Performance of HaiYang-2 Altimetric Data in Marine Gravity Research and a New Global Marine Gravity Model NSOAS22

Abstract

Haiyang-2 (HY-2) missions have accumulated sea surface height (SSH) observations on a global scale for more than 10 years. Four satellites, HY-2A, HY-2B, HY-2C and HY-2D, provide even but differently distributed data, which play a complementary role in marine gravity studies with other missions. Therefore, this paper evaluates the performances of HY-2 altimetric data in marine gravity modeling from the following four perspectives: SSH accuracy, geoid signal resolution ability, vertical deflections and gravity anomaly. First, the centimeter-magnitude accuracy level of HY-2 data is proved by analyzing SSH discrepancies at crossover points within a certain time limit. Second, the spectral analysis of repetitive along-track data sequences in a time domain shows a geoid resolution range from 18 to 24 km. Taking HY-2 exact repeat missions (ERM), for example, the resolution could be remarkably enhanced by stacking repetitive cycles. Third, validation with an XGM2019 model showed that vertical deflections were reliably computed for all HY-2 missions, but HY-2A performed slightly worse than the other HY-2 missions. Meanwhile, HY-2C and HY-2D with a ~66° orbital inclination obviously had an improved ability to capture east–west signals compared to HY-2A and HY-2B. Finally, we constructed global marine gravity results based on three input datasets, HY-2 dataset only, multi-satellite dataset without HY-2 and multi-satellite dataset with HY-2. Validations were performed using published models and shipborne gravimetric data. The results showed that the HY-2 dataset is capable of improving marine gravity anomaly recoveries and that the accuracy of NSOAS22 with incorporated HY-2 data is comparable to DTU21 and SS V31.1. Furthermore, HY-2 observations should not be the only input dataset to construct a 1’ × 1’ resolution marine gravity model.

1. Introduction

Over the past half century, satellite earth observation has dramatically helped people understand and model the planet. Among them, satellite altimetry has proved to be an effective tool for observing earth shapes over oceans, icesheets and inland water surfaces [1,2]. It is, thus, feasible to calculate marine gravity anomalies on a global scale based on geoid heights or vertical deflections, which can be deducted from sea surface height (SSH), measured by onboard altimeters [3,4,5,6]. Improved marine gravity recovery can be expected from incorporating altimeter observations with enhanced range precision, denser spatial coverage and diverse track orientations [7]. These geodetic missions (GM) with denser spatial coverage provide the primary data for mapping the marine gravity field. Exact repeat missions (ERM) are also critical according to a relatively higher accuracy level by averaging nonunique, repetitive cycles. Meanwhile, waveform retracking has proven to be effective in improving the range precision of existing altimeter observations, especially over marginal seas and sea-ice-covered regions. Along with altimeter datasets accumulated over the past decade, there have been many publications on analyzing improvements in gravity recovery, either by adopting waveform retrackers or incorporating updated data [8,9,10,11,12,13,14,15,16,17,18].

Marine gravity can be modelled directly from the geoid heights or vertical deflections. The geoid heights are obtained by the difference between SSH measurements and the corrections of dynamic ocean topography. The SSH should be accurately determined by introducing extra procedures, such as a crossover adjustment, to suppress the effect of radial orbit error. In contrast, vertical deflection can be calculated from the geoid-height slopes. The along-track difference procedure in calculating slopes has the advantage of constraining long-wavelength error components. Consequently, the crossover adjustment is unnecessary for the slopes-based method [19] and several corrections are negligible, dry tropospheric path delay, solid earth tide correction and the geocentric polar tide effect [20]. In addition, the slopes-based method is more sensitive to seafloor topography and performs better when calculating marine gravity field elements over the open ocean. Therefore, this study aimed to evaluate the performance of HY-2 series data from the perspective of the slopes-based method.

Currently, China’s ocean dynamic environment monitoring network has been established with a series of successfully launched satellite missions. Among them, HY-2A has provided reliable SSH observations with uniform data coverage on a global scale for more than 10 years, especially data with denser geographical distribution for the latest geodetic mission (GM), since March 2016. HY-2B is performing an exact repeat mission and has collected more than 3 years of high-quality measurements along ground tracks that are similar to those of HY-2A. The orbital inclination of ~99° has a larger global coverage scope, which is closer to the north and south poles. In addition, HY-2C and HY-2D were successfully launched into pre-designed ~66° inclined orbits. Similar to the Jason series, these altimeter measurements would have performed better obtaining the east components of vertical deflection. Consequently, the HY-2 series of measurements is extremely valuable for recovering marine gravity anomalies because of their different spatial distribution compared to other altimeter missions.

Zhu et al. [17] verified that HY-2A provides similar marine gravity recovery to other altimeters on the basis of 1 Hz measurements. Meanwhile, HY-2A GM data were also proved to be reliable for calculating vertical deflections on regional and global scales [21,22]. Zhang et al. [16] and Liu et al. [23], respectively, indicated that 20 Hz waveform data records of HY-2A GM can derive a refined regional marine gravity field model. In contrast, HY-2B, HY-2C and HY-2D have a significantly lower application rate for constructing marine gravity models even though there are plans for them to be placed in drifting orbits in a future geodetic mission. In addition, the along-track analysis of the resolution capabilities of the HY-2 series missions is rare.

Under this circumstance, this paper fully evaluated the performance of four HY-2 series missions (HY-2A, HY-2B, HY-2C, HY-2D) to construct a global marine gravity model. The two-step waveform retracker has proved to be effective for improving the range precision of HY-2A measurements by a factor of 1.6 [24]; thus, this two-step retracking strategy was used in this study. The performances of SSH measurements for HY-2 series missions were uniformly evaluated for accuracy and resolution; then, the vertical deflections at crossover points and along-track points were calculated and evaluated. Finally, after incorporating the data from these HY-2 missions, the derived global marine gravity models were fully evaluated by ship-borne gravity data and the latest published models.

It is well known that such an analysis cannot cover all possible conditions, but we aimed to discuss this for a number of study areas. For data integrity, 30° × 30° rectangle study regions over six open ocean areas were selected: the North, South, and Mid- Pacific; Indian; and the North and South Atlantic. The specific scopes are located, respectively, at (150–180°E, 10–40°N), (150–180°W, 10–40°S), (105–135°W, 15° S-15°N), (60–90°E, 10–40°S), (30–60°W, 10–40°N) and (0–30°W, 10–40°S). Hereafter, the study areas are renamed, respectively, as Region 1, Region 2, Region 3, Region 4, Region 5 and Region 6. In these regions, regional and along-track directional experiments were launched. The former contained an accuracy assessment of SSH and vertical deflection at crossover points; for the latter, groups with similar ground tracks were selected to carry out resolution analyses and directional vertical deflection calculations.

The remainder of this paper is organized as follows. Section 2 provides a general description of the dataset (e.g., altimeter data and retracking), as well as other gravity models for comparison purposes. The quality assessments for both accuracy and resolution are presented in Section 3. In Section 4, the performance of the 4 HY-2 series missions in calculating vertical deflections is investigated and compared. Section 5 evaluates the HY-2 series-derived global marine gravity model using marine data, including the latest published models and shipborne measurements. The final 1′ × 1′ results will be titled with NSOAS22 and available for download. For better understanding, the methods of evaluating HY-2 ’s performance in marine gravity research were summarized in Figure 1.

2. Data Description

2.1. HY-2 Altimetry Data

All level 2 products of HY-2 series missions were administrated and distributed by China’s National Satellite Ocean Application Service (NSOAS, http://www.nsoas.gov.cn/). The specific FTP address is osdds-ftp.nsoas.org.cn and a user account is required. We obtained 7 GM cycles of HY-2A, 88 ERM cycles of HY-2B, 56 ERM cycles of HY-2C and 23 ERM cycles of HY-2D sensor data records (SDRs) in standard NETCDF format. The general information of the acquired HY-2 datasets is listed in

Table A1. The general information for each cycle of acquired HY-2 altimeter datasets.

Cycle	Time Scope (YYMMDD)	No. of Pass	Cycle	Time Scope (YYMMDD)	No. of Pass	Cycle	Time Scope (YYMMDD)	No. of Pass
HY-2A Geodetic Mission			E053	201026–201109	386	E022	210421–210430	274
G001	160324–160908	4332	E054	201109–201123	386	E023	210430–210510	274
G002	160908–170223	4373	E055	201123–201207	275	E024	210510–210520	274
G003	170223–170810	4360	E056	201207–201221	386	E025	210520–210530	274
G004	170810–180115	4399	E057	201221–210104	385	E026	210530–210609	274
G005	180115–180712	4386	E058	200104–210118	386	E027	210609–210619	246
G006	180712–181227	4333	E059	210118–210201	384	E028	210619–210629	233
G007	181227–190613	4278	E060	210201–210215	386	E029	210629–210709	227
HY-2B Exact Repeat Mission			E061	210215–210301	386	E030	210709–210719	274
E003	181126–181210	376	E062	210301–210315	386	E031	210719–210728	272
E004	181210–181224	381	E063	210315–210329	386	E032	210728–210807	246
E005	181224–181225	12	E064	210329–210412	386	E033	210807–210817	243
E006	190111–190121	293	E065	210412–210426	384	E034	210817–210827	220
E007	190121–190204	386	E066	210426–210510	386	E035	210827–210906	274
E008	190204–190218	351	E067	210510–210524	386	E036	210906–210916	274
E009	190218–190304	386	E068	210524–210607	369	E037	210916–210926	274
E010	190304–190318	370	E069	210607–210621	356	E038	210926–211006	251
E011	190318–190401	386	E070	210622–210705	361	E039	211006–211016	274
E012	190401–190415	386	E071	210705–210719	386	E040	211016–211026	274
E013	190415–190429	386	E072	210719–210815	386	E041	211026–211104	274
E014	190429–190513	386	E074	210818–210830	328	E042	211104–211114	274
E015	190513–190527	386	E075	210830–210913	379	E043	211114–211124	266
E016	190527–190610	386	E076	210913–210927	381	E044	211124–211201	193
E017	190610–190624	386	E077	210927–211011	386	E045	211204–211214	270
E018	190624–190708	386	E078	211011–211025	386	E046	211214–211224	274
E019	190708–190722	386	E079	211025–211108	386	E047	211224–220103	274
E020	190722–190805	386	E080	211108–211122	383	E048	220103–220113	274
E021	190805–190819	370	E081	211122–211206	369	E049	220113–220123	262
E022	190819–190902	321	E082	211206–211220	381	E050	220123–220202	270
E023	190902–190916	386	E083	211220–220103	373	E051	220202–220212	274
E024	190916–190930	386	E084	220103–220117	386	E052	220212–220222	274
E025	190930–191014	386	E085	220117–220131	386	E053	220222–220303	274
E026	191014–191028	386	E086	220131–220214	385	E054	220303–220313	274
E027	191028–191111	384	E087	220214–220228	386	E055	220313–220323	273
E028	191111–191125	380	E088	220228–220314	385	E056	220323–220402	273
E029	191125–191209	386	E089	220314–220328	385	HY-2D Exact Repeat Mission
E030	191209–191223	371	E090	220328–220409	330	E010	210827–210904	172
E031	191223--200106	386	HY-2C Exact Repeat Mission			E011	210904–210913	230
E032	200106–200120	386	E001	200924–201004	274	E012	210913–210923	271
E033	200120–200203	386	E002	201004–201014	274	E013	210923–211003	270
E034	200203–200217	386	E003	201014–201024	273	E014	211003–211013	274
E035	200217–200302	384	E004	201024–201103	274	E015	211013–211023	274
E036	200302–200316	386	E005	201103–201112	255	E016	211023–211102	274
E037	200316–200330	386	E006	201117–201123	139	E017	211102–211112	246
E038	200330–200413	386	E007	201123–201203	274	E018	211112–211122	274
E039	200413–200427	386	E008	201203–201213	274	E019	211122–211202	274
E040	200427–200511	376	E009	201213–201223	274	E020	211202–211212	272
E041	200511–200525	386	E010	201223–210102	245	E021	211212–211222	208
E042	200525–200608	384	E011	210102–210111	274	E022	211222–121231	256
E043	200608–200622	386	E012	210111–210121	274	E023	211231–220110	274
E044	200622–200706	386	E013	210121–210131	274	E024	220110–220120	272
E045	200706–200720	386	E014	210131–210210	274	E025	220123–220130	188
E046	200720–200803	386	E015	210210–210220	273	E026	220130–220209	242
E047	200803–200817	386	E016	210220–210302	268	E027	220209–220219	273
E048	200817–200831	386	E017	210302–210312	274	E028	220219–220301	274
E049	200831–200914	386	E018	210312–210322	274	E029	220301–220311	275
E050	200914–200928	358	E019	210322–210401	265	E030	220311–220321	270
E051	200928–201012	386	E020	210401–210411	274	E031	220321–220331	164
E052	201012–201026	359	E021	210411–210421	275	E032	220331–220409	274

in Appendix A.

Generally, each cycle of HY-2A had 4630 pre-designed passes in the geodetic mission phase and the ground tracks were nearly identical if their pass numbers were the same. HY-2B, HY-2C and HY-2D have 386, 274 and 274 passes, respectively, for each cycle in the exact repeat mission phase. Although the spatial coverage rate of HY-2 series missions (HY-2A GM ~16 km at equator; HY-2B ERM ~200 km; HY-2C and HY-2D ERM ~300 km) was obviously lower than for the Jason (~4 km) and SARAL/AltiKa (~5 km) geodetic missions, the ground tracks could still be a great supplement to providing different track orientations of multi-satellite materials for the recovery of marine gravity fields. The ground tracks of HY-2 missions are shown in Figure 2. In Figure 2a, HY-2B C015, HY-2C C012 and HY-2D C028 were selected to show how HY-2 samples the globe. A magnification over Region 1 was executed for better contrast effects and the ground tracks of measurements from HY-2A GM C004, Jason2 C180 and SARAl/AltiKa C026 are also plotted in Figure 2b,c.

The SDR products provided certain items to calculate along-track SSH, such as altitude, Ku band range and corrections. The correction items were divided into three groups: instrumental corrections, path-delay range corrections (dry and wet tropospheric and ionospheric correction and sea state bias) and geophysical environmental corrections (geocentric ocean and pole tide height, solid earth tide and inverted barometer correction). Among them, instrumental corrections were taken into account in the provided range items, while the latter two groups need to be considered further when calculating SSH. All the above-mentioned correction items are provided in level 2 products. Detailed sources of applied corrections for estimating SSH are listed in

Table 1. List of parameters used to estimate the SSH for HY-2 series missions (ECMWF, NCEP and GIM are the European Centre for Medium-Range Weather Forecasts, National Centers for Environmental Prediction, and Global Ionospheric Map).

Contrastive Parameters	HY-2A	HY-2B, HY-2C and HY-2D
Dry troposphere correction	ECMWF	NCEP
Wet troposphere correction	ECMWF	NCEP
Ionospheric correction	Dual-frequency	GIM
Sea state bias	NSOAS empirical solution	NSOAS empirical solution
Ocean tide	GOT00.2	GOT4.10c
Solid earth tide	Cartwright and Tayler tables	Cartwright and Tayler tables
Pole tide	Wahr [25]	Wahr [25]
Inverted barometer correction	NCEP	NCEP

.

2.2. Two-Step Waveform Retracking

Waveform retracking is an effective tool for improving marine gravity recovery by using waveform-oriented algorithms developed either on fitting functions or empirical statistics. The two-step waveform retracker with a two-step fitting procedure that took into account the correlation between the waveform parameters for calculating two-way arrival time and significant wave height (SWH) was proven to be applicable for most conventional radar altimeters. During the first step, the waveforms were fitted by a three-parameter Brown model (arrival time, rise time and amplitude) with invariable constants describing thermal noise and antenna mis-pointing angles. The rise time parameter was then smoothed along track, before retracking the waveforms a second time using a two-parameter Brown model (arrival time and amplitude) with the rise time being fixed to the smoothed value. Either the former three parameters or the latter two parameters will be solved during the weighted least squaring procedure, while the constants (α and P0) for describing antenna mis-pointing angles and thermal noise are critical before the model fitting procedure. A previous study of ours indicated that the two-step waveform retracker was also effective for HY-2A Ku-band waveforms and reduced the noise level by a factor of 1.6 [24]. Similar analyses about selecting suitable parameters for the HY-2B, HY-2C and HY-2D two-step waveform retracker are not discussed in this study as they would be repetitive. Instead, we checked whether the parameters needed to be adjusted for the other HY-2 missions.

First, α is the most crucial parameter for fitting the waveform shapes and its optimal value can be iteratively determined by considering the misfit standard deviation values between a normalized and modeled waveform. Based on randomly selected sample passes from the four HY-2 missions (HY-2A C0001P4632, HY-2B C0003P0003, HY-2C C0001P0251, HY-2D C0010P0001), the best-fit α values were 0.0105, 0.0100, 0.0092 and 0.0100, respectively. According to the similar background noise level and the maximum power level of echoes between HY-2A and the other missions, we inherited the constant P0 value (5500) and a threshold value of 0.015, which provided an initial estimate of unknown arrival time t0 to accelerate the iterative process in model fitting [24].

Certain criteria had to be met to indicate a failed model fitting. A maximum of 10 times was required to suspend the iterative process if these criteria were not met. The editing threshold was established by constructing histograms of amplitude, chi-square misfits and SWH versus standard deviation of the arrival time parameter. For the four HY-2 missions, valid results had model amplitudes between 50,000 and 65,000, while the chi-squared misfit measurement could not exceed 800. Moreover, we removed waveforms having an SWH outside of a range of 0–10 m to exclude observations over extremely unusual sea state conditions. Considering the robustness of the statistical waveform retracking strategy, the estimated results by selected threshold value of 0.015 will also be adopted as a backup once the model fitting procedure failure happens.

To evaluate the performance of two-step waveform retracking, we performed a statistical analysis on the retracked data for the four HY-2 missions. The noise was estimated as the standard deviation of the SSH with respect to the mean difference from the EGM2008 model computed over a 1 s interval. Figure 3 shows the noise levels for the HY-2 sample passes as a function of SWH. For the four HY-2 missions, the range precision is improved from 59 mm (average of 60.5, 56.1, 60.8 and 57.9 mm) before retracking to 39 mm (average of 39.4, 36.6, 38.8 and 39.5 mm) after retracking for 2 m of significant wave height (SWH), shown as the red and blue curves in Figure 3 for individual points (dots in left figures) and for the median over 0.5 m SWH bins (lines in right figures).

2.3. Typical Gravity Models

We introduced three models for comparison and verification. First, the recently published gravity field model XGM2019 was used to compare vertical deflections computed by the HY-2 missions. It is a high-resolution model (d/o 2190) that integrates multiple satellite data sources and ground observations: gravity, elevation and topographic information [26]. Over the oceans, the model exhibited enhanced performance (equal or better than preceding models), which was confirmed by the authors.

The second introduced model was the DTU21 gravity model, which was recently released by the Technical University of Denmark (DTU). The third model, V31.1, the latest version of the S&S series model and released by the Scripps Institution of Oceanography (SIO), was also used for comparison and verification. Incidentally, the DTU and S&S series models are the most well-known and widely accepted in the altimetry community as typical altimetry-derived marine gravity models. With the acquisition of additional data from satellite altimetry, new reference gravity fields, waveform retracking and improved data processing, DTU and SIO continuously publish new versions of global marine gravity models, so to some extent, DTU21 and S&S V31.1 represent the highest attainable accuracy [27,28,29].

3. Quality Assessment

3.1. Crossover Analysis

The quality of altimetry measurements is commonly evaluated and validated by calibration based on a statistical analysis at crossover points. Smaller crossover discrepancies indicate better stability and internal conforming accuracy for onboard instruments. A certain premise is to determine the position of crossover points precisely. To do this, we introduced a quadratic equation and a linear interpolation to calculate the initial location and precise location, respectively. This accuracy assessment was similarly executed in the six selected regions for the 4 HY-2 missions, while along-track SSH measurements were obtained based on procedures mentioned in Section 2.1. These study areas are shown in blue rectangles in Figure 4. The typical ground tracks inside these rectangles were used for along-track directional experiments during the steps described in Section 3.2 and Section 4.2.

To keep the amount of crossover points consistent, we designed an experiment of accuracy assessment on the basis of one complete HY-2A GM cycle and eight complete ERM cycles from HY-2B, HY-2C and HY-2D. All the crossovers were defined as positions where each mission crossed its own ground position. The time difference between the measurements of ascending and descending passes might somehow have been related to the corresponding statistical values. Hence, we calculated the standard deviations (SDs) of crossover discrepancies with time limits of 1, 5, 14 and 168 days, respectively. Meanwhile, gross errors were detected and removed according to the criteria of triple-standard deviation. The specific results are listed in

Table 2. Statistics of crossover discrepancy with different time limits for four HY-2 missions: Region 1, North Pacific; Region 2, South Pacific; Region 3, Mid-Pacific; Region 4, Indian Ocean; Region 5, North Atlantic; Region 6, South Atlantic.

Experimental Region	Time Limit	HY-2A		HY-2B		HY-2C		HY-2D
		Num	SD (m)	Num	SD (m)	Num	SD (m)	Num	SD (m)
1	<168 days	8370	0.1942	8564	0.1935	7413	0.1482	7566	0.1352
	<14 days	1314	0.1289	2024	0.1339	2369	0.1136	2432	0.1034
	<5 days	463	0.1060	812	0.1134	901	0.0990	920	0.0953
	<1 day	166	0.0666	241	0.0910	256	0.0726	264	0.0599
2	<168 days	8315	0.1679	8492	0.1082	7418	0.1183	7195	0.1331
	<14 days	1324	0.1506	2010	0.0900	2390	0.1091	2309	0.1102
	<5 days	472	0.1214	807	0.0855	905	0.0945	869	0.1036
	<1 day	168	0.0823	236	0.0559	255	0.0793	243	0.0621
3	<168 days	7286	0.1236	7695	0.1154	5608	0.1071	5647	0.1220
	<14 days	1216	0.0936	1821	0.0875	1760	0.0883	1772	0.1064
	<5 days	481	0.0778	705	0.0877	682	0.0831	687	0.0993
	<1 day	—	—	—	—	173	0.0697	175	0.0640
4	<168 days	8055	0.1810	8457	0.1641	7712	0.1386	7461	0.1368
	<14 days	1271	0.1365	2008	0.1273	2482	0.1153	2407	0.1036
	<5 days	452	0.1176	810	0.1202	941	0.1033	911	0.0889
	<1 day	154	0.0929	238	0.0538	260	0.0755	261	0.0745
5	<168 days	8215	0.1913	8741	0.1516	7769	0.1346	7836	0.1311
	<14 days	1326	0.1628	2074	0.1074	2490	0.1083	2508	0.1161
	<5 days	471	0.1052	830	0.0921	941	0.0912	944	0.1034
	<1 day	167	0.0681	242	0.0664	259	0.0760	269	0.0610
6	<168 days	8150	0.1446	8611	0.0903	7674	0.0973	7745	0.1099
	<14 days	1292	0.1248	2037	0.0822	2458	0.0884	2491	0.0966
	<5 days	455	0.1352	820	0.0706	928	0.0855	944	0.0862
	<1 day	159	0.1205	240	0.0507	259	0.0666	269	0.0577

.

Generally, the SD values ranged from 5 to 20 cm for HY-2 missions, representing a reliable accuracy level compared with other altimetry missions [24]. Moreover, the measurements within the shortest time limits had the best coincidence in each situation. While the time limit was the longest (168 days), the SD values were within a range of 9–19 cm. The HY-2C had the smallest SD, with an average value of 12.4 cm, and HY-2A had the largest value of 16.71 cm. HY-2B and HY-2D had slightly larger average SDs at 13.71 cm and 12.8 cm, respectively. Noticing that the time spans of eight complete ERM cycles were ~112, ~80 and ~80 days, respectively, the SD might increase if the time span extended to 168 days. Hence, the first case proved that all four HY-2 altimetry missions have a similar and reliable accuracy level. When the time limit narrowed to 14 days in the second case, the average SD values improved 22.37%, 23.63%, 16.26% and 17.15% for HY-2A, HY-2B, HY-2C and HY-2D, respectively. In addition, the average SD values declined along with the transition to the third case within five days’ time limit, but the improvement percentage was relatively slower. When the time limit was less than 1 day, the SD shrank to 5–12 cm and its average values were within 6–9 cm. This represented a centimeter accuracy level of HY-2 measurements when the time-varying effect is relatively insignificant in the fourth case with the shortest time limit. All these results proved that these HY-2 missions provided reliable and high-quality SSH measurements. Considering that the characteristics of spatial distribution were relatively complementary between HY-2 and the other missions, more stable computations might be expected if HY-2 series data were incorporated into a multi-satellite dataset to recover global marine gravity models.

3.2. Resolution Capability

Spectral analysis is an effective method for studying the frequency characteristics of signals or systems. The coherence function and power spectrum reflect the degree of correlation between signals and the distribution density of power with frequency. The purpose of power spectrum estimation is to describe the frequency component distribution of signals and random processes based on limited data sequences. The coherence function can judge the similarity between two repeated periodic signals to deduce the resolution. Marks defined the standard for judging the wavelength resolution of geoid, which is the corresponding spatial wavelength when the mean square consistency reaches 0.5 [30]. We inherited this standard to analyze the resolution capability for the four HY-2 series missions uniformly.

Both the HY-2 GM and ERM provided multiple cycles with repetitive ground tracks, which provided suitable data sources for analyzing the along-track resolution capabilities of geoid signals. To avoid possible data break and interruption, we selected data sequences with an extremely low data loss rate in the six open-ocean study regions. In addition, the ground tracks of selected data sequences were similar among the HY-2 missions. The spatial distribution of sample data with marked pass number is shown in Figure 4. Detailed information of the sample data sequences—date, quantity and average distance—is listed in

Table 3. Summary of information for sample data sequences in 6 research regions: Region 1, North Pacific; Region 2, South Pacific; Region 3, Mid-Pacific; Region 4, Indian Ocean; Region 5, North Atlantic; Region 6, South Atlantic.

Experimental Region	Satellite	Track Direction	Track Number	Cycle	Date	Data Quantity	Average Distance between Two Cycles
1	HY-2A	ascending	1851	005	2 April	9285	1.101 km
				006	17 September	9283
	HY-2B	ascending	0095	032	10 January	10,532	0.086 km
				033	24 January	10,571
	HY-2C	descending	0248	019	13 March	11,668	0.242 km
				020	10 April	11,667
	HY-2D	descending	0108	025	24 January	11,633	0.156 km
				026	3 February	11,676
2	HY-2A	ascending	2815	004	20 November	9242	0.695 km
				006	22 October	9261
	HY-2B	ascending	0315	014	11 May	10,504	0.195 km
				015	25 May	10,522
	HY-2C	descending	0056	019	24 March	11,672	0.276 km
				020	3 April	11,644
	HY-2D	descending	0190	026	6 February	11,672	0.025 km
				028	26 February	11,676
3	HY-2A	ascending	0661	005	18 February	9254	0.131 km
				006	5 August	9251
	HY-2B	ascending	0283	017	21 June	10,479	0.230 km
				019	19 July	10,477
	HY-2C	descending	0024	011	2 January	11,337	0.011 km
				012	12 January	11,349
	HY-2D	descending	0076	028	22 February	11,364	0.249 km
				032	2 April	11,318
4	HY-2A	ascending	4477	003	4 August	9218	0.113 km
				005	6 July	9249
	HY-2B	ascending	0351	019	21 July	10,453	0.054 km
				020	4 August	10,518
	HY-2C	descending	0174	017	8 March	11,636	0.159 km
				019	28 March	11,637
	HY-2D	descending	0034	027	10 February	11,633	0.156 km
				028	20 February	11,676
5	HY-2A	ascending	2171	005	13 April	8689	1.369 km
				007	15 March	8693
	HY-2B	ascending	0167	012	7 April	10,529	0.146 km
				013	21 April	10,567
	HY-2C	descending	0018	015	11 February	11,663	0.271 km
				016	21 February	11,663
	HY-2D	descending	0152	025	26 January	11,661	0.160 km
				028	24 February	11,674
6	HY-2A	ascending	0847	004	9 September	9256	0.444 km
				006	11 August	9236
	HY-2B	ascending	0221	014	9 March	10,524	0.084 km
				015	18 March	10,520
	HY-2C	descending	0182	017	9 March	11,665	0.130 km
				018	18 March	11,672
	HY-2D	descending	0042	027	11 February	11,145	0.119 km
				028	20 February	11,628

. Sample data are basically acquired in the same season, except HY-2A, since it runs in the GM phase. A full sampling rate of SSH measurements at ~20 Hz was extracted, while the 1 Hz correction items mentioned in Section 2.1 were linearly interpolated to high-rate sampling points. In addition, the mean dynamic topography items provided in HY-2 SDR products were subtracted to obtain along-track geoid heights. Then, we executed a resample procedure in a time domain to constrain the effect of missing data and uneven spacing between adjacent measurements. Space intervals were uniformly set to 0.5 km spherical distances; moreover, 5 km low-pass filtering was adopted to weaken the influence of random noise.

After the resampling in a given time domain, the along-track measurements were evenly distributed. During this procedure, interpolations were unavoidable because of missing data and data sequences had to align with others at a fixed step size. As shown in Figure 5, the along-track geoid heights smoothly changed with latitude after extraction, resampling, filtering and alignment. Red, green, yellow and blue curves, respectively, represent HY-2A, HY-2B, HY-2C and HY-2D data with repeat ground tracks. The variation trends were basically consistent among the four HY-2 missions in each study region. HY-2A and HY-2B ran along the solar synchronous orbits with an inclination of ~99°, while orbits of ~66° inclination were adopted for HY-2C and HY-2D. Hence, similar orbit inclinations led to better consistency in these two groups.

We applied a Hanning window to constrain the effect of spectrum leakage caused by the limited length of the data sequence. Through the averaging effect of the Hanning window, the main lobe reduced the fluctuation in the power spectrum and then achieved a smaller estimated variance. A wider main lobe is advantageous for a more effective averaging effect and smaller variance but has decreased resolution. A stronger side-lobe effect caused more errors in signal recognition and weak signals were indistinguishable. Hence, the size of the Hanning window had to be optimally determined by considering both the main- and side-lobe effects. A 64 km Hanning window was used in this study and the power spectrum was estimated by the Welch method. The resolution capabilities were uniformly determined while the coherence dropped to 0.5, indicating a signal-to-noise ratio of 1 [31]. As shown in Figure 6, all the HY-2 missions had stable resolution capabilities in the 50+ km wavelength range. The coherence was close to 1, indicating that the signal occupied this band. While the coherence was around 0.5, different variations occurred for different missions and regions. The specific results are listed in

Table 4. Summary of resolution capabilities for HY-2 data sequences: Region 1, North Pacific; Region 2, South Pacific; Region 3, Mid-Pacific; Region 4, Indian Ocean; Region 5, North Atlantic; Region 6, South Atlantic.

Experimental Region	Satellite	Resolution
1	HY-2A	27.27 km
	HY-2B	24.51 km
	HY-2C	23.92 km
	HY-2D	25.12 km
2	HY-2A	23.36 km
	HY-2B	22.58 km
	HY-2C	17.74 km
	HY-2D	19.95 km
3	HY-2A	21.03 km
	HY-2B	16.44 km
	HY-2C	20.09 km
	HY-2D	16.71 km
4	HY-2A	20.51 km
	HY-2B	18.58 km
	HY-2C	14.99 km
	HY-2D	15.53 km
5	HY-2A	19.71 km
	HY-2B	17.76 km
	HY-2C	17.33 km
	HY-2D	16.18 km
6	HY-2A	27.40 km
	HY-2B	20.01 km
	HY-2C	22.05 km
	HY-2D	20.40 km

. Generally, the recent HY-2 missions (HY-2B, HY-2C and HY-2D) have smaller resolution capabilities than HY-2A, which indicates stronger detective abilities of the geoid signals. HY-2D had the smallest average value of 18.98 km, while slightly higher values of 19.98 km and 19.35 km were derived from HY-2B and HY-2C. HY-2A data were from a geodetic mission with longer, seasonal time intervals. Time variations, as well as data quality, might have caused the largest average resolution capability of around 23.21 km.

Stacking is a common processing method for repeated periodic orbital altimetry data, which can effectively suppress the time-varying noise and further improve observational accuracy. At present, HY-2A carried out GM with 169-day repeat tracks for more than five years. HY-2B, HY-2C and HY-2D are running ERM and accumulated cycles of data with 14-day, 10-day and 10-day intervals. All these HY-2 missions have repetitive data sequences that can be stacked.

We acquired more than 50 cycles of HY-2B ERM data that were split into two stacking data groups with relatively longer periods. The following experiments were designed to prove enhanced data quality after stacking in the six research regions. This procedure adopted weighted averaging for all repetitive passes. Resolution analysis was also carried out to assess the stacking results of 5, 10, 15 and 20 cycles. As shown in Figure 7, the resolution capabilities were remarkably enhanced along with the increased number of stacking cycles. Relevance and coherence were all improved at both long and short wavelengths.

Table 5. The average resolution capabilities and improvement effect of HY-2B stacking data.

HY-2B Stacking Cycles	Resolution of Mean Geoid	Effect of Every Cycle Improving	Improving Percentage
Reference pass	19.98 km	——	——
5 cycles	14.65 km	1.332 km/cycle	26.67%
10 cycles	10.94 km	1.004 km/cycle	45.24%
15 cycles	9.28 km	0.713 km/cycle	53.55%
20 cycles	7.77 km	0.610 km/cycle	61.11%

indicates that the effect of every cycle improvement remained stable within 10 cycles; then, a slightly slower rate of improvement occurred, involving more and more cycles, indicating that stacking within 10 cycles was more efficient. In addition, HY-2A provided seven complete cycles of GM observations, which inspired us to incorporate stacking GM data in future studies.

4. Vertical Deflections

4.1. Directional Components at Crossover Points

Sandwell presented a method for determining the directional component of vertical deflection at crossover points [3]. The latitude velocities along ascending and descending passes are symmetrical and, thus, have the same numerical value but the opposite sign. In contrast, the longitude velocities along the ascending and descending passes are approximately equal. Following Section 3.2, we extracted latitude, longitude, measuring time and geoid height information from all series of HY-2 SDR products. The derivative of geoid height with respect to time was calculated by a first-order difference formula. Considering the approximate relationship between velocities along the ascending and descending passes, the meridional (north–south) and prime (east–west) components of vertical deflections at crossover points were calculated. The results were then validated with an XGM2019 gravity field model, as shown in

Table 6. Validation for directional components of vertical deflection at crossover points with XGM2019 model (unit: arcsec): Region 1, North Pacific; Region 2, South Pacific; Region 3, Mid-Pacific; Region 4, Indian Ocean; Region 5, North Atlantic; Region 6, South Atlantic.

Experimental Region	HY-2A		HY-2B		HY-2C		HY-2D
	NS-SD	EW-SD	NS-SD	EW-SD	NS-SD	EW-SD	NS-SD	EW-SD
1	1.8158	5.8641	1.8111	3.6730	1.4160	2.4234	1.4950	2.8535
2	1.8060	6.0823	1.3033	3.8435	1.3650	2.6606	1.4224	2.6351
3	1.2695	4.8377	0.9614	3.0489	0.9961	2.3396	0.9435	2.1194
4	1.8012	6.0917	1.4912	4.3776	1.5263	2.3788	1.3558	2.4246
5	1.9329	5.8834	1.5815	3.3820	1.7654	2.4545	1.5086	2.2948
6	1.8720	5.6527	1.4436	3.7008	1.5509	2.1746	1.6584	2.1735
Average	1.7495	5.7353	1.4320	3.6709	1.4366	2.4052	1.3972	2.4168

.

It showed the accuracy of meridional components at the crossover points’ directional vertical deflection ranges from 0.94 to 1.93 arcsec. In contrast, the accuracy of the prime component was slightly lower, between 2.11 and 6.08 arcsec. The meridional component accuracy for HY-2A GM was about 1.7 arcsec, while the SD for HY-2B, HY-2C and HY-2D ERM was about 1.4 arcsec. The accuracy discrepancies of the prime component were more obvious. The accuracy for HY-2C and HY-2D was about 2.4 arcsec and HY-2B was about 3.7 arcsec; HY-2A was the worst, at 5.7 arcsec. Of these, the accuracy of the meridional component for HY-2D was the best and HY-2C had the best performance in calculating the prime component.

The obviouss accuracy discrepancies between the meridional and prime components were mainly due to the approximate north–south distribution of continuous measurements. The ~45° orbital inclination was ideal for obtaining perpendicular ground tracks and determining the equal accuracy level of directional components. From the perspective of spatial coverage, the inclinations for these altimetry missions were designed to be from 66° to 108°. For HY-2A and HY-2B with an inclination of 99°, the accuracy of the prime component was limited. In contrast, HY-2C and HY-2D, with an inclination of 66°, had obviously improved east–west signal-tracking ability. Hence, the accuracy of the prime component for HY-2C and HY-2D was relatively higher and closer to the accuracy of the meridional component.

As a follow-on satellite of HY-2A, HY-2B greatly enhanced the accuracy of the prime component from 5.73 to 3.67 arcsec. We attributed this improvement to better data quality due to the same inclination and higher accuracy in the meridional component. In addition, HY-2C and HY-2D showed consistent reliability with HY-2B in calculating the meridional component of vertical deflections: ~1.4 arcsec.

Table 6 also shows that the statistics in Region 3 were superior to those in the other regions for all HY-2 missions. This was related to the magnitude and feature changes in vertical deflection in different areas. Hence, the distributional proportion of the directional components are calculated and plotted in Figure 8. Compared with the other regions, the meridional components of vertical deflection in Region 3 (Mid-Pacific) were relatively concentrated at 0–6 arcsec and the variation amplitude was smaller. Furthermore, the distributional proportions of the meridional component for the four HY-2 missions were consistent in each region, although the crossover point locations were different. Except for the prime components, HY-2C and HY-2D remained consistent while the distributional proportion for HY-2A and HY-2B varied, indicating gaps in the ability to determine the east–west vertical deflections.

Similar to Figure 8, the distributional proportions of the validation results with the XGM2019 model are plotted in Figure 9. The discrepancies between the meridional components and XGM2019 mainly fell in a range of ±1 arcsec for all HY-2 missions. This regularity was especially prominent in Region 3 (Mid-Pacific), where the proportion of less than 1 arcsec was about 45%. The distribution was more dispersed for the prime components and a reduction of 10–20% occurred in the six study regions for the proportion of small discrepancies within ±1 arcsec. For HY-2C and HY-2D, the proportion of less than 2 arcsec was still more than 50%. In contrast, HY-2B’s ability to determine the prime components of vertical deflection was inferior to that of HY-2C and HY-2D. In addition, HY-2A had the worst performance in tracking east–west signals and abnormal results with large discrepancies were over 5%.

4.2. Along-Track Vertical Deflection

Compared with the targeted vertical deflection information at 1’ × 1’ grid points, the spatial distribution of crossover points determined in Section 4.1 was relatively sparse. A more common approach is to directly construct the relationship between directional components at grid points and the along-track vertical deflection at each measuring point. The latter is obtained in a differential operation among the along-track adjacent measurements of geoid heights. The difference calculation effectively suppressed the influence of long-wavelength error items, such as radial orbit, instrumental and atmospheric propagation errors, as well as inaccurate geophysical environmental corrections. Hence, the accuracy of along-track vertical deflections depended on, but also was distinguished from, the accuracy of SSH observations. In this section, the performance of the HY-2 missions is evaluated from the perspective of along-track vertical deflection.

The experimental data for calculating along-track vertical deflections were exactly the same as the data sequences in Section 3.2. The XGM2019 (d/o) gravity model was also used for validation. Moreover, a triple-standard-deviation criterion was adopted by comparing model values to eliminate gross errors in measurement. The statistical results are listed in

Table 7. Comparison of determined along-track vertical deflections with XGM2019 for HY-2 missions (unit: arcsec): Region 1, North Pacific; Region 2, South Pacific; Region 3, Mid-Pacific; Region 4, Indian Ocean; Region 5, North Atlantic; Region 6, South Atlantic.

Experimental Region	HY-2A		HY-2B		HY-2C		HY-2D
	Num	SD	Num	SD	Num	SD	Num	SD
1	1057	1.4130	1058	1.3100	1167	1.0749	1166	1.2805
2	1060	1.4783	1060	1.0544	1168	1.0571	1167	0.9400
3	1054	1.2890	1054	0.8987	1135	1.0829	1137	0.8626
4	1059	1.7919	1059	1.1186	1168	1.1093	1161	1.0600
5	994	1.4002	1056	0.9773	1166	1.1223	1167	0.9631
6	1060	1.3742	1060	0.9170	1051	1.0810	1167	0.9359
Average	6284	1.4578	6347	1.0460	6855	1.0879	6965	1.0070

, indicating that all HY-2 missions obtained reliable along-track vertical deflections. The standard deviations of discrepancies with the XGM2019 model range from 0.86 arcsec to 1.79 arcsec over six study regions. The average accuracy for HY-2A, HY-2B, HY-2C and HY-2D is, respectively, 1.46, 1.05, 1.09 and 1.01 arcsec. Recent missions, including HY-2B, HY-2C and HY-2D, obviously had better performance compared to HY-2A, which was the first experimental satellite and no longer distributed data.

Moreover, we analyzed the distributional proportion of the along-track vertical deflections as well as their discrepancies with the XGM2019 model over the six research regions, which are, respectively, shown in Figure 10 and Figure 11. In Figure 10, the distributional proportion is, respectively, consistent in prograde orbit group (HY-2C and HY-2D) and retrograde orbit group (HY-2A and HY-2D). Histograms of the two groups were approximately symmetrical along the axis of 0 value. Since we selected ascending passes for the retrograde orbit group and descending passes for the prograde orbit group, the calculated along-track deflections at similar locations had similar values but opposite signs.

Figure 11 shows that the discrepancies with the XGM2019 model are concentrated at 0 arcsec with approximate normal distribution. The proportions of model discrepancies within 1 arcsec were over 30%, of which the performance of HY-2A is far below average. In addition, HY-2B and HY-2D accorded better with XGM2019 in Region 3 (Mid-Pacific) and Region 6 (South Atlantic) and the corresponding proportions were larger than 40%. HY-2D was superior in most cases, while HY-2C performed better in Region 1 (North Pacific). Generally, the determined along-track vertical deflections were reliable for all HY-2 missions, but there were quality gaps between HY-2A and the other HY-2 missions.

5. Marine Gravity Recovery

5.1. Model Construction

The data qualities of the four HY-2 missions were proven from many perspectives in previous sections: SSH discrepancies at crossover points, resolution capabilities of geoid signals and calculation of vertical deflections. The goal of this paper was to evaluate the performance of the HY-2 missions in constructing a 1’ × 1’ marine gravity model. Considering the sparse distribution of HY-2 measurements in cross-track directions, a high degree of interpolation was unavoidable when relying only on HY-2 measurements. Hence, we prepared three groups of input datasets for the model construction. The first was only the HY-2 measurements mentioned in Section 2.1. The second was a 2016 accumulated dataset for constructing the WHU2016 model: Geosat GM and ERM, ERS-1 GM and ERM, Envisat 30d and 35d, Jason-1 ERM and GM, CryoSat-2 and SARAL/AltiKa ERM data [32]. The third group is a combination of the two.

The slopes-based method, which is based on the Laplace equation-deduced relationship between vertical deflections and gravity anomalies, was adopted in this study, for which a series of joint processing procedures was necessary. First, raw waveforms were retracked using the two-step waveform retracker and high-rate observations along profiles were uniformly resampled into 5 Hz to enhance the accuracy and density of available measurements. Second, along-track SSH measurements were calculated by adding correction items provided in the standard products to constrain the corresponding effects for path delay and geophysical environment. Third, the along-track SSH gradients were calculated, whereas the along-track gradients from the EGM2008 model were interpolated for preliminary verification to detect outliers.

Fourth, Parks–McClellan low-pass filters were used uniformly to constrain the amplified high-frequency noise during the difference procedure. Fifth, the along-track residual vertical deflections were computed, during which the DOT2008A_n180 [33] and EGM2008 model are selected, respectively, to remove the effects of sea surface topography and geoid height by interpolating and subtracting from along-track observations. Sixth, the directional components of residual vertical deflection at gridding points were calculated. Then, the residual gravity anomalies were calculated using the FFT method according to the relationship formula between gravity anomaly and vertical deflection. Finally, a global marine gravity model over a range of 80° S–80° N with a 1′ × 1′ grid interval was inverted after restoring the removed reference model.

Following the previous procedure, the marine gravity model results were obtained based on three input datasets. For convenience of description, these three results derived from HY-2 missions only, multi-satellite dataset without HY-2 missions and multi-satellite dataset with HY-2 missions: GRAHY2, WHU16 and NSOAS22. The NSOAS22 gravity model is shown in Figure 12.

Figures for the three final marine gravity anomalies after the remove–restore procedure were hard to distinguish. Hence, we directly plotted the residual gravity anomalies in the six study regions for better discernment. As shown in Figure 13, the lowercase letters a, b and c represent GRAHY2, WHU16 and NSOAS22, while the Arabic numbers 1 to 6 indicate the study regions. The residual gravity anomalies of GRAHY2 have maximum noise level, but the general trend of the gravity signals can be detected and was basically consistent with the other two results. An initial conclusion was that the HY-2 dataset could not be the only input observations to construct a 1’ × 1’ resolution marine gravity anomaly model due to limited spatial resolution. In addition, the multi-satellite altimetry-captured gravity signals derived from WHU16 and NSOAS22 had more consistent distribution, but the latter had smoother features and smaller trajectory characteristics. Taking the 35°–40° N part of Region 1, for example, the signal distribution of positive and negative anomalies was more integrated in Figure 13(c-1) than Figure 13(b-1). Another initial conclusion was that the tracking effect was effectively constrained and the final captured gravity signals were enhanced by incorporating HY-2 series measurements into a multi-satellite dataset. In other words, the HY-2 missions played an irreplaceable role in modeling global marine gravity. Therefore, validating the NSOAS22 model with HY-2 measurements is our main goal in the next section.

To further highlight HY-2’s contribution, the difference between NSOAS22 and WHU16 was also plotted as column d in Figure 13. The difference grid consistently reflected typical gravity signals in the six study regions, indicating that incorporating HY-2 data was helpful to capture more signals during the marine gravity recovery modeling. From the perspective of numerical value, statistical and histogram analysis of discrepancies between two models was executed. As shown in

Table 8. Statistics of the differences between NSOAS22 and WHU16 in the six study regions (unit: mGal): Region 1, North Pacific; Region 2, South Pacific; Region 3, Mid-Pacific; Region 4, Indian Ocean; Region 5, North Atlantic; Region 6, South Atlantic.

NSOAS22-WHU16	Region 1	Region 2	Region 3	Region 4	Region 5	Region 6
Max	89.0	180.4	74.6	109.4	91.0	120.0
Min	−150.6	−209.4	−93.2	−95.2	−99.2	−138.8
Mean	−0.056	0.017	0.235	0.067	0.082	−0.102
SD	9.092	9.068	6.865	8.625	8.192	7.644

and Figure 14, the discrepancies are concentrated within ±20 mGal with approximate normal distribution and the largest deviation, respectively, reached 180.4 and −209.4 mGal. The SD values indicated the difference in captured gravity signals, as well as caused errors by incorporating HY-2 altimetric data over the six study areas.

5.2. Model Validation

To assess the accuracy level of global marine gravity model NSOAS22, we performed comparisons and a “standard deviation of misfit” with published altimetric marine gravity models and shipborne gravity measurements. First, two previously mentioned models, DTU21 and SS V31.1, were introduced for comparison. The SD values are listed in

Table 9. Statistics of difference between NSOAS22, DTU21 and SS V31.1 (unit: mGal): Region 1, North Pacific; Region 2, South Pacific; Region 3, Mid-Pacific; Region 4, Indian Ocean; Region 5, North Atlantic; Region 6, South Atlantic.

SD(Model)	Region 1	Region 2	Region 3	Region 4	Region 5	Region 6
SD (DTU21-NSOAS22)	2.1984	2.1382	1.2778	1.8634	1.7846	1.7260
SD (V31.1-NSOAS22)	2.2945	2.2386	1.4460	1.9923	1.8450	1.7619
SD (V31.1-DTU21)	1.2943	1.4832	1.1498	1.3503	1.2564	1.1988

. The maximum SD of model discrepancies over the six study regions was smaller than 2.23 mGal. Since NSOAS22, DTU21 and SS V31.1 all adopted EGM2008 as a reference model, the SD results also indicated the difference among the altimetry-derived residual gravity anomalies. Moreover, NSOAS22 had slightly better consistency with DTU21 than V31.1, while the SD of misfit between V31.1 and DTU21 was obviously smaller. This was attributed to the fact that NSOAS22 did not incorporate Sentinel-3A, Sentinel-3B or the latest CryoSat-2 data. Generally, NSOAS22 provided reliable marine gravity anomalies but with a slightly higher noise level than DTU21 and SS V31.1.

The second validation was based on a comparison with shipborne gravity anomaly measurements along irregular tracks. This comparison provided a more independent assessment of model accuracy but was limited to special areas where high-quality shipborne data were available [34]. The accessible observations distributed by the national geophysical data center (NGDC) were globally distributed and each study region had several cruises and several thousand measurements. In addition to NSOAS22, other models also carried out the same validation as comparisons, which contained EGM2008, SS V31.1, DTU21, WHU16 and GRAHY2. The gravity anomaly of each model was interpolated according to the location of the shipborne measurements by the GRDTRACK command in generic mapping tools (GMT) and the difference was calculated for further statistics [35].

The initial and processed shipborne datasets were validated and the statistics are listed in

Table 10. Statistics of difference between shipborne measurements and typical gravity models (unit: mGal): Region 1, North Pacific; Region 2, South Pacific; Region 3, Mid-Pacific; Region 4, Indian Ocean; Region 5, North Atlantic; Region 6, South Atlantic. (The upper table shows the original statistics, while the lower table shows the accuracy results after a gross error elimination of shipborne measurements.).

Experimental Region	Ship Num	EGM2008	V31.1	DTU21	NSOAS22	WHU16	GRAHY-2
Comparison with original NGDC shipborne data
1	154307	8.9969	8.4055	8.4498	8.6485	8.7617	9.5523
2	223215	6.7261	6.1551	6.0829	6.4873	6.4206	7.9747
3	183428	7.0148	6.5585	6.5938	6.6597	6.6070	7.2480
4	100646	5.8313	5.2628	5.2111	5.2717	5.3690	6.6649
5	77656	8.2512	8.0527	8.0360	8.2659	8.1806	8.8939
6	106374	10.0158	9.6352	9.6369	9.9101	9.7588	10.5042
Average	-	7.8060	7.3449	7.3350	7.5405	7.5162	8.4730
Comparison with NGDC shipborne data after a gross error elimination
1	130048	4.8749	4.3306	4.3088	4.6215	4.8435	6.0770
2	197434	4.8451	4.2817	4.2640	4.3932	4.5215	6.2772
3	153836	4.1413	3.9048	3.8843	3.7998	3.9424	4.6622
4	68245	4.8880	4.3855	4.3446	4.1921	4.4148	5.8480
5	55325	5.5377	5.2586	5.2472	5.2753	5.3840	6.1715
6	88587	4.3236	3.6457	3.6818	4.0168	3.8482	5.1531
Average	-	4.7684	4.3011	4.2884	4.3831	4.4924	5.6981

. The former dataset was derived from NGDC shipborne measurements with the default value eliminated, while the latter was further processed following two steps. In step one, we eliminated the shipborne measurements according to discrepancies with the reference model larger than 15 mGal. The threshold was set by considering that the accuracy of altimetry-derived marine gravity over open oceans is usually 3–5 mGal and large deviations should not exceed 15 mGal under the triple criteria. For step 2, the shipborne data beyond three-times standard deviation, as checked by NSOAS22 for each cruise, were removed.

The upper part in Table 10 shows that the average SD of discrepancies was larger than 7 mGal. It should be noted that most of the NGDC shipborne measurements were accumulated 40 years ago, so the data quality could not be guaranteed. Furthermore, these altimetry-derived models generally have a smaller SD than EGM2008, indicating that the altimetry measurements captured new gravity signals after a remove–restore procedure. Meanwhile, GRAHY2 had the highest noise level due to a limited spatial distribution and excessive interpolation. The SD values in the lower part of Table 10 showed similar variation regularity with a smaller numerical magnitude. Moreover, the number of shipborne measurements involved in validating decreased significantly. The eliminating rates were, respectively, 15.7%, 11.5%, 16.1%, 32.2%, 28.8% and 16.7% in the study regions. Due to the constraints of navigational and observational accuracy before the 1990s, the relatively large rejection ratio was also acceptable. In conclusion, the validation accuracy between NSOAS22 and NGDC shipborne data was between 3.8 and 5.3 mGal, which was superior to EGM2008, WHU16 and GRAHY2. The accuracy gap among our latest result and the worldwide recognized models (DTU21 and SS V31.1) was still detectable but tiny.

6. Conclusions

Satellite altimetry provides the most comprehensive images of a marine gravity field, with accuracies approaching typical shipborne gravity data. Among them, the HY-2 series missions have realized networking observations and provided SSH measurements on a global scale for more than 10 years. The spatial distribution of these measurements is complementary with other altimetry missions for marine gravity recovery research. Therefore, this paper evaluated the performances of HY-2 series data in marine gravity modeling from the following four aspects: accuracy, resolution capability, vertical deflection and gravity anomaly.

First, the performance of HY-2 data for four missions (HY-2A GM, HY-2B ERM, HY-2C ERM, HY-2D ERM) proved to be reliable according to the statistics of SSH discrepancies at crossover points. Second, the spectral analysis of repeated along-track data sequences showed that the geoid resolution capacity of HY-2 missions ranged from 18 to 24 km. Meanwhile, resolution capabilities were remarkably enhanced along with the increase in the number of stacking cycles. Numerical tests showed that stacking from 5, 10 and 20 cycles brought an improvement of 26.7%, 45.2% and 61.1%, respectively. Third, along-track vertical deflections and directional components of the vertical deflections at crossover points were calculated and validated with the XGM2019 model. Results showed that the along-track vertical deflections were reliable for all the HY-2 missions, but there were acceptable quality gaps between HY-2A and the others. The accuracy of the east–west component for HY-2C and HY-2D was relatively higher than for HY-2B and HY-2A. Finally, we constructed and validated the global marine gravity models based on three groups of input datasets: HY-2 missions only, multi-satellite dataset without HY-2 missions and multi-satellite dataset with HY-2 missions. To retain more short-wavelength gravity information, we incorporated high-sampled 20Hz HY-2 waveforms instead of 1Hz measurements and uniformly retracked them with a two-step waveform retracker. These retracked 20Hz measurements were averaged to 5Hz data by using a low-pass FIR filter to constrain high-frequency noise. Residual vertical deflections were then obtained through adding corrections, differentiating, eliminating outliers, low-pass filtering and removing sea surface topography and geoid height. Finally, marine gravity anomalies with 1’ × 1’ resolution were inverted by a remove–restore procedure, with EGM2008 as a reference model.

Two approaches were used to evaluate the accuracy of the new gravity model, altimetry-derived marine gravity models and NGDC shipborne gravity data. Verifications revealed that NSOAS22 provided reliable marine gravity anomalies but with a slightly higher noise level than DTU21 and SS V31.1. Taking into account the uncertainty in NGDC shipborne data, a gross error elimination is necessary and the validation accuracy of NSOAS22 was around 3.8–5.3 mGal. The HY-2 dataset was capable of improving marine gravity anomaly recoveries since its complementary spatial distribution may have provided detectable signals over previously unreached areas. However, the HY-2 dataset should not be the only input observation to construct a 1’ × 1’ resolution marine gravity model due to its limited spatial resolution.