China State Bureau of Surveying and Mapping, Beijing 100039, China
The effort for the geoid determination in China has lasted for nearly half a century. A large-scale measurement of astro-gravimetric leveling network was carried out from 1950s to the 1970s. The average resolutions of the network are 1.5°´1.5° in the eastern area of China and 2.5°´ 2.5° in the west. The China (Quasi) Geoid (CQG80) derived from this network shows that the difference between the remote measured points of this network and Xi’an Geodetic Origin is estimated at ±2.7 m in average.
It is obvious that CQG80 is far from meeting the needs of China’s economic development and national defense as well as the research on geo-science in this country. Now a new (quasi) geoid (CQG2000) was computed in early 2001 to meet the needs of the first part of the new century, which is with high accuracy, high resolution and covers all the territories of China. The accuracy of CQG2000 has been upgraded one magnitude, from 3-5 m of the CQG80 to the grade of centimeter. And its average resolution has been raised from 2°´2° to 15′´ 15′. Additionally the coverage of CQG2000 has extended to all the mainland and maritime areas of China. CQG2000 has passed the field test and production check and accepted by National Bureau of Surveying and Mapping, so it has been officially issued for public use in the whole country.
Three technical problems were encountered during the computation of CQG2000. The first is the research and computation of China’s mainland geoid. The second is the research and computation of China’s maritime geoid. And the third one is the research and computation of the match of China’s mainland and maritime geoids.
I.RESEARCH AND COMPUTATION OF CHINA’S MAINLAND GEOID
The usual way internationally to solve this problem is to make a combined adjustment of a local gravity geoid with a GPS leveling network, which is improved meantime through local gravity and topographic materials. This is a feasible way to deal with such problems in China at present. The major problem in China is that the resolution of the gravimetric data is rather low and the data are unevenly distributed. In view of the situation, the following project is taken to compute China’s mainland gravity geoid after research and experiment:
1. Computation Project for China Mainland Gravity Geoid
（1）Remove/restore method is adopted for this project
As the best global gravity potential model for time being and the high-resolution digital terrain model (DTM) are available in China, then the so-called remove/restore method can be used for the computation of China mainland geoid, which belongs to a local gravity geoid.
Using the overlapping property of gravity potential fields the method deals with the contributions of different wavelengths separately and then overlaps them directly to restore the local gravity potential field, which is to be approximated. The key aim of this method is to use local gravity and DTM data to improve the local (quasi) geoid (essentially to improve its short wave lengths), which is derived by global gravity potential model.
After the comparison in the testing computations, the geoid height is divided into two parts in the computation. The first part is the model (quasi) geoid height and model gravity anomaly, obtained from the global gravity geopotential model. The second part is the residual (quasi) geoid height, which is obtained by the residual gravity anomaly. The latter is the surface gravity anomaly after removing the model gravity anomaly in the first part.
（2）The gravity geoid is computed with the classic Stokes’ formula and Molodensky’ series (taking account of the 1st order term).
As the FFT/FHT techniques applied to those formula and series are quite mature, the classic formulas are usually the first choice by most countries when computing their gravity geoid.
（3）The classic terrain isostasy model (Airy-Heiskanen system) is used for gravity reduction.
It is due to the following factors: there are vast mountainous areas, low gravity resolutions in China, and the topographic isostatic anomaly is smoother than the Bouguer anomaly, there is no systemic effect of Bouguer anomaly in the areas with good isostatic compensation, then the estimated interpolation error in the classic isostasy model is smaller.
（4）Strict 1D FFT technique is used for the computation.
Generally speaking, the resolutions of the ground gravity data in China are low and the distribution of them is uneven. Terrain isostatic anomaly is used for interpolation and gridded, and then restored to gridded value of terrain (or geoid) free air gravity anomaly. After the gravity model value is subtracted from the anomaly, Stokes formula and Molodensky formula with G1 term correction are used to compute the residual geoid height and height anomaly. All the computation is based on 1D FFT technique.
2．Mathematical Model for China Mainland Gravity (Quasi) Geoid
（1）The computation of gravity geoid undulation
To determine the geoid height, theoretically, the gravity anomaly over the entire earth must be known. But in practice, only the terrain gravity data of a local area can be obtained. Therefore, the determination of a local geoid is usually done with lower pass filter principle, i.e. to compute it by combining the global gravity potential model with local gravity and topographic data. For this purpose the height of local geoid is divided into two parts. Its first part is computed from the global gravity potential field, and the second part is obtained from Stokes integral with 1D FFT/FHT techniques.
（2）The computation of height anomaly
The height anomaly is obtained by substituting Faye gravity anomaly into Stokes formula. In addition, in view of the fact that China’s land stretches as far as about 50o from north to south, so ellipsoid correction has to be taken into account in the computation of the (quasi) geoid.
3. Height Anomaly Control Network in China (HACN2000)
（1）Accuracy evaluation for the A-order network in HACN2000
One of the important bases for the computation of CQG2000 is the height anomaly control network (HACN2000) established with GPS leveling technique. According to the requirement of CQG2000, the network is established in two orders, A-order and B-order. The A-order in HACN2000 is established in accordance with the standards of the national A-order GPS positioning and national 1st or 2nd-order leveling control network. The major objective of the A-order HACN2000 is to transfer accurately the height anomaly in a large span across the country in order to reduce error accumulation. So far 30 points of the A-order HACN2000 have been accomplished which are evenly distributed over China with an average distance of 700 km. Its relative accuracy is at the grade of 10-9.
Another part of HACH2000 is the B-order height anomaly control network. It is measured with the national B-order GPS positioning standards and national 2nd or 3rd-order leveling network. There are 750 points of the B-order HACN2000, which are distributed in the east, central, and west parts of China with the resolutions of 80, 130 and 250 km respectively. The accuracy evaluation formula for the A-order HACN2000 (against Xi’an geodetic origin) is
where L is the accumulated length from the point along the A-order network to Xi’an geodetic origin. Taking the furthest point into account, let L =3000 km, then =±0.21 m.
（2）Accuracy evaluation for the B-order network of HACH2000
The formula of the accuracy evaluation of the B-order HACH2000 against Xi’an geodetic origin is
(central and East China),
If the most remote point of B-order in West China is taken into consideration, i.e. let L=3000 km. Substitute it into the above formula, then the accuracy of the height anomaly against Xi’an geodetic origin is .
As far as the B-order points in central and east China are concerned, the corresponding accuracy is not worse than ±0.23 m if the above formula is used in the computation. It can be seen from this that the accuracy of height anomaly at the corresponding point of HACN2000, which contains China A-order and B-order GPS leveling networks, is no worse than ±0.3 m.
4. Fitting of Gravity (Quasi) Geoid to HACN2000
The systemic or random differences between the gravity (quasi) geoid and the (quasi) geoid obtained from the GPS leveling are caused by various factors. For example, the geoid computed by use of terrain gravity data, potential model and DTM is not connected with certain coordinate system. While the computation of GPS leveling is strictly consistent with the specified coordinate system, potential model and height datum. These factors will lead to the differences of the two kinds of geoids (Table 1).
Table 1. Differences of Height Anomalies between Gravity Geoid and HACN2000
Num. of points
The results of test computation show that the systemic part of the differences between height anomalies of China’s gravity (quasi) geoid and those of HACN2000 can not be fitted with each other by coordinate transformation parameters. The differences can not be effectively improved by coordinate transformation of 3 parameters. Therefore quadric multinomial is used to make the fitting, which is important for reducing the systemic error and match the gravity geoid into a specified coordinate framework.
In order to test the accuracy of the above fitted geoid (the mainland part of CQG2000), we chose 80 GPS leveling points from China Crustal Movements Observation Network to conduct the field check, which are evenly distributed in China mainland and with 10-9 relative accuracy. The test results of the field check demonstrate that the height anomaly of the mainland part of CQG2000 has reached the precision of decimeter. The RMS in the area east to 102°E is no larger than ±0.3 m. And the in the area west to 102°E , the RMSs are ±0.4 m north to 36°N and ±0.6 m south to 36°N respectively. The nominal resolution of CQG2000 is 5’×5’ in the mainland areas, practically no worse than 15’×15’ and 30’ ×30’ in the east part and west part respectively.
II. RESEARCH AND COMPUTATION OF CHINA MARITIME GEOID
1. Scheme for the Computation of China Maritime Geoid
The following scheme is adopted for the computation of China maritime geoid after research and test computation.
（1）Three kinds of data obtained from Geosat, TOPEX/POSEIDON(T/P) and ERS-2 are used for joint data processing. Deflections of the vertical are considered as the basic (input) data to determine the marine gravity field.
（2）The crossing points are solved by total combination of the three kinds altimeter satellite orbits in pairs. The deflections of the vertical are taken at the crossing point.
（3）GRS 1980 and ITRF93 (reference frame for T/P orbit) are taken as the reference frame for China maritime geoid. The orbit of ERS-2 also belongs to ITRF system. However the biases in the Geosat data caused by the reference frame have to be corrected
（4）Molodensky formula for geoid height inverse from the deflections of the vertical is used to compute the maritime geoid. The gravity anomalies derived from inverse Vening-Meinesz formula are used for internal check with those from Stokes formula, and the gravity dada measured by ship are compared with those obtained from altimeter as an external check.
2. Testing of the Computed Results of China Maritime Geoid
（1）Accuracy test of marine gravity anomalies inverse solved from satellite altimetry
The gridded deflections of the vertical in China maritime area are computed from satellite altimeter with 1500 Gb vast data, then they are inverted into the gravity anomalies. Compared and checked with about 600 000 ship measured gravity anomalies, the RMS and standard error are ±9.35 mGal and ±9.34 mGal respectively. The results computed with accurate 2D plane convolution formula are compared with that computed with 1D rigorous convolution formula, and their maximum differences are –1.7 mGal and 1.8 mGal respectively.
（2）Accuracy estimation for the maritime geoid height inversed from satellite altimeter
To test the reliability of the geoid computation, one geoid is computed with Stokes formula by the gravity anomalies derived from inversion. Another geoid is inversed from the deflections of the vertical. Comparing the two geoids, the standard error is ±0.025 m, it means the confirmation of the precision of the computation results.
III. THE MATCH OF CHINA MAINLAND GEOID AND MARITIME GEOID
1. Principles for the Match of China Mainland and Maritime Geoids
It is rather complicated to identify the causes of the differences happening in the two geoids, one is determined by the ground gravity data, and the other determined by altimeter. Besides, it exists a gap of gravity data between the conjunction area of mainland and the shallow sea of China. Considering the practical situation in China, the following scheme is put forward for the match of the mainland and maritime geoids.
（1）The resolution and accuracy should keep consistent after the match of the maritime geoid with mainland geoid. The RMS of maritime geoids should be better than ±0.3 m after the match, then it is necessary to ensure the new geoid CQG2000, which covers the total territory of China, reaches the accuracy at the magnitude of centimeter.
（2）Mainland geoid should remain unchanged after the match. The reason of using satellite altimetry data to extend the mainland geoid to the maritime areas is that the density and accuracy of the gravimetric data in east part of China are rather high, which can reduce the effect of the systemic error caused by altimetry data over the sea area.
（3）The match should meet the requirements of the potential theory. The geoid after match (CQG2000) should keep the original nature of equipotential surface. Therefore the match should be satisfied with Laplace equation.
（4）Earth gravity model EGM96 is used as the reference gravity field for the match, which will control the smooth match of the two geoids at middle and short wave lengths. Meanwhile GRS1980 is used in the match in order to keep consistent to the international reference ellipsoid.
2. Scheme for the Match of China Mainland and Maritime Geoids
Suppose the computations of China mainland geoid and the China maritime geoid obtained from altimeter have already been completed respectively before the match. Then the gridded mean gravity anomaly of China’s coastal area and the gridded mean gravity anomaly of China’s maritime area along the coastal area, which is inversed from the altimeter data, are selected and computed together with Stokes formula to determine a local gravity geoid. The selected coastal area should be as equal to or a bit larger than the adjacent maritime area. As a matter of fact the local gravity geoid only includes the above mentioned selected areas, i.e. a part of China mainland and China’s maritime area, so hereinafter referred to it as China “mainland-sea local gravity geoid”. Then the mainland area of the “mainland-sea local gravity geoid” is fitted to the overlapped part of the already completed mainland geoid. The derived fitting parameters are used to correct the total gravity geoid of China’s total maritime areas, while the completed mainland geoid remains unchanged. Having strong control over the altimeteric gravity data in maritime areas at this stage, the mainland gravity data are used to calibrate the altimetric maritime geoid in order to reduce the systemic errors in the altimetric maritime geoid.
The purpose of the above mentioned scheme is to determine a unified “mainland-sea local gravity geoid” by calculating the gridded mean gravity anomaly of the mainland along China’s coastal areas, and the gridded mean gravity anomaly was inversed from the altimeter data in China’s maritime areas adjacent to the coastal area. Theoretically, the scheme is deliberate and reliable. The weakness of the scheme is that the gravity anomaly of EGM96 has to fill the gravity data gaps in the adjacent areas between mainland and sea of China, which may lead to the low precision of the maritime geoid at short wave length. In addition, in lacking of external (practical) check on the systemic errors possibly existed in the altimetric marine gravity anomaly, and it is difficult to estimate the real accuracy of geoid in the maritime area. In order to avoid the influence on the mainland geoid coming from maritime geoid in such match, the computed mainland geoid keeps unchanged and only the altimeter maritime geoid is corrected in the match.
3. Mathematical Model of the Match of China Mainland-Sea Geoids
After the determination of the “mainland-sea local gravity geoid” is computed with Stokes formula, it needs to be fitted to the overlapped part of the already computed mainland geoid. The fitting parameters in the computation are used to correct the marine gravity geoid. After comparison of the fitting models, a quartic multinomial is used to fit the two geoids by the least-square method to reduce and eliminate the differences between them.
The scheme presented in this paper takes the gravity data on the mainland and the marine data inversed from altimeter as a whole to derive a unified “mainland-sea local gravity (quasi) geoid”. Then China mainland (quasi) geoid is used as a control factor to match the “mainland-sea local gravity (quasi) geoid”. An adjusted China’s unified mainland-sea (quasi) geoid (CQG2000) is then determined accordingly. The influence of systemic error caused by the altimeter marine gravity data can be controlled as the measured data (surface gravimetric data and GPS leveling data) are playing important roles during the procedure for the determination of CQG2000 in this scheme.
1. Utilization of High Resolution DTM and Surface Gravity Data of China
High-resolution DTM and surface gravity data of China are used to compute China mainland gravity (quasi) geoid by remove/restore method on the basis of EGM96. The geoid is then fitted to China’s height anomaly control network (HACN2000) to determine the mainland (quasi) geoid, i.e. the mainland part of the new geoid CQG2000.
2. Utilization of Vast Altimeter Data for China Maritime Geoid Determination
The vast satellite altimeter data of China’s maritime areas are used to inverse China maritime geoid through the computation of the deflection of the vertical in order to reduce the systemic errors. The gravity values measured by ship are used to check the corresponding values derived from the altimeter data, which demonstrates all the computation is right.
3. The Least Square Method Is Used to Fit the Mainland and Maritime Geoids
The least square method is used to fit the mainland and maritime geoids and then to determine the new (quasi) geoid CQG2000.
4. Strict ID FFT or FHT is Used for Integral Computation
Strict ID FFT or FHT is used for integral calculation. Ellipsoid correction or G1 term is considered in the integral formula.
5. CQG2000 Has Been Tested by Production Practical
CQG2000 has been tested by production practical that the accuracy of height anomaly of CQG2000 has reached the designed value, at the level of centimeter. The RMS in the area east to 102°E is no worse than ±0.3 m. And that in the area west to 102°E, the mean square errors are ±0.4 m north to 36°N, and ±0.6 m south to 36°N respectively.
CQG2000 has covered China’s total territory, i.e. all the mainland and maritime areas including the special economic zones. The resolutions in mainland are 15’×15’ in the east area and 20’×20’ in the west. Therefore the accuracy and resolution of CQG2000 has all been upgraded at one level Compared with China’s existing geoid CQG1980.
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