1. Introduction
The Global Navigation Satellite System—Acoustic (GNSS-A) positioning technique combines the GNSS sea surface dynamic positioning technique and the underwater acoustic positioning technique [
1,
2], which is the important basis of marine science research and engineering surveys. As the main method of marine geodesy, the GNSS-A positioning technique can measure the displacement at plate boundaries [
3]. Several great plate faults lie at the bottom of the ocean [
4]. The information provided by the seafloor’s precise positioning technique can not only be used to monitor the natural processes of the ocean, such as seafloor earthquakes, undersea volcanic eruptions, and tsunamis, but also enrich the modelling data of earth system science. The GNSS-A positioning technique also provides important technical support for marine oil and gas exploration and engineering construction [
5]. The long baseline positioning system is the underwater acoustic navigation and positioning system with the widest coverage and the highest accuracy, whereas its application also requires the adoption of the GNSS-A positioning technique for the precise position calibration of its seafloor array [
6,
7,
8,
9].
At present, the accuracy of the seafloor position obtained by the GNSS-A positioning technique is limited by systematic errors, which mainly include the offset error of lever arms between shipborne devices and the bias of sound velocity measured in seawater [
10,
11]. The lever arm is the geometric vector in the ship-fixed coordinate system [
12,
13], which is used to transmit the GNSS dynamic positioning results to the acoustic transceiver centre. Because the actual installation position of the GNSS and acoustic devices is not completely the same as the planned position and direction in the ship’s fixed coordinate system, there is an offset error in the lever arm installation [
14]. Due to the influence of dynamic ocean processes, such as tides and internal waves, the sound velocity underwater has temporal and spatial changes, which is especially irregular in the shallow sea [
15,
16,
17,
18]. Different from the electromagnetic waves of GNSS, which directly use the velocity of light as the velocity approximation, the velocity of the sound signal in seawater needs to be measured by professional equipment (the sound velocity profiler or CTD) [
19]. Therefore, the systematic bias of the sound velocity is composed of an expression error, and a measurement error is introduced.
For transceiver lever arms, professional ships for marine geodesy install acoustic transceivers in the instrument well at the bottom of the middle of the ship [
20], and the total station is used to measure the lever arm between the transceiver and GNSS positioning reference point with millimetre accuracy. Meanwhile, some teams attach transceivers and GNSS equipment to two ends of a pole that attach to one side of the ship [
21,
22], and it is assumed that there is only a vertical lever arm offset between the two sets of equipment, and the offset is considered to be measured precisely with no error. Generally, the transceiver and GNSS antenna are fixed at different positions of the ship [
23,
24,
25]. In this situation, the transceiver is attached at the bottom of the pole on one side of the ship, while the GNSS is installed at the top of the ship. The lever arm between the two devices cannot be directly measured precisely, so the offset in the calculated range value is large.
For a sound velocity error, researchers in marine geodesy arrange the acoustic transponder array on a seafloor circle with the radius of water depth and can estimate the position changes of the array centre (virtual reference station) with centimetre-level accuracy by using shipborne GNSS and acoustic equipment [
26,
27]. Although the research of the virtual station is useful for the determination of the plate motion, which meets the needs of the geodetic survey, it is not specific to the precise position of each entity transponders; thus, its results are hard to be directly used in engineering construction and marine environment monitoring. Some scholars studied the positioning technology of the seafloor entity station composed of a single acoustic transponder, and proposed methods of optimising sailing tracks, processing methods of measurement differencing, and using distances between seafloor stations to build an adjustment model [
28,
29,
30], which reduces the temporal and spatial correlation error and improves the positioning precision. However, few studies specifically discuss the systematic biases of the velocity of sound, including sound velocity measurement errors and temporal and spatial changes.
This paper deeply analysed the function model of the GNSS-A positioning technique for the single entity station and focused on the method of correcting systematic errors. Aiming at the problem of the high correlation between vertical coordinate/offset parameters and the bias of sound velocity measurements in the GNSS-A positioning, a sample search method was proposed to fix the lever arm offset in the vertical direction and detect the measurement error of sound velocity. By improving the GNSS-A positioning model, the sound velocity error and lever arm offset in equipment installation can be determined. Using shallow sea trial data, the GNSS-A positioning model was verified, and the transceiver lever arm offset and sound velocity bias were effectively corrected. Finally, the positioning accuracy of the seafloor station reached the centimetre level.
2. GNSS-A Positioning Model
GNSS-A positioning is a technique based on a range intersection, and the process is as follows. The position of the GNSS antenna can be obtained by utilizing GNSS technologies, and after transformation to the shipborne transceiver acoustic centre, the three-dimensional (3-D) coordinates of the seafloor station (transponder) can be calculated through the Euclidean distances converted from the travel time between the surface and seafloor acoustic equipment.
Figure 1 shows the fundamental elements of this technique, in which the distance (
) between the shipborne acoustic transceiver and the seafloor transponder can be derived by the following:
where
denotes the round-trip travel time,
means the sound velocity of the sea area during the trial period.
,
and
epresent the coordinate components of the seafloor transponder in the northing, easting, and height of the geodetic coordinate system, respectively, and they are the unknown parameters to be solved.
,
and
enote the transceiver positions in the northing, easting, and height of the same coordinate system, respectively.
As the acoustic transceiver and GNSS equipment are fixed on different positions of the ship, the position information should be transformed from the GNSS reference point to the acoustic transceiver centre. For the transformation, in addition to the GNSS positioning results, the ship attitude (heading, pitch, and roll) and lever arm between the transceiver and the GNSS reference point are also required. The position of the transceiver can be obtained based on the GNSS dynamic position results, lever arms, and rotation matrix:
where
,
, and
indicate the GNSS position results.
,
, and
mean the lever arms from the GNSS equipment to the shipborne acoustic transceiver in the ship-fixed reference frame (with the origin at the GNSS reference point and a forward X-axis, a starboard Y-axis, and an upward Z-axis centrally aligned with the ship body axis).
represents the rotation matrix from the ship fixed frame to the geodetic coordinate system:
If the lever arms between the shipborne GNSS and the transceiver are known, Equation (2) can be substituted into (1) directly, and after linearising, the basic function model of GNSS-A positioning is obtained as follow:
where
represents the initial values of unknown parameters (such as
,
, and
),
means the geometrical distance calculated by
.
,
, and
represent
,
and
respectively, which are the partial derivatives of the observation to each unknown parameter at
.
represents the constant term in the function model.
denotes the residual error of each observation.
The velocity of sound in seawater is about 1500 m/s and varies in space and time. This characteristic affects the distance inversion between the acoustic transceiver and transponder, and then the sound velocity bias becomes one of the leading systematic errors for underwater acoustic positioning. Generally, both the shipborne acoustic transceiver and GNSS equipment are installed on site. Thus, the transceiver lever arms are obtained by accumulating measurements in the different directions of the ship fixed frame. Owing to the offset error caused by installation, the lever arms measured are inaccurate, which is another main systematic error. With redundant observation information, the lever arm offset can be set as parameters with sound velocity correction and transponder position. Thus, the observation equation can be expressed as follows:
where the 3-D coordinate of the transceiver contains unknown parameters to be solved,
,
and
:
By linearizing the observation equations, Equation (7) can be derived, and then the improved function model of GNSS-A positioning can be expressed with (9). Based on the least square principle, the parameters can be determined with (10):
3. Experimental Data
For verifying the GNSS-A positioning method, this paper utilised data from a sea trial carried out near Lingshan Island, Qingdao, China, on December 1, 2017. The data mainly included the GNSS dynamic positioning results, ship attitude, travel times of acoustic signals, measured lever arm from the GNSS reference point to the transceiver, and sound velocity profiles. Position and attitude information of the survey ship was obtained using the POS MV 320 system from the Applanix Company. Based on post-processed kinematic technology, the position of the system reference point can be acquired, and the results of ship attitude can be obtained from the inertial navigation system and GNSS of the POS MV. The positioning results illustrate that the mean Standard Deviation (STD) of each epoch in the northing, easting, and height directions were 1.1, 1.0, and 2.8 cm, respectively. For the hull attitude results, the mean STD were 0.02°, 0.01°, and 0.01° for heading, pitch, and roll, respectively. According to the predetermined ship sailing track (a circle with cross shown in
Figure 2), the measurements were collected for the following model discussion. The period of the selected data was about 77 min, during which the sound velocity in seawater was measured by SV PlusV2 sound velocity profiler of the AML Oceanographic Company.
Figure 3 shows the sound velocity profiles, where the blue points represent the original sound velocity observed during the trial period, and the red dotted line represents the weighted mean sound velocity (1454.067 m/s), which was calculated from the original observations. Considering our trial was carried out in the shallow sea (about 25 m), it is safe to fix the weighted mean sound velocity as the prior value in subsequent calculations. The sea state on the trial day was fine, from Calm to Smooth, but there was a sound velocity spring layer (about 2.5 m/s) at the depth 5–10 m from the sea surface, and the sound velocity increased rapidly from 1447 m/s to 1455 m/s, which may be caused by the large sea current in the trial area.
4. Questions Raised
The two main errors, i.e., the offset errors of the shipborne equipment and sound velocity bias, exist in the measurements, whereas the ranges of such errors are unclear. To discuss the influences of these errors on the position results of the seafloor transponder, they were estimated as unknown parameters with the seafloor station’s 3-D position. Assuming the lever arms and velocity of sound have no error, then the parameters that need to be estimated are just 3-D coordinates of the seafloor transponder. In this case, the error equation is (4), and this solution model is called the NEH model, where N, E, and H represent the coordinate components of the transponder position in the northing, easting, and height directions, corresponding to , , and , respectively. If errors exist in the transceiver lever arms, but no error exists in the sound velocity, the three offset components of the lever arms can be solved together with the 3-D position of the seafloor station. In this case, the error equation is (7) without the parameter, and this solution model is called the NEH+dxyz model, where dxyz means including three transceiver lever arms in the model and x, y, and z correspond to , , and , respectively. When an error exists in the sound velocity, but no error in the lever arms, the sound velocity correction parameter and the 3-D coordinates of seafloor transponder should be solved together. Then, the error equation can be written as (7) without the lever arm parameters, and this solution model is called the NEH+dv model, where dv indicates including the parameter of the velocity of sound in the model, corresponding to . It should be noted that for those variables not estimated in certain models, they are fixed in the observation equation as initial values (measured during the trial).
The observation data were processed by the NEH, NEH+dxyz, and NEH+dv models, and their results are summarised in
Table 1. According to the table, significant differences in the three model’s results were observed, especially in the vertical direction. The NEH model’s results feature a higher STD (about 0.12 m in horizontal and 0.24 m in vertical). Adding the lever arm parameter can reduce the STD, but the results of the vertical parameter (vertical components of transceiver lever arms and seafloor transponder) in the NEH+dxyz model appear to be abnormal, which may be due to the high correlation of vertical parameters. The results of the NEH+dv model demonstrated a considerable acoustic velocity correction parameter, suggesting a problem with the acoustic velocity value, which may be caused by the un-calibration of the acoustic velocity profiler. As the experiment was carried out in a shallow water area, where the ship track covered a small area and lasted a short period, the acoustic velocity was deemed to exhibit a relatively small variation in both time and space. As the ship sails along the predetermined symmetrical tracks, the effects of sound velocity errors in the horizontal direction can be mostly eliminated in adjustment, and residual influences can be considered as fixed values in the vertical direction.
6. Conclusions
As the infrastructure of seafloor navigation and positioning, the seafloor geodetic station relies on the GNSS-A positioning technique to obtain its precise position. However, under a situation with a large offset of transceiver lever arms and a significant bias in sound velocity measurement, the positioning accuracy of the GNSS-A technique will be deteriorated by these two systematic errors. To cope with these problems, this paper analysed the function model of GNSS-A positioning for the seafloor station and studied the processing methods of the offset error of transceiver lever arms and the bias of sound velocity. By adding the corresponding parameters to be solved, the high correlation between the vertical lever arm parameter and seafloor station height was revealed. Then, a sample search method was proposed for offset parameter determination in the vertical direction, which obtained significant estimation improvements. To correct the bias of the sound velocity profile and reduce the influence of the unmodeled error, the sound velocity correction parameter was added in the estimation, and the positioning precision was remarkably improved. Finally, by combining the sample search method and adding parameters for the horizontal lever arms and sound velocity correction, the accuracy of the seafloor position parameters was improved from 6 to 9 cm in the horizontal position and 13 to 16 cm in the vertical position to better than 1 cm, which verified the proposed data processing methods.