Determination of Indonesia’s Marine Deflection of Vertical from Multi-Mission Altimetry Satellite Data

The deflection of vertical (DOV) is important for studying the changes in the Earth’s mass, improving the accuracy of geoid models, and the accuracy of the Earth’s gravity field. Compared to traditional methods such as astrogeodesy and astrogravimetry measurements, altimetry satellites have the advantage of observing DOV due to uniform global distribution, denser measurement points, consistent precision, less time required, high temporal resolution, and can be used in all weather conditions. In this study, multi-mission altimetry satellites including TOPEX/Poseidon, Jason-1, Jason-2, Jason-3, ENVISAT, Geosat Follow On (GFO), Sentinel-3A, Sentinel-3B, Sentinel-6A, Cryosat-2, and SARAL were used to determine DOV in Indonesia’s marine with a spatial resolution of 1’ x 1’. The Hwang et al. (1998) method is used to calculate DOV from satellite altimetry data by least-squares adjustment with weights using the second-order inverse distance weighting (IDW) method for each grid. The results show that the variance of the north-south and west-east components resulting from the gridding results are 0.05 arcseconds and 1.2 arcseconds. The evaluation results with the global model from EGM2008 show that the standard deviation values for the north-south and west-east components are 0.63 arcseconds and 1.96 arcseconds, respectively. From the evaluation, it can be concluded that the gridded DOV model is close to EGM2008, but there are still challenges for improvement, especially for the west-east DOV component.


Introduction
The deflection of vertical (DOV) is important for studying the changes in Earth's mass, improving the accuracy of geoid models, and the accuracy of the Earth's gravity field [1,2].DOV represents the angular deviation formed between the direction of gravity at any point on the Earth's surface and the normal direction of the reference ellipsoid [2].Based on the data collection approach, DOV is divided into land DOV and marine DOV.This research will focus on marine DOV.The marine DOV can be obtained by traditional methods such as astrogeodesy, astrogravimetry, and gravimetry.However, the interference of ocean waves, currents, tides, and the need for a lot of energy and a lot of time make it difficult to be conducted [3,4].Currently, satellite altimetry is the most important technology for obtaining marine 1245 (2023) 012046 IOP Publishing doi:10.1088/1755-1315/1245/1/012046 2 DOV due to its uniform global distribution, denser measurement points, consistent precision, less time required, high temporal resolution, and all-weather use [5][6][7][8][9][10].
With its radar altimeter, an altimetry satellite can measure sea surface height (SSH) [11].DOV can be obtained from SSH in three ways proposed by Sandwell [12], Olgiati et al. [13], and Hwang et al. [14].By comprehensively comparing the three methods, the Hwang et al. method [14] was chosen because it is theoretically more precise [15].Subtraction between two adjacent SSH points then divided by the spherical distance between the two points can effectively attenuate longwave errors such as satellite orbital errors, signal propagation errors, and topographic errors [16,17].DOV will later be used in the determination of ocean gravity.There are several methods to derive the ocean gravity from DOV, such as the inverse Vening Meinesz (IVM) formula [17], Laplace equation [18], and least-squeres collocation (LSC) method [19].IVM is a commonly used method today to obtain ocean gravity from altimetry satellite data [6].
In this study, Indonesia and its surroundings (90°W-150°W, -20°S-20°N) are selected as the study area and multi-mission altimetry satellite data are used to calculate DOV with a resolution of 1' x 1'.There are 11 altimetry satellites used, including TOPEX/Poseidon, Jason-1, Jason-2, Jason-3, ENVISAT, Geosat Follow On (GFO), Sentinel-3A, Sentinel-3B, Sentinel-6A, Cryosat-2, and SARAL.The global DOV model from EGM2008 is used to evaluate the results of the DOV calculations that have been done.

Data
2.1 Altimetry Satellite SSH data of altimetry satellites were obtained from the Radar Alimeter Database System (RADS).RADS is an altimetry database first developed by Delft University of Technology and later followed by the National Oceanic and Atmospheric Administration (NOAA) and the European Organization for the Exploitation of Meteorological Satellites (EUMETSAT) [20].The database provides mathematical operation services so that users can make corrections to SSH freely according to their needs.The list of SSH corrections used in this study can be seen in table 1.
Table 1.List of corrections used in SSH.For some corrections, more than one model is written and sorted in order of priority, if the topmost order is not available, the next order is used.There are 11 altimetry satellite missions that are used as the main data in this study.For certain satellite missions, more than 1 phase will be used.There are at least 4 types of phases on altimetry satellites, such as the Excat Repeat Mission (ERM), Geodetic Mission (GM), Interleaved Mission (IM), and Drifting Phase (DP).In the ERM phase, altimetry satellites are designed to go through the specified paths repeatedly according to the repetition period.While the GM phase is the opposite where it will not return to the path it has been on, the distance between paths will be as close as possible.IM is considered as ERM due to repetition while DP of SARAL is considered as GM.The satellites can be seen in table 2. adjusting the SSH and interpolating the slope on the intersecting track along the track between the two crossing points; (3) grids along the sea surface gradient path are transformed into north-south and westeast DOV components by least-squares where all available sea surface gradients along-track are used as observations and different weights are assigned to each gradient; (4) sea surface gradients can also be derived from Mean Sea Surface Height (MSSH) and independent sea surface topography models.In this study, the fourth procedure is used by modeling the MSSH using 11 multi-mission satellite altimetry data (see table 1) so that the data density is very high and the time variation of the ocean signal and data noise can be reduced [17,23].
In the MSS calculation process, collinear analysis was first performed on SSH data so that sea level variability can be eliminated and can be combined with other missions to produce longer time series such as ENVISAT and SARAL missions.For EGM and IM data, collinear analysis can be done by creating time series on the same path in each cycle.As for GM and DP where there is no repetition of paths, Sea Level Anomaly (SLA) generated by ERM data is used.Furthermore, ERM and IM data were homogenized by analyzing their crossing points with reference missions (T/P, J1, J2, J3, and 6A).Then the subtraction analysis is carried out to all satellites including the satellite itself.For GM and DP data, homogenization is performed by performing subtraction analysis at the crossover point with all previously homogenized ERM data.Once all the data has been collinearly analyzed and homogenized, it is ready for the along-track DOV calculation.

Along-Track DOV
By definition, the DOV along the track is the geoid gradient with the opposite sign [17].Since MSSH data is used in this study, the DOV can be obtained by calculating the gradient between two adjacent MSSH points along the path as follows.
Where  is along-track DOV,   and   are the MSSH at point i and MSSH at point j respectively that are located consecutively and on the same track, and   is the distance between the two points.

Gridded DOVs' Derivation
The DOV consists of two directional components, the meridian or north-south component and the prime vertical or west-east component.The relationship can be seen in the following equation.
Where  is DOV,  the DOV azimuth that has a direction along the trajectory of the altimetry data used, ξ the north-south component of the DOV, and η the west-east component of the DOV.By calculating the entire DOV of all the multi-mission altimetry satellite data used and its azimuth, the north-south and east-west components can be calculated in the following way.
With  being the number of observation points around the grid point to be calculated,   ,   , and   , are the residual, azimuth, and DOV of the i-th observation, respectively.From equation (3) the following matrix can be formed.
With least-squares adjustment, equation ( 4) can be solved as follows.
Where  is weight matrix and   is the distance of the i-th point to the calculated grid point.The weighting method used is second-order inverse distance weighting (IDW).The weight given by IDW is inversely proportional to the distance between the prediction location and the sample location.The weight will decrease as the distance increases.Using least-squares adjustment, the variances of the northsouth and east-west components can be known.
Where  2 is the variance of the estimated true error,  is number of measurements,  is number of parameters, and ∑   s variance matrix whose main diagonals   2 and   2 are the variances of the northsouth and west-east components of the DOV, respectively.
In this study, least-squares adjustment with weights using the second-order IDW method was used for each grid.With this technique, the calculation of the DOV component of each grid does not only include points within the grid, but a sub-region is created with the same origin point as the grid point with a size larger than the grid in order to provide the benefit of providing more points and so that the resulting surface is continuous because there are overlapping calculation areas between adjacent grids.The size of the sub-region used in this study is 5 arc-minutes or about 9 km at the equator.After the least-squares is performed with the combination, interpolation is performed to fill in the empty grid due to the sparse number of points such as those found in areas close to the coast.Interpolation is done with the SURFACE function in GMT.

Results and Analysis
Using the least squares adjustment with weights using the second-order IDW method and a sub-region size of 5 arc-minutes, the DOV results for the north-south and west-east components are obtained as shown in figure 1.It can be seen that the range of DOV values is -50 to 50 arcseconds.The variance value for the northsouth component is 0.05 arcseconds and the variance for the west-east component is 1.2 arcseconds (see table 3).This shows that the gridding using least-squares adjustment with weights using the second order IDW and choosing a sub-region size of 5 arc-minutes process performed has run well so that the results produced are considered good.For a better understanding of the gridding results, a comparison was made with the EGM2008 global model.The maximum, minimum, average, and standard deviation values were subtracted from each grid between the two models.The results can be seen in table 4.  2 and visualization its distribution of values can be seen in figure 3. From figure 3, the residual values are distributed to zero, for the north-south component of the DOV is about 3 arcseconds and west-east is about 6 arcseconds, two times larger than the north-south component.This is because the inclination of the altimetry, which is usually north, causes the northsouth component of the DOV derived from altimetry to be more accurate than the west-east component [24].These results show that overall the derived DOV model is close to the EGM2008 global.

Conclusion
With 11 multi-mission altimetry satellite data of various phases including TOPEX/Poseidon, Jason-1, Jason-2, Jason-3, ENVISAT, Geosat Follow On (GFO), Sentinel-3A, Sentinel-3B, Sentinel-6A, Cryosat-2, and SARAL, the Hwang et al. method [14], and least-squares adjustment with weights using the second-order IDW method, a 1' x 1' resolution model of marine DOV in Indonesia and its surroundings can be derived.The results show that the variance of the north-south and west-east components resulting from the gridding results are 0.05 arcseconds and 1.2 arcseconds.The evaluation results with the global model from EGM2008 show that the standard deviation values for the north-south and west-east components are 0.63 arcseconds and 1.96 arcseconds, respectively.From the evaluation, it can be concluded that the gridded DOV model is close to EGM2008, but there are still challenges for improvement, especially for the west-east DOV component due to the inclination of the altimetry.

Figure 1 .
Figure 1.North-south component of the DOV (left) and west-east (right) of gridding result in arcsecond Table 3 Variance (in arcsecond) for the north-south and west-east component of the DOV Deflection of Vertical   North-South 0.05 West-East 1.12

Figure 2 Figure 3
Figure 2 Residuals between gridding and EGM2008 results in arcsecond.North-south (left) and west-east (right) DOV components

Table 2 .
(2)t of altimetry satellite missions used One of the functional models is the DOV model.The EGM2008 DOV model in the research area is used as a comparison to the gridding DOV model.Bao et al. explained that there are 4 types of procedures in deriving sea surface gradients that will be used in calculating DOV, including: (1) applying various types of corrections to the altimetry satellite-derived SSH, calculating the gradient along-track by numerical subtraction, and transforming it into the meridional component and the prime vertical component of DOVs at the crossing point;(2) [21]008 is a spherical harmonic model of the Earth's gravitational potential developed by Pavlis et al.[21].The data used in making the model is a combination of terrestrial, altimetry, and airborne gravity data.

Table 4 .
Comparing DOV gridding results with EGM2008 (unit: arcsecond) From table 4 we can see that for the north-south component of the DOV model, the standard deviation is 0.93 arcseconds.This shows that the north-south component of the DOV model is very close to the north-south component of the EGM2008 DOV model.Then for the west-east component, the standard deviation is 1.96 arcseconds which shows it is quite close but no closer to the EGM2008 model than the north-south component.Visualization of the residual, which is the difference between the gridded DOV model and the EGM2008 model, can be seen in figure