modelling Level 4

Bathymetric LiDAR and Sonar Mapping

How deep is this coastal water? What does the river bottom look like? Bathymetric LiDAR uses green laser (532 nm) penetrating clear water, while sonar uses acoustic pulses for deeper/turbid water. This model derives water column correction, implements multibeam sonar geometry, and maps underwater topography for navigation, habitat, and hazard assessment.

Prerequisites: underwater acoustics, light attenuation, sonar ranging, refraction

27 Feb 2026 Updated 12 Mar 2026 12 min read

1. The Question

What’s the depth and bottom topography of this shallow coastal bay?

Two technologies for underwater elevation:

Bathymetric LiDAR (ALB - Airborne Lidar Bathymetry):

Green laser (532 nm) penetrates water
Measures from aircraft
Shallow water (<50m, clear conditions)
Simultaneous topography (infrared) + bathymetry (green)

Sonar (Sound Navigation and Ranging):

Acoustic pulses (10-500 kHz)
Boat/ship-mounted or autonomous underwater vehicles
Deep water (thousands of meters)
Works in turbid water

Applications:

Nautical chart production
Coastal hazard assessment (tsunami inundation)
Habitat mapping (coral reefs, seagrass)
Dredging and navigation
Underwater archaeology
River bathymetry
Reservoir volume calculation

2. The Conceptual Model

Bathymetric LiDAR

Dual-wavelength system:

Infrared (1064 nm):

Reflects off water surface
Measures water surface elevation

Green (532 nm):

Penetrates water column
Reflects off bottom
Return delayed by water travel time

Water column correction:

\[d_{\text{water}} = \frac{c_{\text{air}} t_{\text{total}} - c_{\text{water}} t_{\text{water}}}{2}\]

More precisely:

\[z_{\text{bottom}} = z_{\text{surface}} - \frac{c_{\text{water}}}{2} (t_{\text{green}} - t_{\text{IR}})\]

Where:

$c_{\text{water}} = c_{\text{air}}/n$ (n = refractive index ≈ 1.33)
Speed of light in water: ~2.25 × 10⁸ m/s

Limitations:

Turbidity (Secchi depth):

Beer’s Law attenuation:

\[I(z) = I_0 e^{-Kz}\]

Where:

$K$ = attenuation coefficient (m⁻¹)
Typical: K = 0.05-0.5 m⁻¹

Maximum depth:

Approximately 2-3 × Secchi depth.

Clear water: 40-50 m
Turbid water: 5-10 m
Very turbid: No penetration

Multibeam Sonar

Principle:

Transmit acoustic pulse, receive echoes from multiple angles simultaneously.

Sound speed in water:

\[c = 1449 + 4.6T - 0.055T^2 + 0.00029T^3 + 1.34(S - 35) + 0.016z\]

Where:

$c$ = sound speed (m/s)
$T$ = temperature (°C)
$S$ = salinity (PSU)
$z$ = depth (m)

Typical: c ≈ 1500 m/s (much slower than light!)

Two-way travel time:

\[d = \frac{ct}{2}\]

Multibeam geometry:

Transducer array spans ~120-150° swath
Simultaneously measures depth at 100s of points
Swath width: ~3-4 × water depth

Example: 100m depth → 300-400m swath width

Refraction Correction

Snell’s Law at water surface:

\[n_1 \sin\theta_1 = n_2 \sin\theta_2\]

For air ($n_1 = 1$) to water ($n_2 = 1.33$):

\[\sin\theta_{\text{water}} = \frac{\sin\theta_{\text{air}}}{1.33}\]

Off-nadir beams refract toward normal.

Effect: Horizontal position shifts inward.

Correction essential for accurate positioning.

3. Building the Mathematical Model

Bathymetric LiDAR Depth Calculation

Surface return time: $t_{\text{IR}}$

Bottom return time: $t_{\text{green}}$

Water travel time:

\[\Delta t = t_{\text{green}} - t_{\text{IR}}\]

Depth below surface:

\[d = \frac{c_{\text{water}} \Delta t}{2} = \frac{c_{\text{air}} \Delta t}{2n}\]

Where $n = 1.33$ (refractive index of water).

Example:

$\Delta t = 200$ ns

\[d = \frac{3 \times 10^8 \times 200 \times 10^{-9}}{2 \times 1.33} = \frac{60}{2.66} = 22.6 \text{ m}\]

Total elevation:

\[z_{\text{bottom}} = z_{\text{surface}} - d\]

Turbidity Limitation

Signal-to-noise ratio:

\[SNR = \frac{I_{\text{bottom}}}{I_{\text{noise}}}\]

Bottom return intensity:

\[I_{\text{bottom}} = I_0 R_{\text{bottom}} e^{-2Kd}\]

Where:

$R_{\text{bottom}}$ = bottom reflectance (~0.1-0.4)
Factor of 2: Down and back through water column

Detection threshold: SNR > 3 (minimum)

Maximum depth:

\[d_{\max} = \frac{\ln(I_0 R_{\text{bottom}} / I_{\text{threshold}})}{2K}\]

Example: $K = 0.1$ m⁻¹ (moderate clarity)

\[d_{\max} \approx \frac{\ln(1000)}{0.2} = \frac{6.9}{0.2} = 34.5 \text{ m}\]

Multibeam Positioning

Range to bottom:

\[R = \frac{ct}{2}\]

Beam angle: $\theta$ (from vertical)

Horizontal offset:

\[x = R \sin\theta\]

Vertical depth:

\[z = R \cos\theta\]

Sound speed correction:

If $c$ varies with depth (common):

Ray tracing through water column needed.

\[\frac{dx}{dz} = \frac{\sin\theta}{\cos\theta} = \tan\theta\]

With $\theta(z)$ from Snell’s Law:

\[c(z) \sin\theta(z) = c_0 \sin\theta_0\]

Iterative solution required for accurate positioning.

Swath Coverage

Swath width:

For flat bottom at depth $D$, beam angle $\pm \theta_{\max}$:

\[W = 2D \tan\theta_{\max}\]

Example: $D = 100$ m, $\theta_{\max} = 60°$

\[W = 2(100) \tan(60°) = 200 \times 1.73 = 346 \text{ m}\]

Survey efficiency:

Line spacing = swath width → 100% coverage

Tighter spacing → overlap for quality assurance

4. Worked Example by Hand

Problem: Calculate bathymetric LiDAR depth measurement.

Data:

Water surface elevation: $z_{\text{surface}} = 2.5$ m (above datum)
Infrared return time: $t_{\text{IR}} = 1000$ ns (round-trip to surface)
Green return time: $t_{\text{green}} = 1400$ ns (round-trip to bottom)
Refractive index: $n = 1.33$

Calculate bottom elevation.

Solution

Step 1: Water column travel time

\[\Delta t = t_{\text{green}} - t_{\text{IR}} = 1400 - 1000 = 400 \text{ ns}\]

Step 2: Speed of light in water

\[c_{\text{water}} = \frac{c_{\text{air}}}{n} = \frac{3 \times 10^8}{1.33} = 2.26 \times 10^8 \text{ m/s}\]

Step 3: Water depth

\[d = \frac{c_{\text{water}} \Delta t}{2} = \frac{2.26 \times 10^8 \times 400 \times 10^{-9}}{2}\] \[= \frac{90.4}{2} = 45.2 \text{ m}\]

Step 4: Bottom elevation

\[z_{\text{bottom}} = z_{\text{surface}} - d = 2.5 - 45.2 = -42.7 \text{ m}\]

(Negative: below datum)

Summary:

Water depth: 45.2 m
Bottom elevation: -42.7 m (relative to datum)

Note: This is near maximum depth for bathymetric LiDAR in clear water.

5. Computational Implementation

Below is an interactive bathymetric mapping simulator.

Technology: Water clarity (Secchi depth, m): 15 Bottom type:

Max measurable depth: -- m

Swath width: -- m

Points measured: --

Mean depth: -- m

Observations:

Blue area represents water column
Brown area shows measured seafloor
Orange dotted line marks maximum measurable depth
LiDAR limited by water clarity (turbidity)
Sonar works in deep/turbid water but from vessel
Clear water (high Secchi) enables deeper LiDAR penetration
Bright bottoms (sand) return stronger signal than dark (mud)
Data gaps occur where depth exceeds technology limits

Key findings:

Bathymetric LiDAR optimal for shallow clear water (<50m)
Sonar required for deep or turbid conditions
Water clarity and bottom reflectance control maximum LiDAR depth
Complementary technologies cover full range of conditions

6. Interpretation

Coastal Zone Mapping

NOAA bathymetric LiDAR program:

Systematic coastal mapping for:

Nautical chart updates
Tsunami inundation modelling
Habitat assessment
Emergency response

Example - Hurricane Sandy (2012):

Post-storm bathymetric LiDAR:

Documented massive sediment redistribution
Inlet breaches
Dune erosion
Critical for recovery planning

Coral Reef Mapping

Challenges:

Complex 3D structure at fine spatial scale.

LiDAR advantages:

Complete coverage (vs. transects)
Rugosity measurement (surface complexity)
Change detection (growth, bleaching, storm damage)

Example - Florida Keys:

Repeated surveys detect:

Coral decline
Algal overgrowth
Storm impacts

Management application:

Prioritize restoration sites, track recovery.

River Bathymetry

Traditional surveys:

Boat-based sonar in navigable reaches only.

Bathymetric LiDAR:

Maps entire river including:

Shallow riffles (boats can’t access)
Bank-to-bank coverage
Floodplain + channel in one dataset

Applications:

Salmon habitat (pool-riffle structure)
Flood modelling (channel capacity)
Sediment transport (bedform mapping)

Archaeological Applications

Underwater sites:

Shipwrecks, submerged settlements, harbor structures.

LiDAR detection:

Subtle features on seafloor visible in high-resolution bathymetry.

Example - Submerged prehistoric landscapes:

8000-year-old settlement sites now offshore mapped with bathymetric LiDAR.

7. What Could Go Wrong?

Suspended Sediment

Backscatter from water column:

Sediment particles reflect laser/acoustic signal.

LiDAR: False bottom returns from turbid layers.

Sonar: Reduced penetration, noise.

Worst case: Impossible to detect true bottom.

Solution:

Post-processing filters
Survey during low-turbidity conditions
Combined interpretation with water column imagery

Surface Waves

Rough sea state:

Water surface moves during measurement.

LiDAR error: ±0.5-1m in moderate waves.

Sonar: Motion compensation systems on vessel.

Solution:

Survey in calm conditions
Average multiple measurements
Motion reference unit (MRU) on aircraft

Refraction Uncertainty

LiDAR off-nadir beams:

Snell’s Law correction requires knowing:

Beam angle
Water surface orientation
Refractive index (varies with salinity, temperature)

Uncertainty propagates to horizontal position.

Solution:

Minimize scan angle
Measure water properties
Cross-line overlap for validation

Bottom Classification

Sonar backscatter varies with:

Grain size (sand vs. mud)
Roughness
Biology (seagrass, coral)

Challenge: Distinguishing features from noise.

Solution:

Backscatter mosaics
Ground-truth sampling
Multi-frequency analysis

8. Extension: Side-Scan Sonar

Different technology: Measures backscatter, not just bathymetry.

Geometry:

Transducers aimed sideways (perpendicular to track).

Creates acoustic image of seafloor.

Applications:

Object detection (shipwrecks, boulders, debris)
Seafloor classification (sediment type, bedforms)
Pipeline/cable inspection
Search and recovery

Advantages:

High-resolution imagery (cm-scale)
Wide swath coverage

Disadvantages:

Doesn’t directly measure depth
Geometric distortion
Interpretation requires expertise

Often combined with multibeam sonar:

Bathymetry from multibeam
Texture from side-scan
Comprehensive seafloor characterization

9. Math Refresher: Wave Attenuation

Beer-Lambert Law

Exponential decay:

\[I(z) = I_0 e^{-\alpha z}\]

Where:

$I(z)$ = intensity at depth $z$
$I_0$ = incident intensity
$\alpha$ = attenuation coefficient (m⁻¹)

Applicable to both light and sound in water.

Attenuation Coefficient

Light:

\[\alpha_{\text{light}} = \alpha_{\text{absorption}} + \alpha_{\text{scattering}}\]

Dominated by:

Pure water absorption (increases with wavelength)
Chlorophyll absorption (green algae)
Suspended sediment scattering

Sound:

\[\alpha_{\text{sound}} = \alpha_0 f^2\]

Where $f$ = frequency.

Higher frequency → more attenuation → shorter range

Trade-off: Resolution vs range.

Two-Way Attenuation

Laser/sonar pulse travels:

Down to bottom: attenuated by $e^{-\alpha z}$
Back to sensor: attenuated again by $e^{-\alpha z}$

Total: $I_{\text{return}} = I_0 R e^{-2\alpha z}$

Factor of 2 in exponent critical for range calculations.

Summary

Bathymetric LiDAR uses green laser (532 nm) penetrating clear water to ~50m depth
Dual-wavelength measurement separates water surface from bottom returns
Water column correction accounts for slower light speed and refraction in water
Turbidity limits LiDAR depth via exponential attenuation following Beer’s Law
Multibeam sonar uses acoustic ranging for deep or turbid water mapping
Sound speed variation with temperature/salinity requires ray tracing for accuracy
Swath width typically 3-4× water depth for multibeam systems
Applications span coastal mapping, coral reef monitoring, river surveys, archaeology
Challenges include suspended sediment, surface waves, and refraction correction
Complementary technologies cover full range from shallow clear to deep turbid water
Critical for navigation safety, habitat assessment, and hazard modelling