Marine Compass Data Quality Control Method Based on Robust Surface Fitting

Towed streamer positioning is a key step in offshore seismic exploration, and its positioning accuracy directly affects the imaging quality of subsequent seismic data. Accurate and clean magnetic compass observations are an important guarantee for high-precision towed streamer positioning. However, the existing compass data quality control methods have the problem of insufficient utilization of spatial distribution prior information, which heavily relies on mathematical statistical characteristics of the data while neglecting the crucial factors of streamer shape, bringing about an inadequate elimination of gross errors. In this work, a compass data quality control algorithm based on robust surface fitting is proposed to improve the accuracy of positioning. Firstly, the towed streamer positioning algorithm and the prior information of the spatial structure distribution of the compass are introduced. Secondly, the robust surface fitting to control the quality of compass data is proposed, and the specific process of the algorithm is given. Lastly, the effectiveness of the algorithm is verified by simulation and measured data. The results of simulation experiments indicate that compared with the traditional single-streamer threshold robust algorithm, the new algorithm has a significant effect on the quality control of compass data, and the positioning accuracy of the towed streamer is significantly improved. The positioning precision of along-line direction is improved by 52.1%, and that of the across-line direction by 52.4%. These demonstrate that the proposed new algorithm is effective for detecting and eliminating gross errors in compass observations in real-time operation scenarios, improving the stability of towed streamer shape calculation, which can better ensure the smooth progress of offshore towed streamer seismic exploration operations.


Introduction
Towed streamer seismic exploration plays a prominent role in global offshore oil and gas prospecting, underscored by its growing economic and technological significance.The cornerstone of seismic exploration lies in attaining precise positions for detectors.Reliable and high-precision positioning data is essential for achieving high-resolution seismic exploration.However, the acquisition environment for towed streamer exploration is intricate, the configurations of sensor equipment is variable, the prior auxiliary information is unavailable, and the mature GNSS (Global Navigation Satellite System) technology is unable to be applied underwater, all these factors collectively pose formidable challenges to obtaining accurate positions of the detectors.To obtain the streamer shape and the accurate position of the detectors on the streamer, the survey system is typically equipped with positioning sensors, including DGNSS (Differential GNSS) units, RGNSS (Relative GNSS) units, gyrocompasses, magnetic compass and acoustic transducers, where the compass is to measure the tangent magnetic azimuth of the underwater streamer at that point.It plays an irreplaceable role in the mathematical model of the streamer, the calculation of the towed streamer position, the coordinate position of the detectors and both the display and adjustment of the streamer shape.However, the complex underwater environment unavoidably causes various gross errors within compass observations, consequently impacting the precision of shot sources and detectors positioning, ultimately influencing seismic imaging resolution and even leading to image degradation.A deviation of just 1° in the compass can lead to an approximate streamer bias of 20 meters [1].
Researchers from different regions have explored methods for quality control of compass data.As for random errors, low-pass filtering and Kalman filtering are typically used during data pre-processing to suppress strong noise within compass observations [2].In the parameter estimation stage, virtual observations can be added to the generalized least squares adjustment calculation to obtain a more accurate streamer polynomial fitting coefficient [3].In view of the systematic error, Yi et al. [1] studied the influence of magnetic declination and meridian convergence angle on the orientation error of the compass, and used the geomagnetic model and improved projection method to finely correct it.Martin et al. [4] proposed to use the tail buoy to rotate a single streamer to eliminate the magnetic deflection angle correction error in the compass observations, so that the node positioning accuracy reached the accuracy level of DGNSS positioning.Schneider et al. [5] proposed the utilization of real-time magnetic declination correction in polar exploration operations to eliminate systematic errors in compass observations.In dealing with gross error, Vidal et al. [6] set different statistical threshold values for various types of observations during data pre-processing and discarded observations exceeding these thresholds.Yi [7] classified gross errors into significant and minor categories, introduced an iteratively weighted robust estimation method to eliminate significant gross errors, and utilized an improved firstorder lag filter to eliminate minor gross errors, achieving better results.Li [2] employed Kalman filtering to automatically identify, eliminate, and interpolate abnormal positioning data.Xu [8] smoothed compass observations using an improved first-order lag filter, and utilized robust estimation to fit curves, thereby detecting continuous gross errors.Wu et al. [9] made full use of the characteristics of low sampling rate of compass, proposed a sliding window data storage structure and utilized methods like extreme value analysis, rate of change analysis, and statistical analysis to detect and eliminate gross errors in observations.
In general, the methods mentioned above have certain limitations, predominantly inheriting concepts from gross errors detection in the field of surveying, and most of them heavily rely on thresholds set by manual, while only utilizing the mathematical and statistical characteristics of compass observations and its time series, neglecting geometric structural characteristics and other prior information of the towed streamer.This limits the comprehensive utilization of data and accurate gross errors identification to a certain extent.Moreover, the small residual gross error was not further processed during the preprocessing stage, which could not eliminate the persistent anomalies in the compass observations, resulting in more residual errors in the later parameter estimation, and increased errors in the subsequent positioning results.Given the insufficiency of current methods in data utilization, their intricate computation steps, and their inability to accurately discriminate gross errors, this paper proposes a robust surface fitting algorithm for compass data that takes into account multi-streamer geometrical features.By engaging spatial distribution, the algorithm effectively detects and eliminates gross errors, offering valuable insights for the development of independent towed streamer positioning algorithms and software in China and thereby propelling the advancement of high-precision offshore seismic exploration techniques.

Marine towed streamer seismic exploration positioning method
The navigation and positioning equipment used in offshore seismic towed streamer exploration, as shown in figure 1.The DGNSS and gyrocompass are installed on the exploration vessel.The DGNSS is used to obtain the absolute position of the vessel, while the gyrocompass provides the vessel's orientation.Both the seismic gun array and the towed streamer are equipped with RGNSS buoys, which measure baseline vectors between them and RGNSS reference stations on the vessel, determining their positions relative to the towed streamer (sometimes RGNSS is also mounted on the front end of the towed streamer) [10]- [12].The underwater streamer is equipped with acoustic transducers and compasses.Among them, the acoustic transducers are used to measure the distance between the acoustic nodes on the streamer, and the gun array and the acoustic devices on the tail buoys together form an acoustic ranging network [13].The compass is used to determine the tangential orientation of the towed streamer at each compass node [14].The basis of obtaining high-precision towed streamer positioning data is the high-precision and robust towed streamer positioning method.Towed streamer positioning methods in offshore applications can be categorized into numerical methods [3], [5], [15], [16] and analytical methods [17]- [21].Assuming that the basic units of the towed streamer are arcs, polynomial curves, and line segments, the towed streamer positioning model can be divided into arc models, polynomial models, and curve integration models [22].The curve integration model is greatly affected by observational errors and may suffer from error accumulation, particularly in the case of compass azimuth observations.To meet precision requirements, shorter integration steps are typically required.Arc models require smooth connections between end nodes of various arc segments and rely heavily on discretization, making them susceptible to the influence of compass observations and error accumulation effects.Polynomial models are currently the most widely studied positioning models, comprising polynomial models based on streamer offsets and tangential azimuth angles [9], and has gradually developed into a polynomial curve fitting with rigorous theory, high computational efficiency, and has the ability to make full use of observation data model [23].
As shown in figure 2, considering the mileage of each sensor on the towed streamer from the starting point as the independent variable and the sensor's coordinates as the dependent variable, a polynomial function relationship between the two can be established, with the model's parameters being the coefficients of the polynomial curve [23].By utilizing approximate parameters obtained through preprocessing, observation error equations can be estimated, followed by the creation of virtual observation error equations, ultimately leading to the estimated values.The relationship between point  on the streamer and its curve offset   is as follows, where the coordinate observation equation as follows.
In the formula,   is the coordinate vector of point ,   T = [  0 ,   1 , ⋯ ,    ] represents the curve offset,  is the order, generally within 7 orders;  = [ 0 ,  1 , ⋯ ,   ] T ,  = [ 0 ,  1 , ⋯ ,   ] T ,  = [ 0 ,  1 , ⋯ ,   ] T are the polynomial coefficient corresponding to the coordinate components of   ,   ,   respectively.The values of  have already been determined for various sensors on the streamer during design and installation, and streamer fitting is the process of streamer positioning.According to the firstorder Taylor series, the corresponding error equation is obtained at the approximate value.The error equation corresponding to the coordinate observations as follows.In the formula,  0 ,  0 is the approximate values of the parameters to be estimated corresponding to the ,  coordinate components of the streamer, and   =     T , where  = diag(0,1,2. . ., N), and    is a constant term.Whether using numerical or analytical methods for towed streamer positioning, the determination of detector positions heavily relies on compass azimuth observations.This paper employs a theoretically rigorous and efficient polynomial positioning model.As Equation 4 clearly illustrates, compass observations significantly impact towed streamer calculations.Therefore, the quality of compass observations directly affects the accuracy of towed streamer node positioning and subsequent seismic data inversion results.

A new method of data quality control for compass observations
Traditional algorithms do not take into account the prior information that adjacent streamers exhibit closely related compass observations on the same horizontal plane.These methods are locally restrictive, and may misjudge the gross errors as the correct observations and fit it, which in turn affects the positioning results of the subsequent parameter solution.
The underwater motion of the towed streamer, owing to its non-rigid physical properties, gives rise to a geometric form resembling a smooth curve.Compass observations on the same streamer show gradual changes.Moreover, adjacent streamers demonstrate high similarity in terms of forces and layout, resulting in streamers maintaining a nearly parallel configuration.Consequently, compass observations on streamers at the same horizontal level are generally similar.Figure 3   The distribution of observations of the compass presents a relatively smooth surface, which can be fitted by polynomials.Considering the three-dimensional coordinates (, , ) formed by the coordinates and observations of compasses on streamers as a point on the fitting surface, solving for the coefficients of the polynomial surface requires establishing a mathematical relationship between compass observations and the compass's coordinates in the local coordinate system.The functional model can be represented is: In the formula, (, ) is the fitted polynomial surface function,   is the coefficient to be established,  and  are the orders of the polynomial  and  respectively.The matrix form is:  =  (6) Expand the matrix.
The surface polynomial coefficients  can be obtained according to the least square principle.In the process of parameter estimation, the IGGIII scheme is used for robust processing [24]. = (  ) −    (8) In the formula,  is the weight of the observations of the compass, which is obtained from the variance-covariance matrix of the observations.The order  in the  direction is 2, and the order  in the  direction is 4, and the order of the cross-terms is 2, which can meet the accuracy requirements.To avoid excessive or insufficient offsets and ensure even distribution of compass observations, normalization of offsets is conducted.Finally, the compass coordinates are substituted into the coefficient to obtain the fitting value of the observation, and then the residual   is obtained according to the fitting value.The workflow of algorithm is depicted in figure 4.

Experiment and result analysis
Normally, the survey vessel has to tow the streamer for sequential operations one by one because of the offshore seismic survey lines are usually arranged in parallel straight lines in the working area, so that it needs to turn to the next survey line after the previous one is completed.To validate the effectiveness of the compass observations robust surface fitting quality control, both simulated and real-world data were employed for experimental verification and analysis.Due to the difficulty in obtaining real coordinates of the streamer network in offshore surveys, we applied a self-developed simulation software SimStr [25] to simulate online operational scenarios and validated the algorithm proposed in this paper using the obtained simulation data.Moreover, a measured data experiment is designed to verify and analyse the actual effect in the real-world application.

Simulated experiment and result analysis
The configuration of the simulation data includes a total of 8 streamers.Each streamer is 7000 m long and the distance between them is 100 m.Also, each streamer is equipped with 25 normal compasses, with one additional compass mounted at the tail buoy.The installation distance between these compasses on the streamers is 300 m.An acoustic network with a front-back configuration is employed, with 2 acoustic nodes at the head of each streamer spaced 75m apart, forming the front acoustic network, and 18 acoustic nodes mounted at the rear part of each streamer from a distance of 3500 m, spaced 200 m apart, together constituting the rear acoustic network.Detector spacing on the streamer is set at 12.5m, resulting in 561 detectors mounted on each streamer from end to end.The speed of vessel is set at 2m/s, assuming a uniform and straight motion without considering ocean currents, with the survey line facing true north.A total of 1000 shot epochs are included.Furthermore, the survey vessel is equipped with DGNSS and a gyrocompass, while both the seismic arrays and the tail buoys are equipped with RGNSS.The real value of the compass observations is 360 °.These observations are added with random errors   ∼ (0,   2 ), where   = 1 °.Moreover, gross errors are introduced randomly, with small gross errors constituting around 8% of the data and being about 10 times the size of random errors, while large gross errors make up approximately 2% of the data and are about 50 times the size of random errors.Meanwhile, continuous large gross errors and small gross errors are randomly added in adjacent epochs.In order to control the variables, no errors have been added to the acoustic distance observations.The simulation experiment simulates a real-time scenario.The positioning model uses a curve polynomial model to calculate the position information of the streamer.The following two schemes are used to control the data quality of the simulation compass observations: Scheme 1: The traditional single streamer fixed threshold robust algorithm is adopted.In the realtime scenarios, the new data obtained is compared with the change rate of the compass observation of the adjacent artillery epoch.If it exceeds the threshold (set to 1°), it is identified as a gross error.
Scheme 2: The robust surface fitting algorithm proposed in this paper is used to process the compass observations.Where the X direction of compass surface fitting function is 2nd order, the Y direction is 4th order and cross-terms is 2nd order.
The compass fitted values obtained through the robust surface fitting algorithm proposed in this paper are compared with the original observations and the real values in the simulated experiment, and the results are statistically analysed.Figure 5 presents a comparison of 150000 compass observations and fitted values for 1000 continuous shot epochs in real-time scenarios.Figure 6 illustrates the absolute average of compass fitting residuals for 1000 continuous shot epochs in the same real-time scenario.Relevant statistical data is provided in table 1. Comparing the simulated compass observations and the fitted values after robust surface fitting across epochs, as shown in figure 5, and evaluating the absolute average of residuals for each shot epoch after surface fitting, as displayed in figure 6, indicates that the algorithm exhibits commendable resistance to gross errors, with fitted values closely approximating real values.
In the formula,  is the number of shot epoch,    and    are the calculated values from the robust surface fitting algorithm,    and    are the reference values, and the real value is used as the reference value in the simulation experiment.To better process data and maintain the accuracy of seismic exploration data, the bias is decomposed into DA bias (distance along-line) and the DC bias (distance cross-line) according to the azimuth angle  of the measuring line.The calculation formula is as follows.From table 2, 3, and figures 7, 8, it's evident that the traditional algorithm does not effectively eliminate gross errors.Additionally, due to the employed front-back acoustic network configuration, which lacks acoustic distance observations in the front-middle part of the streamer, the positioning bias using streamer adjustment algorithms is slightly large.The proposed compass observations robust surface fitting algorithm in this paper is notably superior to the traditional method, maintaining a relatively small average DA and DC bias for each epoch.This shows that the algorithm in this paper can obtain more correct and reliable results, and the new algorithm has good stability.Compared with the traditional single-streamer threshold robust algorithm, the average point deviation of 1000 shot epochs geophones in this algorithm is 4.19 m.The average DA deviation is 2.22 m, of which 92.86% epoch DA deviation is better than 5 m, and 99.09% epoch DA deviation is better than 10 m.The average DC deviation is 3.52 m, of which 78.77% of the epoch DC deviation is better than 5 m, and 96.18% of the epoch DC deviation is better than 10m.

Field experiment and results analysis
The measured experiment data comes from streamer positioning records during a turning scenario in the South China Sea streamer seismic exploration operations conducted in 2023.The surveyed line trends southeast direction, with azimuth angles of 162° or 342° (adjacent survey lines operate in reverse).This operation has a 10-streamer configuration, the length of the towed streamer is about 10km, and the distance between streamers is about 100m.Each streamer carries 23 compasses, with a spacing of about 300m.The tail buoys and the shot arrays are equipped with RGNSS, and the detector installation is set at 12.5 m.The acoustic network is configured in the front-back arrangement.In the real-world turning scenario, the navigation system doesn't record node coordinates, but accurate calculation and display of the streamer shape are crucial.The correct display of the streamer provides a robust reference for online adjustments, steering, turning, streamer shape adjustments, obstacle avoidance, and other operations on the exploration vessel.In complex turning scenarios, the stability of compass observations is not high, and their gross errors have a greater impact on streamer shape calculation and display.To fully validate the robustness in a streamer turning scenario, a section of data with continuous considerable gross errors is selected for testing.Figure 9 presents a comparison of the calculated streamer shapes using two schemes.It is evident that the proposed Scheme 2 accurately calculates streamer shapes that closely resemble the real streamer shape, which is obviously better than the traditional scheme.Figure 10 illustrates the comparison between original raw data and fitted values for one epoch.The proposed robust algorithm effectively detects and eliminates gross errors, and smooths the compass observations.

Conclusion
High-precision towed streamer positioning is an important guarantee for offshore seismic exploration operations, which is also a key step in achieving high-precision imaging of subsequent seismic data.In order to solve the problems of imperfect data quality control methods and insufficient utilization of prior information, such as the spatial distribution of compass observations, this paper combine the spatial distribution characteristics of compass observations to establish a mathematical model of data quality control based on streamer shape, and proposes a robust algorithm based on surface fitting.The algorithm can make full use of the spatial distribution of compass to accurately constrain the observations.Compared with the traditional single-streamer threshold robust algorithm, simulation results show that the proposed algorithm has obvious advantages in the robust effect and the precision of the towed streamer position solution, and significantly improves the positioning precision of the detector.Moreover, the positioning precision along the survey line is better than 3 m, which is increased by 52.1%, and the precision in the cross-line is better than 4 m, which is increased by 52.4%.These results verify the feasibility and superiority of the new algorithm in theory.Also, the experimental results of the measured data demonstrate that the proposed algorithm can accurately display the streamer shape, greatly improve the stability of the streamer shape calculation, and effectively avoid misjudging the knotting of the streamers due to calculation errors caused by gross errors in compass observations.Based on the numerical results of experiments, the proposed algorithm can effectively eliminate gross errors in compass observations and significantly reduce the impact of random errors, thus it can obviously improve the positioning precision of the detector and better meet the needs of offshore seismic exploration operations and subsequent analysis of seismic wave data.
illustrates the spatial distribution of adjacent compass observations in scenarios involving offline turns and online exploration operations, clearly demonstrating their coherent similarity.Therefore, compass observations can be regarded as constituting a comprehensive surface, aligning with an overall surface model.Based on the prior the spatial distribution of the compasses, the surface polynomial can be utilized to conduct surface fitting on all the observations on the streamers to detect and eliminate gross errors.

Figure 3 .
Figure 3. Spatial distribution of compass observations in the offline turn scenario (a) and the online exploration operation scenario (b).

Figure 4 .
Figure 4. Workflow of the compass data quality control algorithm based on robust surface fitting.

Figure 5 .
Figure 5.Comparison of 150000 compass observations and fitted values for 1000 continuous shot epochs in real-time scenarios.

Figure 6
Figure 6The absolute average of compass fitting residuals for 1000 continuous shot epochs in the same real-time scenario.

Figure 9 .
Figure 9.Comparison of streamer shapes calculated by two schemes in the offline turn scenario.

Figure 10 .
Figure 10.Comparison of original compass observations and fitted values in turning scenario.
In the formula, ，，are the corresponding parameter corrections, and    ，   ，   are constant terms.The azimuth observation equation for the compass as follows.In the formula,   is the observed azimuth value of point , and   ′ (  ),   ′ (  ) are the derivatives of polynomial functions of  and  coordinate components at point , respectively.The corresponding error equation as follows.

Table 1 .
Statistics of compass observations surface fitting statistics in real-time scenario.(unit:°)Calculating the positional deviations of streamer nodes, and the calculated coordinates of detectors are converted into plane coordinates under UTM (Universal Transverse Mercator) projection.Considering the absolute coordinate errors in geophysical surveys, the following deviations  bias can be utilized as indicators for algorithm accuracy evaluation.

Table 2 .
Comparison of the average DC bias of all detectors calculated by two schemes in real-time scenario.Detectors deviation results statistics of two schemes in real-time scenario.(unit: m)

Table 3 .
Detectors deviation percentage statistics of two schemes in real-time scenario