879
1 INTRODUCTION
Multibeam Echosounder is nowadays state of the art
for hydrographic surveys, due to the accuracy and
coverage that it offers. Complete MBES system
however requires a set of additional sensors allowing
precise navigation and measurements, like positioning
sensor, motion reference unit or speed sensor. Data
obtained form these sensors influence a lot accuracy of
measured data and consequently of bottom model
obtained. Raw measurements from these sensors are
burden with typical measurement errors and
inaccuracies, therefore these data requires filtration in
processing stage to provide reliable values for the
MBES system. Heading is one of the crucial
information responsible for the direction of beams in
echosounder, allowing proper alignment of the system
in the survey area. Heading is obtained from
gyrocompasses, but more and more often it is based on
GNSS measurements in satellite compasses.
Measurements form both systems may be burden with
inaccuracies and errors however satellite compasses
may additionally be affected by signal propagation
issues (e.g under the bridge or in dense urban
environment). Therefore filtration of raw data is
needed. The goal of it is basically to filter the data by
deleting outlier, removing peaks and generally to
smooth signal distribution over time.
In recent years, machine learning (ML) methods
have emerged as powerful tool capable of modelling
complex relationships in large datasets. Their ability to
learn patterns from data makes them particularly
attractive for tasks involving signal denoising and
smoothing, where classical model-driven approaches
often face limitations. Techniques such as Support
Vector Regression (SVR), deep learning-based models
and recurrent networks (LSTM, GRU, RNN) have
shown promising results in various time series and
signal processing applications, including inertial
navigation, GPS trajectory smoothing, and sensor
fusion.
The aim of the research for this paper was to
analyze and compare several machine learning
methods and their key parameters in the context of
heading data filtering for MBES surveys. This study
Processing of Heading Data with Machine Learning
for MBES Survey
W. Kazimierski
1
& M. Włodarczyk-Sielicka
2
1
Maritime University of Szczecin, Szczecin, Poland
2
Marine Technology Ltd., Gdynia, Poland
ABSTRACT: The paper presents research on using machine learning algorithms for heading signal smoothing
recorded during MBES surveys. Several numerical methods, typically used for time series smoothing and
prediction in Data Science applications are tested, like moving average, Gauss filter, Holt Winters filter and
Wittaker filter. Additionally, recurrent neural networks are analyzed. Data from real use cases are used and
parameters of the methods are verified. The methods are validated against smoothing performance (with variance
analysis) and against original function fitting (with RMSE), allowing the qualitative and quantitative assessment.
Open source python libraries are used. The results shows efficiency of such approach for this problem.
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 19
Number 3
September 2025
DOI: 10.12716/1001.19.03.22
880
aims to assess their effectiveness in reducing noise
while preserving the actual dynamics of vessel motion.
Real-world heading measurements, collected during
hydrographic surveys, will be used for this analysis.
The bathymetric data were collected using an echo
sounder PING DSP 3DSS-DX-450 mounted on the
survey vessel Hydrodron-1. The data were gathered
during the project LIDER/4/0026/L-12/20/NCBR/2021.
Unlike synthetic datasets, real survey data capture the
full spectrum of operational challenges, such as
environmental disturbances, sensor imperfections, and
vessel maneuvers, thus providing a robust benchmark
for evaluating ML-based filtering approaches. The
processing methods were elaborated in the scope of the
project SONARMUS supported by the Foundation for
Polish Science (FNP) in the FENG Proof of Concept
program under grant no. FENG.02.07-IP.05-0489/23
2 LITERATURE REVIEW
Accurate heading data are crucial for the quality of
bathymetric surveys conducted with MBES systems.
Traditional filtering approaches such as moving
average filters, low-pass filters, and Kalman filters
have been widely used in hydrography to mitigate
these effects. Kalman filtering has been popular due to
its optimality under certain assumptions of Gaussian
noise and linear dynamics. Vessel motion can be highly
non-linear, especially during turns, speed changes, or
under the influence of waves and currents.
Consequently, interest has grown in using data-driven
machine learning (ML) approaches to capture such
complex behaviors. For example Support Vector
Regression (SVR) has been applied successfully in GPS
data denoising [1]. Deep learning methods,
particularly Recurrent Neural Networks (RNNs) and
Long Short-Term Memory (LSTM) models, have
shown capabilities in modeling time-dependent
patterns in many fields related to geodata. A fine
survey on this is given in [2].Despite their successes in
related fields, ML methods have not yet been widely
adopted in MBES data processing workflows. Recent
studies suggest that they may offer significant
advantages, especially in cases where traditional
models fail to effectively filter heading data without
introducing delays or signal distortions [3]. Given the
growing interest in applying machine learning
methods in hydroacoustic, the following section
presents a literature review on their use in processing
data from MBES systems.
2.1 Machine Learning in MBES Data Processing
Machine learning (ML) has been a growing tool in
multibeam echosounder (MBES) data processing in
recent years. ML techniques have been widely
explored to improve feature detection, classification,
noise reduction, and point cloud denoising.
Ling et al. used neural networks to denoise point
cloud data from MBES systems. Their approach, based
on score-based generative models and 3D point cloud
processing techniques, effectively detects and removes
noise in MBES data. [4]
For feature detection, Snijder and Lekkerkerk
introduced the Multibeam Object Detection Inferencer
(MODI), a convolutional neural network (CNN)
specifically trained to identify seabed features such as
shipwrecks and geological formations automatically.
This work demonstrates the increasing feasibility of
deep learning for autonomous interpretation of MBES
datasets [5].
Beyond object detection and denoising, semi-
supervised ML methods have gained traction for water
column target detection [6] and for matching MBES
data with side scan sonar [7].
ML-based approaches have also been used in post-
processing step, which significantly improved
accuracy and repeatability in identifying noise and
artifacts [8], for example with Convolution Neural
Networks [9, 10]. A complementary review by Gauchia
et al. emphasized the need for hybrid automatic and
semi-automatic data cleaning strategies in
hydrographic workflows [11].
Interesting approaches for seafloor classification
and spectral analysis can be found in zones [12] or [13].
2.2 Heading estimation and navigation integration
Heading data plays a central role in ensuring precise
georeferencing of MBES measurements. Traditional
model-based approaches, such as Kalman filtering, are
widely used, but recent ML advancements offer
promising alternatives for improved heading
estimation.
Dahan and Klein introduced GHNet, a deep
learning framework capable of regressing heading
angles using GNSS-derived velocity data, even at low
speeds. This approach surpassed conventional
methods in accuracy and robustness [14]. Furthermore,
Engelsman and Klein explored learning-based
gyrocompassing to estimate heading from low-
performance gyroscopes without needing long-term
integration or model-based corrections [15].
In the context of autonomous ship navigation,
Wright examined the integration of multi-sensor
inputs—including heading, speed, and orientation—
using deep learning for dynamic vessel control. These
techniques are increasingly crucial in hydrographic
operations involving unmanned surface vessels
(USVs) [16].
TransNav has also featured studies on machine
learning-driven navigation systems, such as using
NeuroEvolution of Augmenting Topologies (NEAT)
for ship handling optimization [17] and ML-based
methods for maritime risk assessment [18]. Both
studies emphasize the importance of accurate heading
data as a critical input for safe and efficient vessel
operations.
Heading data estimation, being a part of pre-
processing stage, have been also analyzed in wider
context of acoustic data curation. Thompson, Li, and
Garcia (2023) assessed various preprocessing strategies
for echosounder data used in ML applications. Their
study emphasized that consistent normalization and
segmentation protocols have a direct impact on model
performance and generalization capabilities [19].
Similarly, Thompson et al. (2022) conducted an
evaluation focused on fisheries acoustics and found
that the choice of preprocessing strategy can
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substantially affect the interpretability and
effectiveness of ML models [20]. Interestingly, Ling et
al. in [21] presented a benchmarking study using both
classical and deep learning methods, demonstrating
that while machine learning (ML)-based approaches
are promising, traditional methods like Generalized
Iterative Closest Point (GICP) still provide superior
accuracy in fine alignment stages.
The above review shows that, some work have been
already done and using of ML for MBES is becoming a
hot topic. However despite significant advances,
several gaps remain in the application of ML to MBES
data and heading processing, showing future
directions, like non-linear noise sources, real-time
heading estimation or integration with existing
systems and software. Addressing these gaps can
accelerate the adoption of ML in MBES workflows and
improve both the efficiency and accuracy of
hydrographic surveys.
Therefore in these paper we are showing research
on utilization of ML and numerical methods for
heading data processing with the use of typical open
libraries used in data science to prove their usability in
assumed scenarios.
3 FILTRATION METHODS
The filtration, understood as removing outliers and
smoothing of navigational data (e.g. heading), can be
made with the use of methods traditionally used in
data science. For the needs of this paper we can divide
them into two categories – numerical methods and
machine learning methods. In this paragraph we
provide a very brief description of the methods used in
our research, which in fact is only a part of available
options.
3.1 Numerical filtration
Heading filtration during MBES survey can be
understood as time series filtration. This approach may
include a large variety of available filters, used for
time-series data science. In many cases they are used to
predict future values and to find trends (e.g. stock
exchange or weather prediction). In our case the
processing is rather focus on outliers and smoothing.
Various methods can provide various advantages. Low
pass filters (moving average, Gaussian) quickly damp
high frequency noise, while model based smoothers
(ARIMA, Holt Winters, Whittaker) adaptively track the
underlying data with adjustable stiffness. In our
research numerical filters are used as a benchmark for
comparing with ML approach. In this case we used
moving average and Gauss filters as examples of low-
pass filters and Holt-Winters and Whittaker filters as
examples of model-based smoothers.
A moving-average (MA) filter replaces each sample
by the arithmetic mean over a symmetric window of
width 2k+1, as in equation (1).
1
21
k
t i k t i
yx
k
=− +
=
+
(1)
It is in fact a simple finite-impulse-response (FIR)
low-pass, giving strong attenuation of high-frequency
jitter (e.g., wave-induced yaw) with low computational
cost and simple interpretation [22]. Main drawback is
however a fixed k-sample group delay in causal
operation and edge effects near the start/end of a line.
Another FIR approach is a Gaussian filter in which
a Gaussian window is provided to replace the
rectangular kernel, as given in eq. 2, where sigma is a
smoothing parameter, allowing to adjust filter’s
sensitivity. Thus Gaussian filter minimises ringing for
a given effective width and reduces frequency
sidelobes relative to the MA [22].
(2)
Gaussian filter is preferred when preserving the
curvature of gentle turns and smoothing factor allows
to fit to actual curvature.
In contrast to these fixed-kernel convolutions, Holt–
Winters exponential smoothing formulates the signal
as latent level (and optionally slope/seasonal) states
updated by exponentially weighted averages, e.g.,
additive trend, playing the role of the smoothed output
[23]. However, the effective cutoff is signal-dependent:
during sharp course alterations the filter lags, unless
parameters are adapted or robust variants (e.g.,
bounded-influence update rules) are employed.
The idea of Whittaker (Eilers–Whittaker) smoother
is to cast smoothing as penalised least squares, based
on is the discrete second difference operator. As a
result a cubic spline like smoother with explicit control
of stiffness through single parameter is achieved. It is
computationally light for dense heading logs but
sensitive to the choice of smoothing parameter.
In practice, MA/Gaussian filters serve as fast
baselines and as pre-conditioners for learning
pipelines; Holt–Winters provides an online,
interpretable tracker for low-order dynamics and
Whittaker offers a principled post-processing
smoother with tuneable rigidity. In our case they serve
as comparing benchmark to ML filters.
3.2 Machine Learning filtration
Recurrent Neural Networks and their variations are
the ones among many other Machine Learning
methods, most commonly used for smoothing tasks.
They can act as adaptive, non-linear smoothers also for
vessel heading in MBES surveys. Their usage is for this
purpose means to train them to minimize a
reconstruction or one-step-ahead prediction loss; in
deployment they operate causally for real-time use or
bidirectionally (zero-phase) for post-processing.
A traditional, simple RNN updates a hidden state
with a non-linear recursion and emits a linear readout
as the smoothed estimate (eq. 3, 4). It behaves like a
data-driven IIR low-pass: steady segments yield strong
attenuation, while turns increase effective bandwidth.
( )
1
t xh t hh t h
h W x W h b
−
= + +
(3)
t hy t y
y W h b=+
(4)
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where: ht — hidden state vector at time t; φ — element-
wise nonlinearity (tanh/ReLU); Wxh — input→hidden
weights; xt — input at time t (e.g., heading and
auxiliaries); Whh — recurrent weights; ht-1 — previous
hidden state; bh — hidden bias. ŷt— smoothed or
predicted heading at time t; Why – hidde to output
weights; ht — current hidden state; by— output bias.
The advantages includes minimal parameter count
and memory requirements, while the main limitation
is vanishing/exploding gradients, which restrict
temporal memory.
LSTM and GRU are widely known modifications of
RNN, coping with vanishing gradients. The Long
Short-Term Memory augments the RNN with gating
and a persistent cell state, mitigating gradient
pathologies and extending effective memory. In
practice, small causal LSTMs (1–2 layers, 16–64 units)
run in real time on survey hardware. Regularization
(dropout, weight decay) is recommended to avoid
overfitting short calibration runs. The Gated Recurrent
Unit simplifies the LSTM by merging input/forget
behavior into an update gate and using a reset gate. It
offers comparable accuracy with fewer parameters and
faster convergence, which is attractive under tight CPU
budgets on board. For very long or highly non-
stationary legs, LSTMs may hold a slight edge due to
the explicit memory cell [24, 25].
Based on the popularity for time series tasks, these
networks were selected for the research in this paper.
4 RESEARCH METHODOLOGY
This chapter provides description of the methodology
used in research for this paper. It is based on real data
and post-processing experimental analysis with
various filters.
The goal of the research was to analyse the
performance of ML approach for heading filtration
during MBES measurements. Typical ML filters used
for time series analysis were used and popular
numerical approaches as benchmark. The data for
experiment was acquired with real devices and the
analysis was made in post-processing stage with own
scripts and algorithms.
4.1 Data acquisition and processing
Data for the research were acquired during
hydrographical surveys performed with the
Autonomous Surface Vehicle HYDRODRON by
Marine Technology Ltd. [26]. Hydrographic data were
acquired with PING 3DSS-DX-450 sonar and heading
data with SBG Ekinox2 Subsea, which is an advanced
inertial navigation system providing position, heading,
speed and inertial information. The ASV used for the
research is presented in figure 1.
Figure 1. ASV HYDRODRON-1 by Marine Technology Ltd.
used for the data acquisition
Data were acquired in two areas. The first one was
at the Pomorskie quay in Presidential Basin in Port of
Gdynia, while the other one was located in Zawory in
Kłodno Lake. Both surveys included full MBES with
INS recordings. Data were collected with HYPACK
software pack in raw txt format. The first area is a
harbour area with maintained depth, yet as the area are
not wide, the surveys requires many profiles and
manoeuvres between them. These may influence the
stability of the heading measurements. The other are is
a natural lake and the survey required following
profile patterns. In this paper the first area is included,
after initial analysis, as it covers more turns and
heading fluctuations.
Acquired data included 6 files for the first area with
complete recordings from sensors gathered by Hypack.
The data was provided in txt files, which were then
processed via scripts in Python language. The scripts
were launched in notebooks within Google Colab
platform, which is a hosted Jupyter Notebook service
that requires no setup to use and provides free access
to computing resources, including GPUs and TPUs.
Entire processing, including visualization with
graphs in this paper and statistical analysis was
performed within this platform. Following open-
source libraries were used for data processing and
visualization: pandas, matplotlib, numpy, statsmodels,
scipy, whittaker_eilers, sklearn and tensorflow.
4.2 Evaluation Metrics
The goal of filtering heading data, as well as for other
auxiliary sensors is to provide a reliable, accurate, yet
sufficiently smoothed signal. Therefore for
performance assessment of the filters we were
analysing both – the roughness of the produced
trajectory and the accuracy, understood as the distance
to the unfiltered data. Generally the task is to minimize
the roughness of the function, while simultaneously
maximizing the accuracy.
For roughness measurements we use three metrics:
883
− Standard deviation
− Sum of squared second differences (SSSD)
− Variance of local first differences
Standard deviation of first differences (global) – eq.
(5) is a compact, window free indicator of overall
roughness.
( )
( )
2
2
1
1
N
y t t y
y
N

=
= −
−
(5)
where:
y
— sample standard deviation,
y
—
global mean; N — sample count
Sum of squared second differences (SSSD)
aggregates discrete curvature and highlights residual
oscillations (eq. 6).
( )
2
3 1 2
2
N
t t t t
SSSD y y y
= − −
= − +
(6)
where: yt — heading at discrete time t; N — sample
count
Variance of local first differences (eq. 7) on the other
hand is a windowed statistic, which diagnoses short
scale jitter around a given time index.
( )
( )
2
1
1
1
1
,
t
t
wi
Wt
t
t
iW
i i i i
Wt
t
iW
Var y y y
W
y y y y y
W
−
= −
−
= − =
(7)
where: Wt— centred index window around t with size
Wt|=w; Wt(y)— mean of first differences within this
window.
For assessing the accuracy, we use typical
indicating values:
− Mean Absolute Error (MAE);
− Root Mean Square Error (RMSE).
Mean Absolute Error (MAE) is a scale-preserving
measure robust to occasional outliers, according to
eq. 8:
1
1
N
t t t
MAE y y
N
=
= −
(8)
where: N — number of paired samples; yt— reference
(ground truth) heading at time t; ŷt — model estimate
at time t. In our case real measured value is set as
ground true |·|.
While MAE is less sensitive to large residuals, Root
Mean Square Error (RMSE) emphasises large
deviations by squaring residuals before averaging and
taking a square root (eq. 9):
( )
2
1
1
N
t t t
RMSE y y
N
=
= −
(9).
This set of measurements indicators allows to assess
the quality of the filtration in terms of roughness and
accuracy.
5 RESULTS
The results are presented in two parts. Firstly we
provide results achieved with numerical methods for
the area near Pomorskie quay. Then the same data are
analyzed with neural methods. Such approach led to
comprehensive analysis of the results.
5.1 Pomorskie quay area – numerical methods
In this area, data were acquired from six survey
profiles. Gyro heading along time, for an example
profile is presented in figure 2. Generally the course
was stable, however some rapid fluctuations in some
places arose. These can affect final data and should be
filtered.
Figure 2. Raw gyro heading for one of analysed profiles.
In figure 3 selection from this profile is presented,
showing the efficiency of filtration of numerical
methods.
Figure 3. Numerical filtration for gyro heading – selected part
of the profile.
The metrics for this profile, showing roughness and
smoothness are given in table 1. The analysis of the
table confirms that proposed metrics can be used for
roughness and smoothness assessment. However
standard deviation seems to be less sensitive than
SSSD. In the presented example, the most smoothed
values were achieved with Gaussian filter, which
resulted in small SSSD and other smoothness metrics.
Simultaneously MEA and RMSE were higher than
Holt-Winters, yet still smaller than Moving average.
Based on this example Whittaker filter showed the best
balance between smoothness and roughness. This can
be also observed on the graph in figure 3.
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Table 1. Metrics for numerical filters for example profile.
Method
Standard
deviation
SSSD
Variance of
local diff.
MAE
RMSE
Raw data
2,17
2,34
0,00109
0
0
Whittaker filter
2,12
0,004
0,00069
0,076
0,098
Moving average
2,16
0,021
0,00085
0,211
0,28
Gauss filter
2,10
0,002
0,00062
0,118
0,15
Holt-Winters filter
2,17
2,19
0,00114
0,009
0,013
5.2 Pomorskie quay area – neural methods
In this section the same profiles were analyzed with
neural methods. Then statistics for all six survey
profiles for Pomorskie quay were calculated for
numerical and neural filters.
In figure 4 the same part of the profile as in fig. 3 is
presented for neural filters. It can be noticed that the
signal is smoothed, yet it follows the changes of
heading. Very small fluctuations (about 0,2 degrees)
are filtered out. The size of the smoothness can be
adjusted with filter parameters and analyzed networks
gives similar results.
Figure 4. Numerical filtration for gyro heading – selected part
of the profile.
It should be however noticed that in case of
proposed neural methods, the processing time is
higher as each time, iterative training period is needed.
Roughness and smoothness metrics for all analyzed
filters (numerical and neural) are given in table 2. These
are average values for all analyzed profiles. Interesting
observation is that not all metrics are suitable for joint
analysis of many profiles. Average standard deviation
for all methods is more or less the same, which makes
it not useful for such analysis. However SSSD shows
good discrimination, while variance of local
differences is flatten. MAE and RMSE react similar,
however RMSE is more sensitive to variations. Thus,
SSSD and RMSE are the metrics to analyze roughness
and smoothness of filtered signal.
The analysis of SSSD and RMSE in table 2, shows
that numerical filters generally more significantly
smooths the signal, except of Holt-Winters filter.
Neural filters generally better fits raw data, which
results in smaller RMSE. This could be expected, taking
into account the background of these filters. Generally,
the bast balance was achieved with Whittaker and
Gaussian filters in numerical approach and with GRU
in neural approach.
Table 2. Average metrics for all analysed profiles and
various filters
Method
Standard
deviation
SSSD
Variance
of local
diff.
MAE
RMSE
Raw data
3,323
6,310
0,0033
0,000
0,000
Whittaker
filter
3,275
0,011
0,0024
0,066
0,118
Moving
average
3,338
0,063
0,0028
0,209
0,362
Gauss
filter
3,245
0,007
0,0022
0,105
0,190
Holt-
Winters
filter
3,325
3,374
0,0032
0,009
0,015
RNN
3,389
0,895
0,0030
0,096
0,107
LSTM
3,399
0,747
0,0030
0,080
0,098
GRU
3,378
0,508
0,0029
0,074
0,081
6 CONCLUSIONS
The paper shows initial research on filtration of
heading data during Multibeam Echosounder Surveys.
This process is needed as the data many times is
affected by temporal inaccuracies and sudden jumps of
signal, which affects quality of MBES measurements.
In this research we propose to use Machine Learning
approach known from time series analysis, using three
various neural networks, based on recurrent neural
network. For comparison, we also use traditional
numerical filtration methods. Real data form the
measurements were used for analysis.
The results show that Machine Learning approach
can be used for this purpose with good results – better
than some numerical methods. However, the
drawback of analysed neural network is the need of
iterative training for any new dataset. It means that for
each profile new network parameters needs to be
established, which takes time and efforts.
Simultaneously comparatively good results were
achieved with some numerical filters, namely
Whittaker smoother and Gaussian filter. The neural
approach can be in this situation treated as interesting
alternative, however for real-time implementations,
indicated numerical filters are recommended.
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