The principle of GNSS-R altimetry is to use the distance difference between the GNSS reflection signal on sea surface and the direct signal to the receiver to achieve the sea surface altimetry. In the last few years, several GNSS-R satellite missions including the TechDemoSat-1 (TDS-1) satellite [1
], the Cyclone Global Navigation Satellite System (CyGNSS) mission [2
] and the Bufeng 1A and 1B satellites have been put into operation [4
]. In recent years, the GNSS-R altimetry has achieved advances in specular reflection point positioning, phase altimetry at grazing angles, sea surface height (SSH) retrieval and retracking techniques, etc. [5
]. Compared with the traditional satellite altimeter, GNSS-R altimetry has the advantages of a rich signal source, low payload cost, all-weather, all-day time, low power consumption, and so forth. GNSS-R can acquire multiple sea surface reflection signals at the same time based on multi-channel receiver. Combined with multiple satellites network, GNSS-R can achieve high-spatial-resolution global sea surface altimetry. To take advantage of the high resolution and realize the application, the accuracy of GNSS-R satellite sea surface altimetry needs to reach cm level, which is comparable to SSH obtained by satellite radar altimeter. As one of the main error sources in GNSS-R altimetry, the positioning error of the specular reflection point restricts the improvement of the altimetry accuracy [14
]. The specular reflection point is the point which minimizes the distance across which the GNSS satellite signal reaches the receiver through reflection on the reflection surface [16
]. It is the datum point and reference center of the GNSS-R signal reflection geometry and relevant parameters. The positioning error of the specular reflection point affects the sea surface altimetry accuracy, which is directly related to the path error, on the reference datum [17
]. In 1997, Wu proposed a two-step method to search the position of the specular reflection point on the reference ellipsoid [15
]. In 2009, Gleason proposed the minimum path length (MPL) method which determined the specular reflection point on the WGS84 ellipsoid as an optimization problem [5
]. In 2012, Semmling proposed the osculating sphere method which provided a simpler way to search the specular reflection point on the ellipsoid [6
]. In 2018, Southwell proposed a latitude dependent unit difference positioning method that satisfied Snell’s Law on the WGS84 ellipsoid, this method is more accurate and more computationally efficient than MPL method [10
]. The signal reflection surface of the GNSS-R altimetry is the instantaneous sea surface. However, among the reflection reference surfaces selected by the current specular reflection point geometric positioning methods, the one which is the closest to the instantaneous sea surface is the earth ellipsoid surface. The elevation difference between the ellipsoid surface and the instantaneous sea surface, sea level height (SSH), is not considered in the positioning, resulting in non-ignorable positioning error of specular reflection points [13
] (see Figure 1
). SSH can be decomposed into static-state elevation and time-varying elevation: 1) The former is the elevation difference between geoid and the reference ellipsoid surface—the geoid undulation. It is determined by the earth’s own gravity, with low time variation. 2) The latter is the real-time change of SSH from geoid to the instantaneous sea surface caused by tides, winds, ground rotations, mesoscale eddies, currents, tsunami and other external dynamics, ie, dynamic topography, which varies with time.
The static-state elevation error of the reflection reference surface is the basic error source for the specular reflection point positioning. We have modified this in a previous study by proposing the gravity field-normal projection reflection reference surface combination correction method [13
]. The earth ellipsoid surface is taken as the reflection reference surface to initially position the specular reflection point, and then the geoid undulation is introduced to correct the reflection reference surface to geoid. Besides the positioning error caused by the static-state elevation error, that caused by the normal-radial difference is corrected without approximation at last. The positioning of the specular reflection point is improved by 28.66 m.
Based on the static-state elevation error correction, the sea surface dynamic topography with a global amplitude of ±2 m is the reflection reference surface time-varying elevation error that must be further eliminated to achieve cm-level accuracy sea surface altimetry. The contribution of ocean tides to the dynamic topography of the sea surface is 80% [23
]; it is the most important source of time-varying elevation positioning error of specular reflection points. The standard deviation of ocean tides changes in open seas is 10~60 cm, up to several meters in marginal seas and offshore. The standard deviation contributed by ocean tides in sea level changes is 5~30 cm [24
], which is a non-negligible specular reflection point positioning error for cm-level accuracy GNSS-R sea surface altimetry. Therefore, applying ocean tides to construct the ocean tidal correction positioning method is the key to further refining the specular reflection point positioning, and then to increase the GNSS-R altimetry accuracy. As the geometric positioning of the GNSS-R specular reflection point is independent of the instrument, it can be corrected based on the global tidal model. In the 1980s, the Schwiderski ocean tidal model was used as the most important global tidal model for the correction of ocean tides in various precision geodesy observations [25
]. In 1992, with the success of TOPEX/Poseidon (T/P) satellite mission, satellite altimetry made it possible to quickly acquire global high-accuracy sea surface heights. In recent years, based on the development of satellite altimetry technology and data assimilation technology, a series of high-accuracy, high-resolution ocean tidal models such as Finite Element Solution (FES) [26
], Technical University of Denmark (DTU) model [27
], TOPEX/Poseidon global inverse solution (TPXO) [28
], Empirical Ocean Tide (EOT) model [31
], Goddard/Grenoble Ocean Tide (GOT) empirical model [32
], etc. have appeared. Based on these models, tides can be calculated at any position and time in the global ocean. Comparison of the statistical results of the ocean tide correction by different tidal models and the altimeter error correction shows that the accuracy difference between the main tidal models is below 1 cm, while the error is relatively large in the offshore shallow waters [23
]. The ocean tidal model and data are mainly applied to correct and calibrate the sea surface height data obtained by the satellite altimeters [33
]; it has not been applied in the correction of the GNSS-R specular point positioning, yet. In addition to the important characteristics of time-varying, ocean tides also have different spatial variation characteristics in the offshore and deep seas. In the offshore, the tide difference is larger than that in the deep sea, the tidal wave propagation is more complicated, and complex additional tidal constituents are formed. These differences will have different effects on the accuracy of the tidal elevation correction in the offshore and deep sea. The offshore is the key area for sea surface altimetry with the requirement of accurate altimetry support. Therefore, it is necessary to study the influence of offshore tidal height variation on the accuracy of tidal elevation correction positioning; this is discussed in Section 3
. In summary, the existing GNSS-R specular reflection point positioning methods do not use the instantaneous sea surface with real-time variation as the reflection reference surface, resulting in a non-negligible positioning error for the cm-level accuracy sea surface altimetry. The time-varying elevation correction to the specular point positioning based on the main decision factor of SSH real-time variation, ocean tide, has not been thoroughly studied. Additionally, the different effects of the geographical tidal difference on the positioning have not been considered into discussion.
Differing from previous studies based on ocean tide, in order to reduce the time-varying elevation error of the reflection reference surface and improve the positioning of the specular reflection point, this study proposes the ocean tidal correction positioning method. Firstly, the initial specular reflection point is positioned on the earth ellipsoid. In the iterative positioning process, the geoid undulation calculated based on the earth gravitational model is introduced. The basic static-state elevation difference between the instantaneous sea surface and the ellipsoid surface is corrected. Secondly, on this basis, the ocean tidal model is applied to predict the total tidal height relative to geoid. Then the tidal height is introduced in iterative positioning. The reflection reference surface is corrected to the ocean tidal surface, reducing the positioning error resulted from time-varying elevation difference. Thirdly, based on the elevation correction, the reflection reference surface is corrected from the radial to the normal direction of the specular reflection point without approximation. The positioning error caused by the normal-radial difference is reduced. Fourthly, the different effects of the tidal height gradient modulo in the offshore and deep sea on the positioning correction are discussed.
In addition to the important characteristics of time-varying, ocean tides have different variations in the offshore and in the deep sea. Since the tide difference is affected by tidal forces, topography and other conditions, it varies by time and location and it is larger in the offshore than in the deep sea. Due to the non-linearity of tidal power in the offshore, additional tidal constituents such as overtides or compound tides are formed. These tidal constituents have complexities of nonlinearities, small amplitudes and short wavelength. These tidal constituents interact with other principal tidal constituents (such as the most active M4 tidal constituent interacting with the M2 tidal constituent) with amplitudes reaching a significant range (the M4 tidal constituent reaches 1 cm in some areas of the Atlantic [44
]), which has a negligible effect on tidal height and its changes. In addition, as the depth of the offshore water becomes shallower than in the deep sea, the friction between the tide wave and the bottom of the seabed changes the propagation process [24
], and the propagation of the tide wave is more complicated than in the deep sea. Due to the diversity of offshore straits and bay forms (long straits, semi-enclosed wide bays, narrow long semi-closed bays, etc.), tide waves and tide difference in offshore vary in variety and characteristics, and they are more complex than those in deep seas [45
]. Therefore, the influence of the offshore tidal height variation characteristics on the positioning of specular reflection points needs to be taken seriously. The TPXO8 ocean tidal model assimilates the altimeter data and the model is enhanced in accuracy and resolution in the offshore, it can calculate the global offshore tide amplitude distribution of high quality. In this section, the offshore part of the specular reflection point track is separated from the deep sea part, and then their tidal height variation and the different influences on the tidal correction are compared.
3.2. Division of Offshore and Deep Sea
The most obvious difference between ocean tides in the offshore and deep seas is reflected in the tidal height gradient. According to Equations (8)–(15), the tidal correction gradient modulo is affected by both the tidal height and the geoid undulation. The former differs according to time and geographical location like offshore/deep sea. The latter is determined by gravity anomaly at the location of the specular reflection point. Therefore, the tidal height gradient of the offshore and deep sea will have different effects on the tidal correction gradient. The effects will be reflected in the difference of the tidal correction gradient between the two kinds of sea areas on some level. Due to the continuity and equal spacing of the GNSS-R specular reflection points of the tracks on earth surface, the tidal height gradient along the track is also a sequence of equal-spacing samples. The tidal height gradient is calculated as the tidal height difference between the two adjacent points on one track. Figure 5
shows the tidal height (the red curve) and its gradient modulo (the blue curve) of the specular reflection points by the OTCP method on a track containing an offshore part on the right and a deep sea part on the left. The trend of tidal height along track changes gently and nearly linearly in the deep sea (see the 1st to the 300th points of the red curve). While in the offshore part, the degree of tidal height change is significantly increased, and the increase and decrease of the tidal height is more random (see the points after the 300th of the red curve). In addition, due to the periodic black body correction to the noise reference of the GNSS-R receiver on TDS-1 satellite, the position of the specular reflection point is periodically hopped. The influence of the hopping of TDS-1 data on the positioning accuracy has been discussed [13
]. Although the hopping has little effect on the positioning accuracy of the correction positioning method, it will cause the tidal height along track to jump (see the jumps of the red curve in Figure 5
), so that periodic peaks of the tidal height gradient are produced (see the blue curve). The magnitude of these peaks are generally comparable to those of offshore tidal height gradient mutation peaks; the jumps’ effects on the tidal height gradient and the correction gradient in the offshore are negligible. But the jumps have significant effects on the deep sea segments with gentle change of the tidal height gradient, so these jumps are eliminated in the analysis. The sampling time difference between adjacent specular points on one track is only 1 s, the corresponding error of tidal phase and tidal height is insignificant, the influence on the tidal height gradient can be neglected.
In this study, the offshore and deep sea parts on each track are divided and compared according to the changes of the tidal height gradient of the track, the specific methods are as follows.
Filtering the tracks across both sea and land.
There are 28 tracks cross both sea and land among all the 43 tracks, the sea parts of these tracks are extracted.
The tidal height gradient and the corresponding tidal elevation position correction gradient of each track is calculated and detrended, the average is then subtracted, and the modulo is taken at last.
Division of the offshore and the deep sea segments.
Since some of the offshore tracks repeatedly cross land and sea, or cross islands, peninsulas, etc., one track is often divided into multiple sea sub-tracks by land, and the sub-tracks mostly contain offshore segments. Moreover, due to the complexity of the global coastline and tracks characteristics (length, direction, curvature, distribution, etc. [13
]), the position of the offshore segment in the track is complicated. It could be divided into four cases: the offshore part is at one end of the track, at both ends of the track, in the middle of the track (the middle of the track is close to the land) or the entire segment is in the offshore (generally short tracks). The points where the tidal height gradient modulo in each track (or sub-track) are greater than 3 times the standard deviation η
of the tidal height gradient modulo of the track are extracted. Then, the offshore and deep-sea segments are divided by these points according to the above four specific cases. Choosing too large a multiple of η
would ignore some of the tidal height gradient points with sudden changes in the offshore, so that the offshore segment cannot be completely extracted. If the multiple is set too small, some high frequency peaks would be misjudged as offshore tidal height gradient mutation points. In addition, judging the continuous variation characteristics of the tidal height gradient requires the segment to be long enough, segments with more than 10 continuous points are selected.
Eliminating tidal height gradient jumping points.
The points where the tidal height gradient modulo is greater than 3η in the deep sea segments are eliminated, and the tidal height gradient jumping peaks caused by the TDS-1 data jumping are well removed. According to the above filtering and division, 67 offshore segments are obtained, which contain 2476 specular reflection points, each segment has an average of about 37 points. And 54 deep sea segments are obtained, which contain 5716 specular reflection points, each with an average of approximately 106 points.
3.3. Tidal Height Gradient and the Tidal Positioning Correction Gradient
The average value and standard deviation of the tidal height gradient modulo and tidal correction gradient modulo of the offshore and deep sea segments are shown in Table 2
, and those of each segment are shown in Figure 6
, Figure 7
and Figure 8
. The tidal height gradient modulo and its standard deviation in the offshore segments are significantly higher than those in the deep sea, with the gradient of about 2.5 times and the standard deviation of about 3.5 times. The tidal height change in the offshore is more dramatic than that in the deep sea, and the difference between segments (with different tidal amplitudes and phases) is greater. Compared with the large difference of the tidal height gradient modulo in the offshore and deep sea segments, the changes of the positioning correction gradient modulo in the two sea areas are closer. The correction gradient modulo in the offshore and its standard deviation are respectively 1.2 and 2 times those in the deep sea. The tidal height gradients modulo in the offshore segments have a positive correlation with the positioning correction gradient modulo. They are very close for all the segments, and the difference between them does not increase significantly with the increase of the gradient (see Figure 6
and Figure 8
). This indicates that the variation of the tidal height in the offshore increases, and the change of the tidal correction is increased, the tidal positioning correction gradient modulo has a good response to the tidal height gradient modulo in the offshore. However, the difference between the two in the deep sea is larger, and the difference increases significantly with the increase of the gradient. The direct cause is that the gradient of the tidal positioning correction in the deep sea is higher than the tidal height gradient modulo; it does not decrease correspondingly as the tidal height gradient modulo decreases (see Figure 7
and Figure 8
). It can be considered that the variation of the tidal positioning correction is more consistent with the tidal height gradient modulo in the offshore where the tidal height changes greatly. While the tidal height change tends to be gentle in the deep sea, the change of the tidal positioning correction does not greatly decrease with it. The sensitivity of the response of tidal positioning correction to the change of the tidal height is reduced in the deep sea compared to offshore. It is speculated that the change in terrain and gravity anomaly in the deep sea is more dramatic than in the offshore with the relatively flat continental shelf underwater, which makes the geoid undulation and the total elevation positioning correction change more in the deep sea [45
]. This may contribute to the decreasing of the response sensitivity and the maintenance of high tidal correction gradient in the deep sea, which is consistent with the expression that the elevation positioning correction is affected by both gravity anomaly and tidal height in Section 2.2