Effectiveness of vertical error budget model for portable multi-beam echo-sounder in shallow water bathymetric survey

With the increasing availability of portable survey equipment and platforms, hydrographic data acquisition now offers a diverse range of device options and installation methods. Tailoring the mounting and installation of hydrographic survey devices to fit the boat or platform dimensions has become common practice. The use of portable equipment is essential, given the wide selection of combined options available in the market. Consequently, sensor offsets and device performance differ for each survey conducted, necessitating the constant verification of accuracy performance. This paper examines the capabilities of portable multibeam echosounder (MBES) deployments on various types of survey boats. The investigation focuses on error budget modeling and field tests. A model is developed to estimate the propagated uncertainty resulting from different sources of measurement uncertainties. The model primarily considers vertical sources of error in MBES surveys, as they have the most significant impact on hydrographic data quality. To validate the model, field tests are conducted at two different locations using different boat and device configurations. The results demonstrate that the modeled uncertainties align with the measured standard deviations, particularly in the Pramuka area, where the standard deviation of data acquired at the widest beam angle of ±60° is 0.25m, close to the predicted value of 0.28m. However, in the Patimban area, the model overestimates the uncertainty, predicting 0.27m compared to the measured standard deviation of 0.17m.


Introduction
Accurate and reliable bathymetric data plays a crucial role in various applications, such as offshore construction, dredging operations, navigation safety, and the study of marine geological and 1245 (2023) 012041 IOP Publishing doi:10.1088/1755-1315/1245/1/012041 2 biological systems [1][2][3].The acquisition of this data relies on in-situ surveys using Multi-Beam Echo-Sounder (MBES) technology, which is widely employed for nautical chart production, offshore engineering, and mineral resources exploration [4].MBES systems collect depth data by emitting acoustic pulses and measuring the two-way travel time for a set of predefined beam angles [5].To ensure high-quality depth data, additional sensor data is incorporated to compensate for errors arising from factors such as sound speed, boat attitude, offsets, and latency [6].
While fixed survey systems permanently installed on vessels offer persistent performance, their use is limited in regions with shallow water depths and difficult access to inland seas [7][8].Portable systems, mounted on small craft such as fishing boats, have emerged as an alternative for shallow water surveys due to their adaptability and availability [9].However, portable systems are more susceptible to errors and uncertainties compared to fixed systems, including those associated with sound speed estimation, platform motion, and installation offsets [10].Thus, it is essential for surveyors to realistically estimate the uncertainty, or error budget, of portable MBES systems.
The widely used approach for predicting vertical uncertainty in bathymetry surveys, developed by Hare [11][12] has been modified to account for additional error sources introduced by Doppler frequency shift and baseline decorrelation when using frequency modulated (FM) pulse [13][14].Several studies have examined the performance of MBES bathymetric uncertainty prediction models, focusing on depth measurement uncertainty [13] [15].However, only a few studies have considered the total vertical uncertainty (TVU) for reduced depth measurements acquired by autonomous underwater vehicles [16].
To address the need for realistic estimates of reduced depth uncertainty in bathymetry acquired using portable MBES systems, this study remodels the existing prediction model by incorporating a Jacobian matrix.The Jacobian matrix enables the propagation of depth measurement errors, considering the latest insights into error contributors and facilitating the handling of complex equations and correlations [17].The objective of this study is to evaluate the performance of the model in estimating the TVU of a portable MBES system by comparing modeled reduced depth error budgets with standard deviations from measured datasets obtained in two shallow water areas.
The estimated values obtained from this study can serve as a reference for portable MBES surveys, but they may vary depending on the sensors used and the operating depth.The remainder of this paper is organized as follows: Section 2 provides a description of the model, the approach used for remodeling, and details about the datasets employed.Section 3 presents the results and discusses relevant issues.Finally, the concluding remarks are provided in Section 4.

Depth error budget model description
Propagation of uncertainty deals with the quantification of the effect of the input variables randomness on the uncertainty of function based on them.Most of the equations in this paper are taken from [12] and [18].Assumes that a measurement system is modeled as a function of multiple real input quantities and a single real output quantity.This is represented as:  =  (, +1…, ) (1) In the equation, y denotes the output quantity (measured depth), y represents the real input quantities (individual components in the depth derivation), i is an index for a specific component, and n represents the total count of input quantities.Random errors are treated as independent, with the total determined by squaring the individual errors and computing the square root of their sum: In the existing model, all the error contributors identified above are mapped into an error in the measured depth by applying propagation of errors to equation (3) then using the root-sum square (RSS) to get the total measured depth error  due to the sounder system given by: To get the Reduced depth to reference datum uncertainty, the following reduction errors must be added to the measured depth uncertainty quadratically.
• Heave error   • Dynamic Draught error   and • Water level error   .
Reduced depth is given by the following equation: Where  is the reduced depth to reference datum,  is the measured bathymetry beneath the transducer,  is the heave,  is the water level distance from the transducer,  is the measured water level above the reference datum, which is affected by tide and wave.These components are all independent of each other.The resulting reduced depth uncertainty is obtained as: The general flowchart of this paper is illustrated in figure 1. followed by a detailed description of the errors contributing to reduced depth error will be given in the next section.

Error budget model of the reduced depth
The error budget modeling is based on the variances from the measured depth, heave, dynamic draught, and water level measurement.Equations describing the individual uncertainties are provided herein.

2.2.1.
The Measured Depth Uncertainty.Herein, we apply the Jacobian matrix to map all error contributors in equation (3) to measured depth errors and calculate the total measured depth uncertainty   .The general form of the matrix is given by: variance-covariance The matrix of the partial derivatives in equation ( 7) is known as the Jacobian matrix and is denoted here as .equation ( 7) can be re-written as: Where  = variance of the computed result, .  = Jacobian matrix of the measurement function, , and  = the variance-covariance matrix of the input variables, .

Heave Uncertainty.
To get the total variance of depth due to heave errors, the heave measurement errors, and induced errors are quadratically added, which results in: Where  is the total heave variance, is the heave measurement error, is the heave induced error.

Dynamic Draught Uncertainty.
The dynamic draught variance is calculated by the quadratic sum of the variances of static draught, squat and loading errors, which results in: Where  is the dynamic draught variance, ℎ is the static draught error,  is the squat error and  is the load error.

Uncertainty for Water Level Measurement.
Water level measurement variance WL is divided into two components, measurement uncertainty at water level gauge WLmeas and spatial and temporal prediction (zoning) uncertainty WLzone.

Field Measurement Survey Design
To assess the agreement between the modelled and measured uncertainties, surveys were conducted at Patimban port development area and Pramuka island in February 2019 and June 2022, respectively.Patimban is located on the northeast coast of the Subang District in the West Java Province, while Pramuka island is situated approximately 50km north of Jakarta, the capital of Indonesia, in the Java Sea as shown in figure 2. The data used in this study were collected using the R2 Sonic 2020 and Kongsberg M3 Sonar-500 multibeam echosounders, which were deployed on local fishing boats in Patimban and Pramuka, respectively, operating in Continuous Wave (CW) mode.Additional sensors used during the survey and their specifications can be found in Table 1. and the static offsets of the sensors, as installed at the survey locations, are depicted in Figure 3.

Field Measurement Standard Deviations
The acquired measurement data undergoes processing and filtering using Qimera software to remove spikes and produce a cleaned bathymetry.The resulting bathymetric data is then used to calculate the standard deviations, which are compared with the predictions from the model.It should be noted that the model assumes a flat seafloor, and therefore, only the flat portion of the seafloor, depicted by the red area in Figure 4, is considered for determining the standard deviation.

Computed Reduced Depth Error Budget
Table 2 shows the parameters used to compute the depth error budget.Most of the parameters are from the sensor specifications [19][20] deployed at the survey areas.TVU was computed as a function of beam angle from  = − 60°  60° based on the average water depth in the survey areas, 6 meters at Patimban and 26 meters at Pramuka.  5 display the results of TVU computation, presented as values corresponding to a 95% confidence level.The findings indicate that the smallest TVU values are observed within a beam angle range of 0  ± 15 degrees, where the bottom detection method shifts from amplitude detection to phase detection.As the beam angle increases, the TVU also increases due to the greater distance between the echo sounder and the seafloor.

Comparison of measured and reduced depth error budgets
The bathymetric uncertainties modeled in this study are based on parameters obtained from the sensor specifications deployed during the survey, the settings used during data acquisition, and environmental factors.A CW type was used during data acquisition, the modeled uncertainties do not account for the contributions of Doppler effect and baseline decorrelation.Table 4-4 compares the model estimate and the measurement standard deviations in the two test areas, showing that there is little difference between the uncertainties, particularly in the Pramuka area where the difference is only 0.03m at the widest beam angle of ±60°.However, a larger discrepancy was observed in the Patimban area, where the model overestimates the uncertainty by 0.09m.Furthermore, the uncertainty values revealed a 1 percent increase in uncertainty for every 20m depth interval.Overall, the model presented this study performed well in estimating the reduced depth uncertainty of portable hydrographic survey systems.The bar chart in figure 6 offers a useful way to compare and contrast the data presented in the table by providing a visual representation of the predicted and calculated standard deviations in two distinct areas: Patimban (16m) and Pramuka (26m).The chart clearly displays the magnitude of the difference between the predicted and calculated values in each location, with the use of an appropriate color code to enhance readability.

Conclusion
Based on the study's results, the newly proposed error budget provides a dependable estimation of the uncertainty of a dataset obtained through portable systems deployment in the field.The model estimate and the measured depth's uncertainty exhibit a negligible difference in uncertainties in the Pramuka area, indicating that the model performed well in this area.A considerable discrepancy was observed in the Patimban area, where the model overestimated the uncertainty.This discrepancy may be attributed to the presence of various environmental factors in the area.Overall, the model performed well in estimating the reduced depth uncertainty of portable hydrographic survey systems, suggesting that the model can be trusted by surveyors to predict the error budget of the hydrographic survey system they plan to use in the field.
The present study has several limitations that should be acknowledged.firstly, it is important to note that the model used in this study assumes a flat seafloor during the computation of the error budget.However, it is known that there can be significant variations and local slopes in the seafloor topography in the actual measurement area.Another limitation is that the study did not consider other sources of error, such as environmental factors or operator error, which could impact the accuracy of the survey measurements.Additionally, the study only tested two areas, which may not be representative of all survey environments.Future research could expand the study to include a wider range of survey environments and sensors to validate the results and improve the generalizability of the findings.
Importantly, it must be emphasized that the same parameters were employed to calculate the depth error budget in both study areas, solely based on the similarity in technical specifications of the sensors used.However, this approach overlooks potential variations in sensor characteristics and performance, which could significantly introduce biases or inaccuracies in the error budget calculations.
The findings of this study provide valuable insights into the estimation of uncertainty in portable hydrographic survey systems.Surveyors can leverage the proposed error budget model to enhance their decision-making process and improve the accuracy of survey measurements.By considering the limitations and potential areas of improvement identified in this study, further advancements can be made in the field of hydrographic surveying.

3 MBES
systems determined the surface depth by measuring the travel time of a transmitted acoustic signal.The water depth, d, measured from below the transducer to the seafloor can be determined as: =  cos  cos ( +  + ) =  cos  cos (3)With ,  and  being the across track angle under which the MBES is mounted on the vessel, roll and pitch angle, respectively. =  +  +  represent rotations around the depth-axis.The factors contributing to the error in the measured depth, as represented by equation (3), include:• Range error for sounder 1 • Beam angle and roll error 2 • Pitch error 3 • Depth measurement limitation error  4 • Steering angle shift due to Doppler Effect error 5and • Baseline Decorrelation error 6.

Figure 1 .
Figure 1.Flow diagram illustrating the comprehensive methodology design for the analysis and evaluation of the system performance.

Figure 3 .
Figure 3. Spatial relationship between sensor positions and offset relative to the center of gravity (cog) -Patimban (left) and Pramuka (right).

Figure 4 .
Figure 4. Red portion representing flat area use for measured data standard deviation calculation -Patimban (left) and Pramuka (right)

Figure 6 .
Figure 6.Comparison of modeled and calculated depth error budgets with difference magnitudes.

Table 1 .
Boat dimensions and device specifications for field

Table 2 .
Parameters used for modeled depth error budget computation.

Table 3 .
Modeled depth error budget result.

Table 4 .
Difference between the modeled depth error budget and the calculated standard deviation