2.1. Existing underwater measurement technology

Most underwater 3D scanners use either ultrasound or lasers, applying either time-of-flight or triangulation principles to acquire measurements (Massot-Campos and Oliver-Codina, Reference Massot-Campos and Oliver-Codina2015). In time-of-flight devices, a pulse is emitted and reflected off a surface. By measuring the time it takes for the pulse to travel from the emitter to the surface and back, and knowing the speed of the signal in the medium, the distance to the surface can be calculated (Lurton, Reference Lurton2002; Massot-Campos and Oliver-Codina, Reference Massot-Campos and Oliver-Codina2015). Triangulation, on the other hand, uses the known angles and distances between surveying points to calculate the distance to a target. In laser-based systems, these points typically include the laser (emitter), a camera (receiver), and a known feature or target on the camera image (Massot-Campos and Oliver-Codina, Reference Massot-Campos and Oliver-Codina2015).

In offshore settings, ultrasound measuring devices are the most commonly used and have demonstrated reliability over many years (Coates, Reference Coates1990; Lurton, Reference Lurton2002). The most cost-effective of these devices is the single-beam echosounder, which uses the time-of-flight principle to acquire measurements. Despite certain limitations, single-beam echosounders are widely used in the offshore industry due to their ability to provide accurate measurements over long distances and operate effectively in turbid water and harsh environmental conditions (Coates, Reference Coates1990; Lurton, Reference Lurton2002). Multi-beam echosounders, which consist of an array of single-beam echosounders arranged in a fan pattern, can collect 3D data more rapidly. However, they tend to be bulky, expensive, and prone to signal interference when used in confined spaces or shallow waters, which can result in inaccurate measurements. Consequently, multi-beam echosounders are primarily used for bathymetric data collection (Lurton, Reference Lurton2002). A more economical underwater device for capturing 3D data is the dual-axis mechanical scanning sonar. This system uses two stepper motors to move a single-beam echosounder along two axes to record measurements. By tracking the angles of the motors and the distance measured by the echosounder, the system can generate 3D coordinates. These coordinates are then used to create a point cloud, which visualizes the underwater scene (Clarke, Reference Clarke2009).

Laser distance measurement devices are becoming increasingly popular for underwater inspections, but are still less commonly used than ultrasound devices. They offer distinct advantages, such as faster measurement acquisition and superior spatial resolution compared to ultrasound (Massot-Campos and Oliver-Codina, Reference Massot-Campos and Oliver-Codina2015). Additionally, unlike ultrasound, laser devices are not affected by interference from residual signals (Chemisky et al., Reference Chemisky, Menna, Nocerino and Drap2021). However, their performance is significantly influenced by water’s physical properties (Menna et al., Reference Menna, Agrafiotis and Georgopoulos2018). For example, complex calibration is needed to address refraction at the water-lens interface (Chemisky et al., Reference Chemisky, Menna, Nocerino and Drap2021). Their operational range is limited because laser light is strongly attenuated in water. Moreover, in turbid conditions, scattering from suspended particles further reduces their effective range (Menna et al., Reference Menna, Agrafiotis and Georgopoulos2018). Therefore, their maximum range is significantly lower than that of ultrasound devices. Most underwater laser distance measurement devices belong to two main types: (i) Laser line scanners with triangulation, or (ii) Light Detection and Ranging (LIDAR) devices. Laser line scanners project a laser line onto a surface and use cameras to capture its deformation, enabling the creation of very detailed 3D profiles in a short period of time. These devices are particularly suited for close-range inspections and high-resolution imaging. LIDAR systems, on the other hand, emit pulsed laser beams and calculate distances by measuring the time-of-flight of the reflected light. A summary of the different types of commercially available ultrasound and laser scanners and their characteristics is provided in Table 1.

Table 1.

Commercially available underwater scanners and their typical characteristics

2.2. Bayesian optimization

Bayesian Optimization (BO) is a widely used machine learning method for finding the maximum of an unknown, expensive-to-evaluate objective function g(x), which takes the form:

(1) $$ {x}^{\ast }={\mathrm{argmax}}_x\;g(x) $$

BO has been applied successfully across various domains, including environmental monitoring (Marchant and Ramos, Reference Marchant and Ramos2012), optimization of machine learning model hyperparameters (Wu et al., Reference Wu, Chen, Zhang, Xiong, Lei and Deng2019) and physical devices like lasers (Jalas et al., Reference Jalas, Kirchen, Messner, Winkler, Hübner, Dirkwinkel, Schnepp, Lehe and Maier2021), as well as robot control (Martinez-Cantin, Reference Martinez-Cantin2017). A key strength of BO is its efficiency in minimizing the number of function evaluations required to characterize the objective function (Shahriari et al., Reference Shahriari, Swersky, Wang, Adams and de Freitas2016). This efficiency is achieved by incorporating prior knowledge to strategically select sampling locations. BO balances the trade-off between exploration, which involves sampling areas with high uncertainty, and exploitation, which targets areas where the objective function is expected to be high. When applied to the control of a 3D scanner, BO aims to minimize the number of measurements required to characterize a target surface. This approach significantly reduces the time needed to make reliable predictions of the surface.

BO operates through a series of iterative steps. First, a surrogate model is created to approximate the objective function, with GP regression commonly used for this purpose (Shahriari et al., Reference Shahriari, Swersky, Wang, Adams and de Freitas2016). Next, an acquisition function is employed to decide the most promising location to sample the objective function. The acquisition function leverages the surrogate model’s predictions to identify the point where the objective function is likely to be at its maximum. The objective function is then evaluated at this selected location, and the resulting data is used to update the surrogate model. This process of updating the model and selecting new sampling points continues iteratively until the optimization goal is achieved.

When applying BO to problems where there is a tangible time cost associated with moving to new sampling points (e.g., in physical systems where a device must physically travel to a sampling location, unlike computational simulations where movement costs are negligible), it may be necessary to use a specialized acquisition function that accounts for these costs. Such a function balances the need to select informative points with the need to minimize delays caused by traveling to distant sampling locations. Marchant and Ramos (Reference Marchant and Ramos2012) proposed the distance-based upper confidence bound (DUCB) function to address this issue. This acquisition function incorporates a distance penalty to priorities closer sampling points and reduce delays in data acquisition.

In the current study, the aim is to reconstruct unknown target surfaces with different geometries by strategically selecting sampling locations that maximize information gain. GP regression serves as the surrogate model for the unknown target surface elevations with respect to the sensor location, providing both predictions of the surface shape and estimates of uncertainty. To achieve this aim, the following acquisition function is employed, which selects the next sampling location $ {x}^{\ast } $ as the location with maximum uncertainty:

(2) $$ {x}^{\ast }={\mathrm{argmax}}_x\;\sigma (x) $$

where $ \sigma (x) $ is the GP’s posterior standard deviation at location $ x $ . The acquisition function used in this study does not include a distance penalty (Marchant and Ramos, Reference Marchant and Ramos2012), as initial experiments revealed that selecting distant locations incurs minimal additional cost compared to nearby ones. The values of $ \sigma (x) $ are influenced by the geometry of the target surface; simpler geometries result in lower $ \sigma (x) $ values with few observations, compared to more complex geometries. By targeting regions of high predictive uncertainty, this acquisition function ensures efficient exploration of the domain, leading to improved surface reconstruction with minimal samples. The iterative process of updating the GP model with newly acquired measurements facilitates accurate surface estimation while avoiding unnecessary redundancy in sampled data.

2.3. Gaussian process regression

Gaussian process (GP) regression is a probabilistic machine learning method used for non-linear regression and interpolation. It is widely used due to its flexibility in modeling complex relationships, ability to handle small datasets, and capability to provide uncertainty estimates (Suryasentana and Sheil, Reference Suryasentana and Sheil2023). A detailed explanation of GP regression can be found in Rasmussen and Williams (Reference Rasmussen and Williams2008) and Murphy (Reference Murphy2012). However, a brief explanation of the underlying theory is provided here. GP regression models an unknown function as a GP, which is a set of random variables where any finite subset has a joint Gaussian distribution. GPs can be described as a probability distribution over functions and are fully characterized by their mean function $ m(x) $ and covariance function (also known as the kernel) $ k\left(x,{x}^{\prime}\right) $ :

(3) $$ f(x)\sim GP\Big(m(x),k\left(x,{x}^{\prime}\right) $$

where $ m(x) $ and $ k\left(x,{x}^{\prime}\right) $ are:

(4) $$ m(x)=\unicode{x1D53C}\left[\;f(x)\right] $$

(5) $$ k\left(x,{x}^{\prime}\right)=\unicode{x1D53C}\left[\;\left(f(x)-m(x)\right){\left(f\left({x}^{\prime}\right)-m\left({x}^{\prime}\right)\right)}^T\right] $$

$ m(x) $ represents the expected value of the function at input $ x $ , while $ k\left(x,{x}^{\prime}\right) $ encodes the similarity between the outputs at $ x $ and $ {x}^{\prime } $ . Commonly used kernels include the Matérn kernel $ {\kappa}_{MA} $ and squared exponential kernel $ {\kappa}_{SE} $ :

(6) $$ {\kappa}_{MA}\left(x-{x}^{\prime}\right)={\sigma}^2\frac{2^{1-\upsilon }}{\Gamma \left(\upsilon \right)}{\left(\sqrt{2\upsilon}\frac{\left(x-{x}^{\prime}\right)}{\rho}\right)}^{\upsilon }{\mathcal{R}}_{\upsilon}\left(\sqrt{2\upsilon}\frac{\left(x-{x}^{\prime}\right)}{\rho}\right) $$

(7) $$ {\kappa}_{SE}={\theta}_f^2\exp \left(-\frac{1}{l^2}{\left\Vert x-{x}^{\prime}\right\Vert}^2\right) $$

The squared exponential kernel assumes that the functions are smooth and is therefore well suited for modelling smooth surfaces, whereas the Matérn kernel can model less smooth functions and may be better suited to modelling physical phenomena (Stein, Reference Stein2012). The values of the kernel hyperparameters, such as $ {\sigma}_f^2 $ and $ l $ in Equation 7, are optimized by maximizing the marginal log likelihood of the observed (training) data, given the hyperparameters (Rasmussen and Williams, Reference Rasmussen and Williams2008). In the current study, the squared exponential kernel was used in the surrogate model within the BO process and to predict the target surface.

The prior distribution of the function is a Gaussian distribution, denoted as:

(8) $$ f\sim N\left(f|\boldsymbol{\mu}, \boldsymbol{K}\right) $$

where $ {K}_{i,j}=k\left({x}_i,{x}_j\right) $ and $ \boldsymbol{\mu} =\left(m\left({x}_i\right),\dots, m\left({x}_n\right)\right) $ .

Once trained, the method provides not only predictions at new inputs but also an estimate of uncertainty in these predictions, making it particularly useful in applications where uncertainty quantification is important. Suppose that some observations $ \boldsymbol{y}={\left[{y}_1,\dots, {y}_N\right]}^T $ have been obtained for some inputs $ \boldsymbol{x}={\left[{x}_1,\dots, {x}_N\right]}^T $ and the kernel’s hyperparameters have been learnt from the training process. To predict the output $ {\hat{y}}_{\ast } $ for a new input $ {x}_{\ast } $ , the GP regression model computes the conditional distribution, which is obtained analytically using the standard conditioning rules for the multivariate Gaussian distribution:

(9) $$ p\left(\;{\hat{y}}_{\ast }|{x}_{\ast },\boldsymbol{x},\boldsymbol{y}\right)=N\left({\mu}_{\ast },{k}_{\ast}\right) $$

where

(10) $$ {\mu}_{\ast }=m\left({x}_{\ast}\right)+{{\boldsymbol{k}}_{\ast}}^T{\boldsymbol{K}}^{-1}\left(\mathbf{y}-m\left(\boldsymbol{x}\right)\right) $$

(11) $$ {k}_{\ast }=k\left({x}_{\ast },{x}_{\ast}\right)-{{\boldsymbol{k}}_{\ast}}^T{\boldsymbol{K}}^{-1}{\boldsymbol{k}}_{\ast} $$

(12) $$ {\boldsymbol{k}}_{\ast}={\left[k\left({x}_1,{x}_{\ast}\right),\dots, k\left({x}_N,{x}_{\ast}\right)\right]}^T $$

(13) $$ \boldsymbol{K}=\mathrm{covariance}\ \mathrm{matrix},{K}_{ij}=k\left({x}_i,{x}_j\right) $$

The prediction $ {\hat{y}}_{\ast } $ is a weighted sum of the observed data, where the weights are determined by the trained kernel.

2.4. Prototype 3D scanner

A prototype 3D scanner for the AI-driven sensor concept is constructed using a dual-axis rotating arm equipped with two 180-degree servo motors. These motors are enclosed in waterproof housings. The arm controlled a single-beam echosounder, operating at $ 500 kHz $ , allowing it to scan in the 3D space. An M10 threaded rod is attached to the housing to facilitate mounting of the scanner for the laboratory experiments. Figure 1 provides a schematic diagram of the scanner’s construction alongside a photo of the prototype.

Figure 1.

(a) Schematic diagram of the dual axis ultrasound point scanner. (b) Prototype 3D scanner.

Figure 2 illustrates the construction of the waterproof housings. These housings are fabricated using thick, 3D-printed resin for durability. A rubber O-ring seated in a flange groove ensures a watertight seal at the junction between the housing lid and its main body. To maintain a waterproof barrier around the rotating driveshaft, a rubber rotary shaft seal is used. Additionally, the cables are routed through a protrusion on the housing, referred to as a cable penetrator, which is filled with silicone sealant to prevent water ingress.

Figure 2.

Schematic diagram of the waterproof servo motor housing.

A Raspberry Pi computer controls the operation of the echosounder and servo motors. By recording the angle feedback from the servo motors and the distance measurements from the echosounder, the system calculates the spherical polar coordinates of the target surface for each measured data point. The computer contains both the BO algorithm and the conventional scanning algorithm. The conventional scanning algorithm employs a ‘brute force’ method, systematically sweeping the entire target surface with a series of line scans. During each line scan, the echosounder records measurements at every two-degree increment of the servo motor’s movement along the path. The process starts with the scanner tracing a straight line from the point beneath the echosounder to the edge of the target surface. Once this scan is completed, the scanner rotates two degrees around the vertical axis and conducts the next line scan, moving from the edge back toward the point beneath the echosounder. This alternating pattern continues, with the scanner rotating two degrees around the vertical axis after each pass, until the entire target surface has been fully scanned. Figure 3 illustrates the conventional scanning pattern.

Figure 3.

Schematic diagram of the conventional scanning pattern.

2.5. Laboratory experiments

All laboratory experiments were conducted underwater in an 800 L cylindrical water tank with a height of 1.2 m and a diameter of 1 m. The experimental setup is shown in Figure 4, with the prototype 3D scanner positioned at the center of the tank cross-sectional area. The target surface was placed 50 cm below the scanner.

Figure 4.

(a) Schematic diagram of the experimental equipment. (b) A top-down picture of the experimental equipment.

Three different target surfaces were used during the experiments, each made of 2.5 mm thick PVC and circular, with a diameter of approximately 80 cm. The first surface is perfectly flat and serves as a control to verify the proper functioning of the scanner. The second surface was shaped like a conical hole, mimicking a scour hole. The third surface took a ‘wave-like’ form, a more complex shape used to evaluate the performance of the Bayesian scanning method in areas with multiple critical zones. This wave-like surface also mimicked sediment buildup in rivers and wave imprints on the seafloor. Photos of the different target surfaces used in the experiments are shown in Figure 5.

Figure 5.

Target surfaces used to test the 3D scanner: (a) hole surface, and (b) wave-like surface.

During the laboratory experiments, the following procedure was followed for all three surfaces. First, a full scan of the surface was conducted using the conventional scanning algorithm. This scan ensured the scanner was fully functional and provided a baseline for comparison with the BO-driven scan. Once the conventional scan was completed, the BO-driven scan was performed. As the BO-driven scan has no fixed endpoint, it was left to collect 120 measurements.

To estimate the target surface from the measurements, a GP regression model is used to predict the surface from the measurements, instead of linear interpolation, which is traditionally used. This approach allows for more reliable predictions of the shape of the surface, especially when extrapolating from the measurements. A mathematical representation for each target surface is also available as ‘ground truth’ to evaluate the accuracy of the scans.