 10 Hz–1 kHz, where machinery tonals and broadband cavitation noise propagate furthest. Nodes are assumed to know their positions and share synchronised clocks, as the brief allows. Active (echo-ranging) operation is touched on where it changes a conclusion.

1. The passive sonar equation

A detection needs the signal after processing to exceed a threshold:

$$\mathrm{SNR} = SL - TL - (NL - AG) \;\geq\; DT$$

Every design question in this study moves one of these terms. A wide sensor spacing does not change $AG$ by itself; it changes $TL$ (sensors nearer the target), the geometry available for localisation, and what can be combined coherently. The notebooks keep these effects separate.

2. Array gain and why it depends on coherence

For $N$ hydrophones whose signals add in phase while their noises are independent, delay-and-sum beamforming gives

$$AG = 10\log_{10} N .$$

Two conditions hide in that formula:

1. The noise must be uncorrelated between elements. In isotropic noise this holds for spacing $\geq \lambda/2$; in directional noise (wind noise from above, shipping near the horizontal) the array shape decides how much noise it rejects. This is where vertical and horizontal arrays differ (notebook 03).
2. The signal must stay coherent across the array. The ocean scrambles the phase of a signal over distance (internal waves, multipath, fluctuating paths). The signal coherence length $L_c$ is the separation over which coherence is lost. Beyond it, extra aperture adds noise and signal equally and the gain saturates.

Measured coherence lengths. Carey (1998) combined coherence and array-gain measurements and found horizontal coherence lengths in shallow water of about 20–40 wavelengths, averaging 30 λ, now called the "Carey number" (Acoustics Today review; Carey 1998, JASA 104, 831). The SW06 experiment at 200 Hz found the same 20–40 λ range, falling sharply while non-linear internal-wave trains crossed the path. Deep-water paths are generally more coherent, but by how much varies with range, frequency and path; the simulations therefore sweep $L_c$ instead of assuming one value.

In metres, 30 λ is small at the frequencies most passive sonars use:

At 300 Hz, 30 λ is about 150 m; at 30 Hz it is 1.5 km. Nodes spaced hundreds of metres to kilometres apart are therefore mutually incoherent across most of the passive band. Only in the lowest band (roughly 5–30 Hz), and more so in deep water, can kilometre-scale spacings stay partly coherent. Notebook 02 turns this into gain numbers.

3. The reference systems

Known towed-array limitations: a single line cannot tell which side a sound comes from (left/right ambiguity, e.g. Kaouri, Oxford thesis); it sits at one depth set by speed and cable scope; it hears its own tow platform and flow noise, worst towards the forward endfire; and its shape wanders when the ship turns, blurring the beams.

4. Ambient noise is directional

Wind and waves make noise at the surface; it arrives at a submerged sensor mainly from above, strongest near the vertical. Distant shipping arrives near the horizontal, because only low-angle paths survive long ranges. In downward-refracting water a noise notch forms near the horizontal (Internoise 2014 review; Kuperman & Ingenito 1980). The same review notes that vertical arrays steered horizontally gain much more than horizontal arrays in surface-dominated noise. Notebook 03 models this with a two-component directionality: wind noise $\propto \sin(\text{elevation})$ from above (dipole surface sources) with partial bottom reflection, plus a band of shipping noise near the horizontal.

5. Vertical arrays and source depth

In a waveguide, the field from a source is a sum of normal modes whose shapes depend on depth. A vertical array samples those shapes directly, so it can estimate source depth by matched-field processing or mode filtering, which tells a surface ship from a submarine (Source depth discrimination with a vertical line array, JASA 2016). A horizontal array can do this only when it is long and the source is near endfire, using mode wavenumbers instead of mode shapes (mode subspace projections with an HLA, JASA 2013). Deep vertical arrays can also exploit the reliable acoustic path, a direct refracted path from near-surface sources to a deep receiver (Passive detection using the RAP, JASA 2009).

6. Sparse arrays

Elements spaced much more than λ/2 produce grating lobes: copies of the main beam in other directions. Randomising the positions spreads them into a sidelobe floor whose average power is about $1/N$ of the main lobe (Lo 1964). A network of 9 nodes therefore focuses with sidelobes averaging only about −10 dB even when perfectly coherent. Broadband processing helps: the grating lobes of different frequencies fall in different places and average out, which is why the animations use a broadband transient.

7. Combining incoherent sensors

When sensors cannot be combined coherently, their detector outputs can still be fused. For $M$ equal-SNR independent looks, the required per-look SNR falls by roughly $5\log_{10}M$ (Albersheim's equation gives 6.9 dB for $M = 9$ at $P_d=0.5$, $P_{fa}=10^{-4}$). The optimal weighted fusion used in the coverage maps adds deflections in quadrature, $\mathrm{SNR}_{\text{eff}} = \sqrt{\sum \mathrm{SNR}_i^2}$, the same $5\log_{10}M$ law for equal SNRs. Coherent combination would give $10\log_{10}M$, so incoherent fusion recovers only half the decibels. That holds for a steady signal; when the signal fades (multipath, aspect-dependent radiation) and a high $P_d$ is required, independent looks also bring diversity gain, which can be worth several dB more. Multistatic sonobuoy studies reach similar conclusions: the benefit of many receivers comes mainly from geometry and coverage, not per-receiver gain (Fewell & Ozols, DSTO; design of multistatic sonobuoy fields).

02 · Q1: do nodes hundreds of metres to kilometres apart beat a towed array?

Does an array of sonar transducers, separated by hundreds of metres or kilometres, provide sensitivity advantages for surface or underwater vessel detection, compared with conventional towed sonar arrays?

We separate the question into four parts, each with its own simulation:

1. Array gain. Does spreading sensors out raise the gain against noise? (§1–2)
2. Coherent combination across nodes. When, if ever, can widely spaced nodes add in phase? (§3)
3. Detection coverage. Including the effect of sensors being closer to targets. (§4)
4. Localisation. Where wide spacing clearly wins. (§5)

Every parameter cell is marked Parameters and can be edited and re-run.

1. Array gain depends on the number of coherent elements, not the aperture

Two families of line arrays at 300 Hz (λ = 5 m), in spatially white noise:

• filled: λ/2 spacing, so the element count grows with length;
• sparse: always 16 elements, spread evenly over the same length.

With a perfectly coherent signal, the sparse array's gain stays at $10\log_{10}16 = 12$ dB however large it gets: spacing buys resolution, not gain. With a realistic coherence length the sparse array is worse still, because its elements fall outside each other's coherence.

Reading the plot. A filled 1.5 km line at 300 Hz has about 600 elements and would give 28 dB if the ocean were perfectly coherent; with $L_c$ = 30 λ it gives about 20 dB. The sparse 16-element array starts at 12 dB and, summed coherently (delay-and-sum), falls towards 0 dB (one hydrophone's worth) once its spacing exceeds the coherence length, because a coherent sum of incoherent signals adds noise as fast as signal. Detecting at each element and fusing the outputs incoherently would keep about $5\log_{10}16 \approx 6$ dB instead; either way, spacing buys no gain.

2. How much gain does a real towed array keep?

The long array is where coherence bites: the 1.5 km array gives up 3–9 dB of its ideal gain at 300 Hz and 7–14 dB at 1 kHz, depending on the coherence length. At 30 Hz the same array is only 15 wavelengths long and stays nearly coherent. This is the physical reason surveillance arrays are long and low-frequency.

3. Can kilometre-spaced nodes be combined coherently?

Take 9 nodes on a 3 × 3 grid, 5 km apart. If their signals were coherent, a coherent sum would focus with $10\log_{10}9 = 9.5$ dB of network gain. The fraction actually realised is $\frac{1}{M^2}\sum_{ij}\rho_{ij}$ with $\rho_{ij} = e^{-d_{ij}/L_c}$:

With $L_c$ = 30 λ the coherent network gain across 5 km-spaced nodes is under 1 dB above ~30 Hz; with an optimistic 100 λ it is still ~2.5 dB at 50 Hz and ~0.5 dB at 100 Hz. Incoherent fusion of the node outputs gives about $5\log_{10}M$ (4.8 dB for 9 equal nodes; see notebook 01) without any coherence, so a coherent sum is only worth attempting where it beats that: below ~8 Hz ($L_c$ = 30 λ) to ~28 Hz (100 λ). At those frequencies a 150 m shelf is close to modal cutoff, so the low-frequency end of this plot is a deep-water result.

Conclusion for sensitivity: widely spaced nodes add no array gain beyond what each node has on its own, plus a modest incoherent fusion gain. A towed array with hundreds of closely spaced hydrophones has 15–25 dB more gain than a single-hydrophone node.

4. Detection coverage: the proximity effect

Array gain is not the whole sonar equation. Distributed nodes are closer to wherever the target is, which cuts transmission loss. The model below compares, over a 20 km × 20 km box in 150 m-deep water:

• a 1.5 km towed array at the centre (30 m deep), with its coherence-limited gain in shallow-water ambient noise (wind and distant shipping, normal-mode model; see notebook 03, §2) and an optional self-noise penalty;
• 9 or 25 single-hydrophone nodes on a grid, fused incoherently.

Transmission loss is spherical to 150 m, cylindrical beyond, plus seawater absorption and 0.5 dB/km of bottom loss. Noise 65 dB re 1 µPa²/Hz (moderate shipping and sea state 3), bandwidth 100 Hz, 10 s integration, $P_d \approx 0.6$, $P_{fa}=10^{-4}$, steady (non-fading) signal. Source levels are swept, because modern submarine levels are not public and the comparison is more robust than any single number.

Reading the coverage results. The towed array covers 90% of the box against targets down to about 104 dB. Nine single-hydrophone nodes need about 116 dB, a target 12 dB louder, and even 25 nodes need about 112 dB. Being closer to the target does not make up for the missing array gain of single hydrophones at these ranges and frequencies. That changes when each node is itself an array (notebook 04), when the towed array is noisy (try TOWED_SELF_NOISE_DB = 6), or when transmission loss is steep (high frequency, lossy bottoms: try EXTRA_LOSS = 2).

5. Localisation: where wide spacing clearly wins

A towed array measures bearing. Its angular resolution is about λ/L; its range is unobserved in a single look unless the target is inside the near field ($\sim L^2/\lambda$), and a single line cannot tell left from right. Widely spaced nodes measure position, from arrival-time differences across long baselines.

The animations show both at 15 Hz (λ = 100 m), so the waves are visible at 16 km scale. A transient from the star spreads out; each sensor records it; the recordings are replayed time-reversed, and the re-emitted waves refocus on the source. The running maximum of that field is the delay-and-sum focused beamformer image.

The towed array focuses to a bearing band, not a point, with an identical mirror band on the other side of the line (left/right ambiguity). In the zoom, the band is about 400 m across, which is λ/L × range = 100 m / 1.5 km × 5.5 km, and it runs the full length of the bearing: range is unresolved.

Nine coherent nodes 5 km apart focus to a spot about half a wavelength across (~50 m), with a sidelobe pattern of radial streaks around it: no ambiguity, and the position is fixed in one look. But this assumes the nodes are coherent, which §3 showed is realistic only below ~10–30 Hz. The next case adds random path errors of 40 m (0.4 λ) between nodes, a mild stand-in for the medium decorrelating the signal between sites:

The coherent focus collapses (peak 0.44 instead of 1) and fragments. Without coherence, the nodes still localise by arrival-time differences (TDOA) of the signal envelope, which needs only that each node detect the same event. Accuracy is then set by timing precision, not wavelength. The map below is the Cramér–Rao bound for TDOA localisation with the 9-node grid, assuming each node times the arrival to 30 m-equivalent (20 ms): a pessimistic figure allowing for multipath.

Inside the node field the position error is 20–40 m from a single event, degrading outside it. A single towed array gets bearing only; range needs either several minutes of own-ship manoeuvre (target motion analysis) or a second array.

Answer to Q1

For sensitivity, no. Array gain against noise comes from the number of sensors whose signals add coherently. The ocean keeps signals coherent over only ~20–40 wavelengths in shallow water (≈100–200 m at 300 Hz), so nodes hundreds of metres to kilometres apart cannot be added coherently across most of the passive band. Their outputs can only be fused incoherently, worth about $5\log_{10}M$ dB. Nine single-hydrophone nodes therefore have ~15 dB less combined gain than a 1.5 km towed array with ~600 hydrophones (20 dB coherence-limited), and in the coverage model they need a target ~12 dB louder to cover the same 20 × 20 km box, despite being closer to it.

For the system as a whole, it depends what is at each node. Spreading sensors out does bring real advantages: (i) proximity, which pays off when each node already has meaningful gain (notebook 04) or transmission loss is steep; (ii) localisation: position from one event, with no left/right ambiguity; (iii) freedom from tow-platform self-noise and the option to put sensors where propagation is best (below the layer, on the reliable acoustic path); (iv) for active sonar, multistatic geometry.

Coherent processing across nodes is physically possible only at very low frequency (≲10 Hz in shallow water, ≲30 Hz in deep water for 5 km spacing), where it would give a very sharp focus with ~1/M sidelobes, but still only $10\log_{10}(\text{total sensors})$ of gain.

03 · Q2: how do vertically suspended arrays compare with towed arrays?

What about vertically suspended sonar arrays — how do/would they compare to towed arrays?

A towed array is a horizontal line at one depth; a suspended array is a vertical line spanning part of the water column. The two see different things:

Five comparisons follow: array gain in the free-field (deep-water) picture of directional noise (§1) and in a shallow-water waveguide (§2), bearing (§3), coverage across a thermocline (§4), and telling a surface ship from a submarine (§5).

1. Array gain in directional noise: the free-field (deep-water) picture

Both arrays have 32 elements at λ/2 (300 Hz, 77 m long). The noise field mixes wind noise from above (∝ sin elevation, with a fraction reflected from the bottom) with distant shipping near the horizontal (±6°, all azimuths). The target is a plane wave at broadside for the horizontal array and at the horizon for the vertical array. This is the classic deep-water picture, with a noise "notch" near the horizontal; §2 repeats the comparison in a shallow waveguide.

When wind noise dominates (quiet shipping, rough sea), the vertical array rejects the steep surface noise and gains ~9 dB more than the horizontal array. The horizontal array's broadside beam is a vertical fan that still takes noise from overhead. When distant shipping dominates, noise arrives near the horizontal from every azimuth, straight into the vertical array's horizontal beam, and the horizontal array wins by ~8 dB because it rejects shipping from other bearings. In this picture the answer depends on the noise environment: quiet, windy or remote deep waters favour the vertical array; busy shipping lanes favour the horizontal one. The vertical figures are upper bounds: they assume the whole signal arrives as one plane wave from the horizon, perfectly coherent down the array.

2. Array gain in a shallow-water waveguide

On a shelf both assumptions of §1 fail. The target's signal is a sum of normal modes, arriving at a spread of vertical angles up to the critical angle, so no single vertical beam collects all of it. And wind noise is carried to the array mainly by the same low-order modes (Kuperman & Ingenito 1980), which fills the deep-water notch. The model below uses the Pekeris waveguide of §5 (150 m of water over a sand-like bottom, 300 Hz, 28 modes) for both signal and noise:

• noise: Kuperman–Ingenito wind noise from a sheet of near-surface sources, plus distant shipping (sources beyond 10 km, which reach the array mostly in low-order modes), plus a white floor 20 dB down;
• horizontal array: 32 elements at λ/2, 30 m deep, plane-wave signal with the Carey coherence of notebook 02;
• vertical array: 32 elements spanning 10–140 m, signal = the modelled field of sources at 5–120 m depth and 2–10 km range, processed two ways: a fan of conventional plane-wave beams (best beam kept), or matched-field weights (the modelled field itself, which needs the environment and range to be known).

In shallow water the ranking flips. The horizontal array keeps ~16 dB whatever the noise mix. The vertical array loses its deep-water advantage: with plane-wave beams it keeps only ~7–12 dB (the high end only when shipping dominates), because the signal is spread over many modes and the noise shares those modes; matched-field processing recovers ~13–14 dB, still 2–3 dB below the horizontal array, and only if the environment is known well enough. Real vertical coherence is also imperfect (internal waves), which this model leaves out. So for sensitivity alone, a shelf favours the horizontal array.

3. Bearing: the vertical array has none

The horizontal array steered to 90° has an equal beam at 270°, its left/right ambiguity. The vertical array responds equally to every azimuth, so a single vertical array gives no bearing. It resolves elevation instead, which a horizontal array at broadside cannot. Bearing from vertical arrays therefore needs several of them (notebook 04), a horizontal arm, or directional (vector) sensors.

4. Coverage across a thermocline

The FDTD runs below simulate a 150 m-deep shelf at 100 Hz: a 40 m mixed layer at 18 °C over a thermocline to 10 °C at 80 m, a sandy sediment, and a sea surface. A 48-element vertical array spans 5–145 m at 3.5 km range; a single hydrophone at 30 m stands in for a towed array at a typical tow depth. The right-hand panel of each movie builds up the energy received at each depth.

For the submarine below the layer the energy concentrates in the lower water column: a sensor at 30 m sits about 7 dB below the best depth, but only ~2 dB below the water-column average; for the surface ship it is ~1 dB above average. A vertical array's output follows the average energy over its aperture, so a full-column array is insensitive to where the layer is, while a fixed-depth array gains or loses a few dB depending on which side of the layer the target is. The effect is real but modest at this range; a towed array can also be lowered below the layer (at the cost of targets above it). (2D runs: levels are for line sources, but the depth distribution is the physically meaningful part.)

5. Surface ship or submarine? Depth discrimination

In a waveguide the depth structure of the received field encodes the source depth. A vertical array samples it directly; matched-field processing compares the data with modelled fields for each trial depth. The model is a Pekeris waveguide (100 m water over a fast sediment, 200 Hz, 13 propagating modes), source at 5 km, 0 dB SNR per element, 20 snapshots, range assumed known.

The vertical array's output peaks sharply at the true depth for both sources: it tells the 5 m surface ship from the 50 m submarine directly. The broadside horizontal array's output is flat: every element sees the same field, so it carries no depth information. At endfire the horizontal array can also resolve depth, through the different horizontal wavenumbers of the modes, but only for targets near its axis, which is also where towed arrays suffer most from tow-platform noise. Real environments are less well known than this model, which widens the peaks, but the contrast between the geometries holds (published VLA results: see notebook 01, §5).

Practical comparison

Answer to Q2

A vertical array trades bearing for depth. For equal element counts its gain depends on the water:

• in deep water, where noise has a notch near the horizontal, it can beat a horizontal array by up to ~9 dB when wind noise dominates, and loses by ~8 dB where distant shipping dominates (§1, upper bounds);
• on a shelf, it has 2–3 dB less gain than a horizontal array with matched-field processing and a well-known environment, and 4–9 dB less with conventional beams (§2).

Its real advantages are information, not gain: it spans the water column, so it is insensitive to the layer (a fixed-depth array gains or loses 1–2 dB here); it can tell surface ships from submarines and estimate range from a single array, which towed arrays mostly cannot; and it has no tow noise. It gives no bearing on its own and does not move, so it complements towed arrays rather than replacing them. The combination points to Q3: several vertical arrays, whose shared geometry supplies the bearing.

04 · Q3: what if each distributed node is a vertical array?

What if each of the sonar nodes in Q1 are vertical arrays, does that offer desirable traits?

Notebooks 02 and 03 found complementary strengths and gaps:

• widely spaced single hydrophones localise well but lack array gain, so they cover little area against quiet targets;
• a vertical array has gain, spans the layer and measures source depth, but gives no bearing.

A network of vertical arrays (VLA nodes) should combine the two: each node brings its own array gain and depth information, and the network geometry supplies the horizontal position. This notebook tests that with three simulations: coverage (§1), coverage when the target hides below the layer (§2), and surface/submerged classification from a VLA node whose range comes from the network (§3).

1. Detection coverage with array gain at every node

The same 20 × 20 km box, 150 m shelf, noise level, bandwidth and propagation as notebook 02. Each VLA node has 32 elements spanning 10–140 m. Its gain comes from the shallow-water normal-mode model of notebook 03 (§2), for the chosen noise mix, with two kinds of processing: conventional plane-wave beams, and matched-field processing (needs a known environment). The towed array's gain is computed in the same noise field.

The depth structure from the FDTD runs (notebook 03, §4) is applied on top of the range-only transmission loss: each configuration receives the average energy over its depth span, relative to the water-column average, for a surface ship (5 m) and for a submarine below the layer (90 m).

With matched-field processing, 9 VLA nodes reach parity with the 1.5 km towed array for the surface ship (both ~101 dB) and are ~3 dB ahead for the submarine below the layer, where the fixed-depth towed array sits in the weaker part of the column. These matched-field figures are an upper bound: they assume the environment and range are known and the modal field is perfectly coherent down the array, while the towed array carries the coherence loss measured at sea. With conventional plane-wave beams the VLA nodes need a target ~2–8 dB louder than the towed array (worst for the surface ship, whose energy sits in high-angle modes). Either way they are far ahead of single-hydrophone nodes (~8–16 dB).

Notes on fairness: 9 VLA nodes use 288 hydrophones against the towed array's 601; the towed array here has no self noise (try TOWED_SELF_NOISE_DB = 3 to 6, typical of real tows, which moves the balance towards the VLA network); the towed array moves and can sweep a larger area over time; and the result depends on the noise mix (try SHIPPING_WEIGHT = 3).

2. Three-dimensional localisation

The network measures horizontal position by arrival-time differences (notebook 02, §5); each VLA measures depth, and from the depth structure also range. Together they give a 3D fix from a single event. The fix is also consistent across nodes, which is a strong check against false alarms.

3. Classification: surface or submerged?

A practical benefit of depth: most contacts in busy waters are surface ships. A node that can say "this source is at 5 m" can drop it before it reaches an operator. The Monte Carlo below places sources at random: half surface ships (2–8 m depth), half submarines (30–90 m), at 2–8 km from a VLA node in a Pekeris waveguide at 200 Hz. The network supplies the range with a ±150 m error; the VLA estimates depth by matched-field processing over depth and a ±300 m range window, and the contact is called submerged if the estimate is deeper than 20 m.

At −10 dB per hydrophone (about +4 dB after 24-element processing) the node calls 92% of contacts correctly, and from −5 dB it is essentially perfect in this idealised waveguide; at −15 dB it falls to 66%, heading for chance. In real water, uncertainty in the sound-speed profile and bottom degrades matched-field processing; robust variants (mode filtering, depth discrimination by interference striations) are the published response (notebook 01, §5).

4. Coherent processing across VLA nodes

Nothing about vertical arrays changes the horizontal coherence limit from Q1: nodes kilometres apart combine coherently only below ~8–30 Hz. VLA nodes do help a very-low-frequency coherent network in one way: each node can steer to a chosen vertical arrival angle before the nodes are combined, rejecting surface noise and multipath that would otherwise decorrelate the inter-node signal.

Answer to Q3

Yes, a network of vertical arrays has most of the desirable traits of both parent designs:

• Sensitivity comparable to a long towed array: on a shelf each node brings ~6–14 dB of array gain (plane-wave beams to matched-field), and the network adds proximity and incoherent fusion. In the model, 9 VLA nodes with matched-field processing cover a 20 × 20 km box as well as a 1.5 km towed array, ~3 dB better when the target is below the layer (an upper bound: known environment and range, no vertical-coherence loss); with conventional beams they are ~2–8 dB worse (§1). In deep water with a noise notch near the horizontal the VLA gain, and so the balance, can be higher (notebook 03, §1).
• Insensitive to the layer: every node spans the water column, so the answer does not depend on which side of the thermocline the target is (§1, right).
• 3D localisation from one event: horizontal position from the network, depth (and range) from each VLA.
• Classification: surface or submerged from each node's depth estimate, which cuts false alarms (§3).
• No left/right ambiguity, no tow noise, quiet deep deployment possible (TRAPS-style, notebook 01).

What it does not give is coherent array gain between nodes at normal passive frequencies; that limit is set by the ocean, not the hardware. The sensitivity gain over a towed array is also modest: roughly parity, with the edge depending on processing, noise and tow self-noise. The case for the concept rests on what it adds at equal sensitivity: 3D position, classification and layer robustness. The costs are practical: many moorings or drifting buoys, power and communications per node, suspension motion and strumming, and a fixed geometry that cannot chase a contact.

06 · What drives sonar performance: design parameters

The earlier notebooks compared specific configurations. This one pulls out the parameters that decide which configuration wins, and uses them to sketch what a good system looks like:

1. Coherence sets a ceiling on array gain (§1).
2. Frequency trades gain and resolution against absorption, bottom loss and target signature; for a given aperture and range there is a best frequency (§2).
3. The thermocline matters a lot in some regimes and little in others; a parabolic-equation (PE) model maps it in deep water and on a shelf at 100 Hz, 1 kHz and 3.5 kHz (§3).
4. Rigid (hull) arrays and an ideal-design sketch (§4).

1. Coherence: the ceiling on array gain

The ocean keeps a signal in step over a coherence length $L_c$, which scales with wavelength: about 20–40 λ on continental shelves (Carey), longer on clean deep-water paths. With exponential decorrelation, the gain of a λ/2-sampled line grows as $10\log_{10}N$ while it is shorter than $L_c$ and then flattens towards $10\log_{10}(4L_c/\lambda)$, whatever its length.

What sets $L_c$:

• the medium: internal waves and sound-speed fluctuations scramble phase, more so at longer range, higher frequency and on summer shelves with a sharp thermocline;
• boundaries: rough sea surface and seabed scatter the signal; bottom-interacting paths decorrelate fastest;
• multipath and modes: interference between paths varies from point to point. In the vertical this structure is deterministic and recoverable with matched-field processing (notebook 03); horizontal decorrelation is random;
• the platform: unknown element positions (towed-array shape) act like a loss of coherence; errors must be below about λ/10 (notebook 07, §4);
• time: target manoeuvres, Doppler and medium changes limit how long a narrowband signal can be integrated.

A longer array beyond one or two coherence lengths buys resolution (narrower beams, range focusing), not sensitivity. More sensitivity needs more coherent hydrophones inside a coherence length: more elements per wavelength² (planar or volumetric arrays) or lower frequency (longer $L_c$ in metres).

2. Choosing the frequency

For a fixed aperture the frequency trades several effects against each other:

An illustrative figure of merit captures the core trade for a passive array of fixed size:

$$\mathrm{FOM}(f) = AG(f) + s\,\log_2(f/1\,\text{kHz}) - \alpha(f)R - b\,(f/1\,\text{kHz})\,R$$

$AG$ grows ~3 dB/octave for a line (6 dB for a square planar array) until the array is a coherence length long; $s$ is how fast the target spectrum falls relative to the noise (target falls ~6 dB/octave, wind noise ~5, so ~−1); $\alpha$ is seawater absorption; $b$ is a frequency-proportional bottom loss for shelf water. The best frequency is where the FOM peaks.

Reading the plot. Two regimes appear:

• Coherence-limited apertures (long lines): gain grows only slowly once the array is longer than ~30 λ, so the best frequency sits near where $L \approx 30\lambda$ and falls slowly with range. A 40 m flank array is best at ~0.7–2.4 kHz (5–30 km, deep water); a 1.5 km towed array near 100 Hz in deep water, falling to 40–65 Hz at 30–100 km on a shelf.
• Small apertures (hull arrays): gain keeps growing with frequency until absorption and bottom loss stop it. A 5 m bow array is best at ~3–7 kHz in deep water out to 30 km (~600 Hz at 100 km) and ~0.5–3 kHz on a shelf. A 2 m array on a small hull wants ~6–9 kHz at 5–10 km. On a shelf beyond ~15–60 km the small arrays' curves drop to the bottom of the plot: bottom loss eats all the gain they could add, so at long range they are no better than one hydrophone at low frequency.

Treat these numbers as illustrative. The last printed lines show how strongly they depend on the assumed slope $s$: for the 1.5 km line the best frequency moves from a few hundred Hz ($s$ = 0) to a few tens of Hz ($s$ = −2), and the frequency-proportional bottom loss is unphysical near modal cutoff. The model also leaves out tonals (which favour low frequency for passive detection and classification), the platform's flow noise (which falls with frequency and favours higher frequency on a moving hull), and the thermocline, which §3 adds. What is robust is the structure: small apertures are pushed up in frequency until absorption and bottom loss stop them; long apertures are capped by coherence and gain nothing from going higher.

3. The thermocline, mapped

The PE model (split-step Fourier; against the normal-mode solution for a 200 Hz shelf: bias 0.3 dB, rms 0.6 dB, and converged in range step) computes transmission loss for two environments with the same kind of mixed layer:

• deep water: 5 km deep, a 60 m mixed layer at 18 °C over a main thermocline, sound-channel axis near 1 km;
• shelf: 150 m deep, the 40 m mixed layer and thermocline of the FDTD runs in notebook 03.

Sources are a surface ship (5 m) and a submarine below the layer (150 m in deep water, 90 m on the shelf). The duct's cutoff frequency (Urick's rule) decides which frequencies it traps:

The maps show the regimes directly:

• 100 Hz is below the duct cutoff: the layer barely exists for it. Sound refracts downward from either source, and near-surface receivers see a deep shadow beyond a few km (made worse by the surface's Lloyd's mirror, which cancels low-angle sound near the surface at low frequency). Energy returns by bottom bounce and at the convergence zone (~55 km).
• 1 kHz and 3.5 kHz: the ship's sound is trapped in the duct and stays loud near the surface; the submarine's sound refracts down and away, and only leaks into the duct weakly.

The table puts numbers on it: incoherent average TL over 8–12 km and 25–35 km for a receiver in the layer (hull depth, 10 m), one just below it (150 m in deep water; 90 m on the shelf), and the best receiver depth for the submarine. The 3.5 kHz deep-water run is truncated at 1500 m (no bottom return; the convergence zone is beyond its 40 km). Near the surface at 100 Hz, TL overstates the SNR loss because ambient noise drops there too (notebook 07, §5).

What the numbers say about the layer:

• The last column is the practical question for a hull sonar in the layer: how much harder is a submarine below the layer to hear than a surface source at the same range and source level? In deep water it is about 9–12 dB at 1 kHz and 26–34 dB at 3.5 kHz: the duct keeps the ship loud and the submarine's sound refracts deep. The effect grows with frequency because the duct traps higher frequencies more tightly. (The 34 dB at 25–35 km is an upper bound: that run has no bottom-bounce return.) This is the classic "below the layer" effect, and in deep water at mid frequency it is large.
• Putting a receiver just below the layer does not fix it in deep water: at 10 km the 150 m receiver is itself in the downward-refracted shadow. At 8–12 km the submarine's sound is loudest at 1.4–1.7 km depth, ~12–16 dB louder than at hull depth at 1–3.5 kHz (and ~38 dB at 100 Hz, where the near-surface shadow is deepest); at 25–35 km the best depth is ~3 km. The best depth moves with range, which is the case for deep vertical aperture, a variable-depth sensor, or a deep fixed receiver.
• At 100 Hz the layer is transparent (below the ~210 Hz duct cutoff). Near-surface receivers suffer instead from the Lloyd's mirror and the deep-water shadow zone, for both sources, until bottom bounce and the convergence zone return the energy.
• On the shelf the seabed returns sound through the whole column, and the layer's effect shrinks to ~5–7 dB at 1–3.5 kHz. At 100 Hz the submarine is actually the louder of the two at hull depth, because a source 5 m below a pressure-release surface radiates poorly at low frequency. For a submarine on the shelf the best receiver depth is close to its own depth, 8–10 dB better than 10 m. The FDTD run in notebook 03 (100 Hz, 3.5 km) found a 30 m receiver 7 dB below the best depth, consistent with this.

These are single environments; the size of the effect varies with the layer's depth and strength, sea state (surface roughness leaks the duct), the seabed and the season.

4. Rigid (hull) arrays, and an ideal-design sketch

What makes a good rigid array:

1. Aperture in both dimensions (cylindrical, spherical or conformal): azimuth and elevation, no left/right ambiguity, vertical rejection of surface noise, some depth information.
2. Design frequency set by aperture: about 10–30 λ across, elements at λ/2 at the top of the band. Hull size is what pushes hull sonar up to 1–5 kHz, exactly where submarines are quietest and the layer bites hardest (§3), which is why navies also tow arrays below the layer.
3. Self noise usually decides performance: flow noise grows roughly 15–18 dB per doubling of speed and machinery noise travels through the hull. Arrays forward and away from machinery, decoupling mounts, baffles and acoustic windows, and "sprint and drift" operation.
4. Processing: adaptive beamforming to null own-ship noise and loud contacts; long narrowband integration on tonals; vector sensors (pressure plus particle motion) so each element has directivity and small arrays resolve left from right.
5. Passive ranging needs baselines: a rigid array focuses in range only within $L^2/\lambda$ (about 200 m for a 10 m array at 3 kHz), so hull sonar ranges passively only with widely separated sub-arrays, over a few km at most.

If the goal is maximum sensitivity and range:

In one line: size the aperture to one to a few coherence lengths at a frequency where absorption and bottom loss over the target range are a few dB, then spend the rest of the budget on quietness, vertical aperture and adaptive processing.

07 · Can a formation of USVs beat a towed array at 100 Hz?

For the multi-USV array, is there a system design where we could do better than a towed array by position controlling, say, 10 USVs at around 0.5 wavelengths and targeting around 100 Hz?

At 100 Hz the wavelength is 15 m, so half-wavelength spacing puts 10 USVs 7.5 m apart. This notebook compares that formation, and variations on it, with towed arrays, in the same shallow-water model used in notebooks 03 and 04:

• waveguide: 150 m of water over a sand-like bottom (Pekeris), 9 propagating modes at 100 Hz;
• noise: wind and distant shipping in equal parts (normal-mode noise), plus optional white self noise;
• signal: a submarine at 90 m or a surface ship at 5 m, 5–15 km away, with horizontal coherence $e^{-d/L_c}$, $L_c$ = 30 λ = 450 m (Carey);
• metric: output SNR relative to one quiet hydrophone at 30 m, so an array that puts its hydrophones in a poor place, or adds self noise, is charged for it;
• processing: matched-field weights (the modelled field of the true waveguide, source depth and range: an upper bound that needs a good environment model), or processing that needs no environment model: geometric focusing on the source point with free-space delays, or conventional plane-wave beams (the better of the two is shown).

Position and clocks are assumed known, as in the rest of the report; §4 asks how well they must be known.

1. Ten single hydrophones: spacing buys resolution, not gain

Each USV hangs one hydrophone at 30 m. The plot sweeps the spacing of a straight line of 10.

Half to two-thirds of a wavelength is in fact the best spacing for ten single hydrophones (11–12 dB): the shallow-water noise arrives near the horizontal, and at that spacing it is slightly anti-correlated between neighbours, so the array does a little better than $10\log_{10}10$. Spreading the USVs out narrows the beam but lets horizontal noise in through grating lobes (the dip at one wavelength) and, beyond a few hundred metres, loses signal coherence. Either way the formation stays between about 6 and 12 dB, roughly 10 dB short of a 1.5 km towed array. The spacing is not the problem; the number of coherent hydrophones is. A 10-USV line at λ/2 is a 70 m array, the length of a TB-16-class tactical towed array.

2. More hydrophones per USV: dipped vertical lines

Each USV lowers a vertical line of hydrophones at λ/2 (7.5 m) through the water column, so 10 USVs carry 200 hydrophones, the same as the 1.5 km towed array. The lines are spaced tens of metres apart (the formation no longer needs λ/2 horizontally; the vertical lines supply the element count).

With 200 hydrophones on dipped lines, ten USVs come 3–4 dB short of the 1.5 km towed array for the submarine and ~2 dB short for the surface ship, and only with matched-field processing; twenty USVs (400 hydrophones) come within about 1 dB. Without an environment model the dipped lines lose 4–6 dB, because the modal field is not a plane wave (or a spherical wave) in the vertical. The towed array does not have this dependence: at 100 Hz a 1.5 km array is in its own near field out to $L^2/\lambda$ = 150 km, but simple geometric focusing recovers its full gain.

The towed baseline here is at 30 m and broadside, its best bearing. Lowered to 60–90 m it gains another 1–2 dB against this submarine; 10° off endfire it loses ~8 dB, where a ring formation does not.

So, on ambient noise alone and at equal hydrophone count, the formation does not beat a long towed array; it reaches parity only with twice the hydrophones and a good environment model. The balance can shift through self noise.

3. Self noise decides it

A towed array at a few knots adds flow and platform noise, typically several dB above ambient at low frequency. A USV formation can drift or hold station slowly with propulsion off or on quiet electric thrusters; its hydrophones still pick up cable strum, heave and wave slap unless the lines are decoupled. The plot adds white self noise to each design (relative to the ambient level per hydrophone) against a quiet reference hydrophone.

The plot gives both designs the same self noise. A towed array is hurt more by it than the formation is, because much of its advantage came from rejecting the structured, near-horizontal ambient noise; once uncorrelated flow noise dominates, that advantage goes, and the curves cross at about 1–2 dB above ambient. The printed cases give each platform its own level: with the towed array 3–6 dB above ambient and quiet USVs (10 dB below ambient), the formation leads by roughly 3–5 dB with matched-field processing, but only draws level (−2 to +0.5 dB) without it; with both at ambient level the two are within half a decibel.

Two caveats weigh on this. The USV self noise is modelled as white; real strum, heave and vehicle noise are correlated along a line and strongest in the top elements, so −10 dB is an optimistic design target, not a given. And part of the towed array's larger loss is that it rejected the structured ambient noise better in the first place. Quiet vehicles, decoupled lines and a good environment model are therefore the design drivers, more than the geometry.

4. Position control versus position knowledge

Coherent processing needs each hydrophone's position to a fraction of a wavelength, but the positions need to be known, not held. The formation can wander by metres as long as the beamformer uses measured positions. The plot adds random, unknown 3D errors to every hydrophone.

Random errors up to about 1 m (λ/15) cost almost nothing; 2 m costs ~1 dB; 4 m costs ~4 dB. Correlated errors are more dangerous: a 150 m line tilted by an unknown 2° (5 m offset at the bottom) costs ~3 dB, and 5° costs ~9 dB. The requirement is therefore ~1 m knowledge of every hydrophone, including each line's shape, not 1 m station-keeping. RTK GNSS gives the USVs to centimetres, but a hydrophone 100 m down a line moves with current and heave, so each line needs its own positioning (depth sensors plus an acoustic or inertial fix of the line shape). Clock synchronisation to ~1 ms (λ/10 at 100 Hz) is trivial with GNSS time.

5. Hydrophone depth under a USV

A hydrophone near the sea surface sits in the pressure-release node of every mode (the Lloyd's mirror effect). But ambient noise, which also arrives through the modes, drops there too, so ambient-limited SNR changes less than the signal level does:

In ambient noise a hull-mounted hydrophone loses only a few dB at 100 Hz. The practical reason to go deeper is platform noise: wave slap, hull vibration, propulsion and heave-driven flow are all strongest at the surface. A hydrophone on a compliant (heave-isolated) suspension tens of metres down is the sonobuoy solution to this.

6. Geometry: bearing, ambiguity and range

Gain is not the only output. A straight line of USVs has the towed array's left/right ambiguity; a ring or an irregular 2D pattern does not, and a formation a few hundred metres across can focus in range as well as bearing out to a few km. The animations show both 10-USV geometries at 100 Hz (free-space, perfectly coherent: they show geometry, not detection).

The half-wavelength line focuses to a broad bearing fan (~11° wide) with its mirror image. The 190 m ring puts a single, much tighter spot on the source with ~1/N sidelobes around it: bearing to a few degrees with no mirror, and some range information because the source is inside the ring's focusing distance ($D^2/\lambda$ ≈ 2.4 km).

7. What the gains mean for detection range

At 100 Hz absorption is negligible and spreading beyond the water depth is cylindrical (3 dB per doubling of range), so every 3 dB of gain roughly doubles detection range until bottom loss takes over. The table gives the range at which each design detects a target that the quiet 1.5 km towed array detects at 20 km (0.3 dB/km bottom loss).

Answer

Not with ten single hydrophones. Ten USVs at λ/2 form a 70 m array: about 11 dB of gain, close to the best (12.4 dB, at ~0.7 λ) any spacing of ten single hydrophones achieves, but roughly 10 dB less than a 1.5 km towed array. In the range model of §7 that is about a quarter of the detection range (5 km against 20 km). Position control does not change that; the number of coherent hydrophones does.

Possibly, with a different design and some favourable conditions. The version of the idea that competes is:

1. Many hydrophones per vehicle: each USV lowers a vertical line at λ/2 through the water column (20 hydrophones over 150 m). Ten USVs then carry as many hydrophones as the towed array; twenty carry twice as many.
2. Spacing set by coherence and geometry, not λ/2: vehicles tens of metres apart on a ring or irregular 2D pattern, a few hundred metres across (inside the ~450 m coherence length). This removes left/right ambiguity and gives some range from focusing, and no weak endfire direction.
3. Quiet platforms: drifting or slow electric vehicles and heave-decoupled lines. Against a towed array with 3–6 dB of self noise, a formation 10 dB quieter than ambient is 3–5 dB ahead with matched-field processing.
4. Matched-field processing with a surveyed environment model, to use the vertical lines' full gain (without a model they lose 4–6 dB of it, and the advantage in 3 disappears). Real mismatch costs part of this; it also gives source depth, so surface ships can be rejected (notebook 04).
5. Hydrophone positions known to ~1 m, including each line's tilt and shape, from depth sensors and acoustic tracking; the vehicles only need to keep formation to within metres and avoid each other.

In this model, at equal hydrophone count and on ambient noise alone, the formation is 3–4 dB short of a quiet 1.5 km towed array; it pulls ahead only if its vehicles and lines are much quieter than the tow and the environment is well modelled. Its unconditional advantages are elsewhere: full-column coverage, depth classification, 360° coverage with no endfire weakness, and a 3D position fix. The main risks are practical: line strumming and heave noise, keeping 150 m lines near vertical in current and measuring their shape, and matched-field mismatch.

Model limits for this notebook: isovelocity Pekeris waveguide (no thermocline), perfect vertical coherence, azimuthally uniform shipping noise, white self noise, and no own-ship radiated noise for the tow.

05 · Animations and your own cases

Seven animations are pre-rendered (GIF here, MP4 in renders/). Each is defined in sonarsim/cases.py with every parameter visible; copy one, change fields, and render it.

The pe_* cases (notebook 06) are transmission-loss maps rather than movies; cases.render() saves them as .npz.

Make your own

Every case is a frozen dataclass; .replace(...) returns a modified copy. Plan-view cases are analytic and take seconds (in the browser use quick=True). Section cases run the FDTD solver: about 2 minutes each with desktop numpy; in the browser use quick=True for a coarse preview, or run the notebook locally for full quality.

Example 1: a ring of 6 nodes, 3 km radius, with small path errors. Change any field and re-run.

Example 2: a deeper submarine and a different thermocline. This one is slow in the browser; set RUN = True to render it (use quick=True there).

Fields you can change

PlanCase: layout ('towed', 'distributed', 'both'), towed_length_m, towed_center, towed_heading_deg, n_nodes, node_spacing_m, node_layout ('grid', 'ring', 'random'), target_xy, extent_m, f0_hz, timing_sigma_m, seed.

SectionCase: source_depth_m, source_range_m, f0_hz, cycles, range_m, depth_m, mixed_layer_m, thermocline_bottom_m, t_surface_c, t_below_c, t_bottom_c, sediment_m, c_sed, rho_sed, vla_range_m, vla_top_m, vla_bottom_m, vla_elements, tow_depth_m, t_end_s, ppw (grid points per wavelength), n_frames.

To regenerate every pre-rendered case after changing defaults: python scripts/render_cases.py.