Calibrating the GOCE accelerations with star sensor data and a global gravity field model

A reliable and accurate gradiometer calibration is essential for the scientific return of the gravity field and steady-state ocean circulation explorer (GOCE) mission. This paper describes a new method for external calibration of the GOCE gradiometer accelerations. A global gravity field model in combination with star sensor quaternions is used to compute reference differential accelerations, which may be used to estimate various combinations of gradiometer scale factors, internal gradiometer misalignments and misalignments between star sensor and gradiometer. In many aspects, the new method is complementary to the GOCE in-flight calibration. In contrast to the in-flight calibration, which requires a satellite-shaking phase, the new method uses data from the nominal measurement phases. The results of a simulation study show that gradiometer scale factors can be estimated on a weekly basis with accuracies better than 2 × 10⁻³ for the ultrasensitive and 10⁻² for the less sensitive axes, which is compatible with the requirements of the gravity gradient error. Based on a 58-day data set, scale factors are found that can reduce the errors of the in-flight-calibrated measurements. The elements of the complete inverse calibration matrix, representing both the internal gradiometer misalignments and scale factors, can be estimated with accuracies in general better than 10⁻³.     

An alternative algebraic algorithm to transform Cartesian to geodetic coordinates

The algorithm to transform from 3D Cartesian to geodetic coordinates is obtained by solving the equation of the Lagrange parameter. Numerical experiments show that geodetic height can be recovered to 0.5 mm precision over the range from −6×10⁶ to 10¹⁰ m. Electronic Supplementary Material: Supplementary material is available in the online version of this article at     

Precise alpine geoid determination without digital terrain models

The short wavelength geoid undulations, caused by topography, amount to several decimeters in mountainous areas. Up to now these effects are computed by means of digital terrain models in a grid of 100–500m. However, for many countries these data are not yet available or their collection is too expensive. This problem can be overcome by considering the special behaviour of the gravity potential along mountain slopes. It is shown that 90 per cent of the topographic effects are represented by a simple summation formula, based on the average height differences and distances between valleys and ridges along the geoid profiles, δN=[30.H.D.+16.(H−H′).D] in mm/km, (error<10%), whereH, H′, D are estimated in a map to the nearest 0.2km. The formula is valid for asymmetric sides of valleys (H, H′) and can easily be corrected for special shapes. It can be used for topographic refinement of low resolution geoids and for astrogeodetic projects. The “slope method” was tested in two alpine areas (heights up to 3800m, astrogeodetic deflection points every 170km ²) and resulted in a geoid accuracy of ±3cm. In first order triangulation networks (astro points every 1000km ²) or for gravimetric deflections the accuracy is about 10cm per 30km. Since a map scale of 1∶500.000 is sufficient, the method is suitable for developing countries, too.     

Equatorial radius of the Earth: A dynamical determination

An intrresting variation on the familiar method of determining the earth's equatorial radius a_e, from a knowledge of the earth's equatorial gravity is suggested. The value of equatorial radius thus found is 6378,142±5 meters. The associated parameters are GM=3.986005±.000004 × 10²⁰ cm³ sec-⁻² which excludes the relative mass of atmosphere ≅10⁻⁶ ξ GM, the equatorial gravity γ_e 978,030.9 milligals (constrained in this solution by the Potsdam Correction of 13.67 milligals as the Potsdam Correction is more directly, orless indirectly, measurable than the equatorial gravity) and an ellipsoidal flattening of f=1/298.255.     

On the existence of solutions for the method of Bjerhammar in the continuous case

The method of Bjerhammar is studied in the continuous case for a sphere. By varying the kernel function, different types of unknowns (u^*) are obtained at the internal sphere (the Bjerhammar sphere). It is shown that a necessary condition for the existence of u^* is that the degree variances (σ _n ² ) of the observations are of an order less than n⁻². According to Kaula’s rule this condition is not satisfied for the earth’s gravity anomaly field (σ _n ² =n⁻¹) but well for the geopotential (σ _n ² =n⁻³).     

Botswana gravity reference net

Gravity reference stations for the National Gravity Survey of Botswana have been established at twenty-three sites throughout the country in a net linked to existing bases in South Africa, Kenya and Zambia with an internal accuracy of better than 0.5 gravity units (one gravity unit, gu, equals an acceleration of 10⁻⁶ m.s⁻²). The field procedure and reduction of data are explained and a list is given of the gravity values.     

Recurrence relations for integrals of Associated Legendre functions

Recurrence relations for the evaluation of the integrals of associated Legendre functions over an arbitrary interval within (0°, 90°) have been derived which yield sufficiently accurate results throughout the entire range of their possible applications. These recurrence relations have been used to compute integrals up to degree 100 and similar computations can be carried out without any difficulty up to a degree as high as the memory in a computer permits. The computed values have been tested with independent check formulae, also derived in this work; the corresponding relative errors never exceed 10⁻²³ in magnitude. Contribution from the Earth Physics Branch No. 719     

On the accuracy of modified Stokes's integration in high-frequency gravimetric geoid determination

 Two numerical techniques are used in recent regional high-frequency geoid computations in Canada: discrete numerical integration and fast Fourier transform. These two techniques have been tested for their numerical accuracy using a synthetic gravity field. The synthetic field was generated by artificially extending the EGM96 spherical harmonic coefficients to degree 2160, which is commensurate with the regular 5^′ geographical grid used in Canada. This field was used to generate self-consistent sets of synthetic gravity anomalies and synthetic geoid heights with different degree variance spectra, which were used as control on the numerical geoid computation techniques. Both the discrete integration and the fast Fourier transform were applied within a 6^∘ spherical cap centered at each computation point. The effect of the gravity data outside the spherical cap was computed using the spheroidal Molodenskij approach. Comparisons of these geoid solutions with the synthetic geoid heights over western Canada indicate that the high-frequency geoid can be computed with an accuracy of approximately 1 cm using the modified Stokes technique, with discrete numerical integration giving a slightly, though not significantly, better result than fast Fourier transform. Received: 2 November 1999 / Accepted: 11 July 2000     

Measurement and Scaling of Carbon Dioxide (CO<Subscript>2</Subscript>) Exchanges in Wheat Using Flux-Tower and Remote Sensing

The present study investigates the characteristics of CO₂ exchange (photosynthesis and respiration) over agricultural site dominated by wheat crop and their relationship with ecosystem parameters derived from MODIS. Eddy covariance measurement of CO₂ and H₂O exchanges was carried out at 10 Hz interval and fluxes of CO₂ were computed at half-hourly time steps. The net ecosystem exchange (NEE) was partitioned into gross primary productivity (GPP) and ecosystem respiration (R _e) by taking difference between day-time NEE and respiration. Time-series of daily reflectance and surface temperature products at varying resolution (250–1000 m) were used to derive ecosystem variables (EVI, NDVI, LST). Diurnal pattern in Net ecosystem exchange reveals negative NEE during day-time representing CO₂ uptake and positive during night as release of CO₂. The amplitude of the diurnal variation in NEE increased as LAI crop growth advances and reached its peak around the anthesis stage. The mid-day uptake during this stage was around 1.15 mg CO₂ m⁻² s⁻¹ and night-time release was around 0.15 mg CO₂ m⁻² s⁻¹. Linear and non-linear least square regression procedures were employed to develop phenomenological models and empirical fits between flux tower based GPP and NEE with satellite derived variables and environmental parameters. Enhanced vegetation index was found significantly related to both GPP and NEE. However, NDVI showed little less significant relationship with both GPP and NEE. Furthemore, temperature-greenness (TG) model combining scaled EVI and LST was parameterized to estimate daily GPP over dominantly wheat crop site. (R ² = 0.77). Multi-variate analysis shows that inclusion of LST or air temperature with EVI marginally improves variance explained in daily NEE and GPP.     

Iterative vector methods for computing geodetic latitude and height from rectangular coordinates

 Two iterative vector methods for computing geodetic coordinates (φ, h) from rectangular coordinates (x, y, z) are presented. The methods are conceptually simple, work without modification at any latitude and are easy to program. Geodetic latitude and height can be calculated to acceptable precision in one iteration over the height range from −10⁶ to +10⁹ m. Received: 13 December 2000 / Accepted: 13 July 2001     

Applications of an orbiting gravity gradiometer

Considering present attempts to develop a gradiometer with an accuracy between 10⁻³ E and 10⁻⁴ E, two applications for such a device have been studied: (a) mapping the gravitational field of the Earth, and (b) estimating the geocentric distance of a satellite carrying the instrument. Given a certain power spectrum for the signal and 10⁻⁴ E (rms) of white measurement noise, the results of an error analysis indicate that a six-month mission in polar orbit at a height of 200 km, with samples taken every three seconds, should provide data for estimating the spherical harmonic potential coefficients up to degree and order 300 with less than 50% error, and improve the coefficients through degree 30 by up to four orders of magnitude compared to existing models. A simulation study based on numerical orbit integrations suggests that a simple adjustment of the initial conditions based on gradiometer data could produce orbits where the geocentric distance is accurate to 10 cm or better, provided the orbits are 2000 km high and some improvement in the gravity field up to degree 30 is first achieved. In this sense, the gravity-mapping capability of the gradiometer complements its use in orbit refinement. This idea can be of use in determining orbits for satellite altimetry. Furthermore, by tracking the gradiometer-carrying spacecraft when it passes nearly above a terrestrial station, the geocentric distance of this station can also be estimated to about one decimeter accuracy. This principle could be used in combination with VLBI and other modern methods to set up a world-wide 3-D network of high accuracy.     

Global mean sea surface and marine gravity anomaly from multi-satellite altimetry: applications of deflection-geoid and inverse Vening Meinesz formulae

 Global mean sea surface heights (SSHs) and gravity anomalies on a 2^′×2^′ grid were determined from Seasat, Geosat (Exact Repeat Mission and Geodetic Mission), ERS-1 (1.5-year mean of 35-day, and GM), TOPEX/POSEIDON (T/P) (5.6-year mean) and ERS-2 (2-year mean) altimeter data over the region 0^∘–360^∘ longitude and –80^∘–80^∘ latitude. To reduce ocean variabilities and data noises, SSHs from non-repeat missions were filtered by Gaussian filters of various wavelengths. A Levitus oceanic dynamic topography was subtracted from the altimeter-derived SSHs, and the resulting heights were used to compute along-track deflection of the vertical (DOV). Geoidal heights and gravity anomalies were then computed from DOV using the deflection-geoid and inverse Vening Meinesz formulae. The Levitus oceanic dynamic topography was added back to the geoidal heights to obtain a preliminary sea surface grid. The difference between the T/P mean sea surface and the preliminary sea surface was computed on a grid by a minimum curvature method and then was added to the preliminary grid. The comparison of the NCTU01 mean sea surface height (MSSH) with the T/P and the ERS-1 MSSH result in overall root-mean-square (RMS) differences of 5.0 and 3.1 cm in SSH, respectively, and 7.1 and 3.2 μrad in SSH gradient, respectively. The RMS differences between the predicted and shipborne gravity anomalies range from 3.0 to 13.4 mGal in 12 areas of the world's oceans. Received: 26 September 2001 / Accepted: 3 April 2002 Correspondence to: C. Hwang Acknowledgements. This research is partly supported by the National Science Council of ROC, under grants NSC89-2611-M-009-003-OP2 and NSC89-2211-E-009-095. This is a contribution to the IAG Special Study Group 3.186. The Geosat and ERS1/2 data are from NOAA and CERSAT/France, respectively. The T/P data were provided by AVISO. The CLS and GSFC00 MSS models were kindly provided by NASA/GSFC and CLS, respectively. Drs. Levitus, Monterey, and Boyer are thanked for providing the SST model. Dr. T. Gruber and two anonymous reviewers provided very detailed reviews that improved the quality of this paper.     

National height datum,the Gauss–Listing geoid level value w0 and its time variation 0 (Baltic Sea Level Project: epochs 1990.8, 1993.8, 1997.4)

 A methodology for precise determination of the fundamental geodetic parameter w ₀, the potential value of the Gauss–Listing geoid, as well as its time derivative ₀, is presented. The method is based on: (1) ellipsoidal harmonic expansion of the external gravitational field of the Earth to degree/order 360/360 (130 321 coefficients; http://www.uni-stuttgard.de/gi/research/ index.html projects) with respect to the International Reference Ellipsoid WGD2000, at the GPS positioned stations; and (2) ellipsoidal free-air gravity reduction of degree/order 360/360, based on orthometric heights of the GPS-positioned stations. The method has been numerically tested for the data of three GPS campaigns of the Baltic Sea Level project (epochs 1990.8,1993.4 and 1997.4). New w ₀ and ₀ values (w ₀=62 636 855.75 ± 0.21 m²/s², ₀=−0.0099±0.00079 m²/s² per year, w ₀/&γmacr;=6 379 781.502 m,₀/&γmacr;=1.0 mm/year, and &γmacr;= −9.81802523 m²/s²) for the test region (Baltic Sea) were obtained. As by-products of the main study, the following were also determined: (1) the high-resolution sea surface topography map for the Baltic Sea; (2) the most accurate regional geoid amongst four different regional Gauss–Listing geoids currently proposed for the Baltic Sea; and (3) the difference between the national height datums of countries around the Baltic Sea. Received: 14 August 2000 / Accepted: 19 June 2001     

A comparison of recent Earth gravitational models with emphasis on their contribution in refining the gravity and geoid at continental or regional scale

Since the publication of the Earth gravitational model (EGM)96 considerable improvements in the observation techniques resulted in the development of new improved models. The improvements are due to the availability of data from dedicated gravity mapping missions (CHAMP, GRACE) and to the use of 5′ × 5′ terrestrial and altimetry derived gravity anomalies. It is expected that the use of new EGMs will further contribute to the improvement of the resolution and accuracy of the gravity and geoid modeling in continental and regional scale. To prove this numerically, three representative Earth gravitational models are used for the reduction of several kinds of data related to the gravity field in different places of the Earth. The results of the reduction are discussed regarding the corresponding covariance functions which might be used for modeling using the least squares collocation method. The contribution of the EIGEN-GL04C model in most cases is comparable to that of EGM96. However, the big difference is shown in the case of EGM2008, due not only to its quality but obviously to its high degree of expansion. Almost in all cases the variance and the correlation length of the covariance functions of data reduced to this model up to its maximum degree are only a few percentages of corresponding quantities of the same data reduced up to degree 360. Furthermore, the mean value and the standard deviation of the reduced gravity anomalies in extended areas of the Earth such as Australia, Arctic region, Scandinavia or the Canadian plains, vary between −1 and +1 and between 5 and 10 × 10⁻⁵ ms⁻², respectively, reflecting the homogenization of the gravity field on a regional scale. This is very important in using least squares collocation for regional applications. However, the distance to the first zero-value was in several cases much longer than warranted by the high degree of the expansion. This is attributed to errors of medium wavelengths stemming from the lack of, e.g., high-quality data in some area.     

基于葵花-8卫星大气产品的地表下行短波辐射计算

地表下行短波辐射DSSR(Downward Surface Shortwave Radiation)的准确估算在气候变化研究和地表太阳能估算等领域具有重要作用。新一代静止气象卫星葵花-8(Himawari-8)具有高达10 min的对地观测能力,为DSSR近实时估算提供了新机遇。然而,日本宇宙航空研究开发机构(JAXA)对外公开的葵花-8辐射产品中,没有将其反演的云、气溶胶产品作为DSSR的输入参数,从而没有形成一整套的DSSR估算算法流程,缺乏产品输出的一致性。大气中的云、气溶胶是DSSR的重要影响因子,本文重点考虑云、气溶胶对太阳辐射的影响,基于大气辐射传输模式RSTAR构建了DSSR查找表,开发了DSSR的快速计算方法,进而将JAXA葵花-8二级云、气溶胶产品(光学厚度,粒子有效半径等)作为快速化计算方法的输入参量,计算得到了DSSR。通过与JAXA葵花-8二级DSSR产品(JAXA DSSR)对比,发现两者具有很好的空间一致性。为了进一步评价本文的DSSR计算精度,分别选取了陆地(Yonsei)和海洋(0n_165e)的观测数据验证了2016年4、7、10和12月本文计算的DSSR和同时期的JAXA DSSR产品,验证结果显示两者的DSSR在两个观测站点均具有非常高的相关性(全天空、晴空和云天条件下的相关系数R均大于0.88)。在两个站点云天条件下的验证结果中,考虑了云相态并在冰云模型中使用了非球形冰晶粒子(六棱柱)来计算DSSR,获得了比JAXA DSSR更小的偏差。本文提出的快速化计算方法能快速准确地计算DSSR,可为计算地表辐射收支等研究提供重要数据支撑。     

Assessment of groundwater potential zones using GIS technique

A case study was conducted to find out the groundwater potential zones in Kattakulathur block, Tamil Nadu, India with an aerial extent of 360.60 km². The thematic maps such as geology, geomorphology, soil hydrological group, land use / land cover and drainage map were prepared for the study area. The Digital Elevation Model (DEM) has been generated from the 10 m interval contour lines (which is derived from SOI, Toposheet 1:25000 scale) and obtained the slope (%) of the study area. The groundwater potential zones were obtained by overlaying all the thematic maps in terms of weighted overlay methods using the spatial analysis tool in ArcGIS 9.2. During weighted overlay analysis, the ranking has been given for each individual parameter of each thematic map and weights were assigned according to the influence such as soil −25%, geomorphology − 25%, land use/land cover −25%, slope − 15%, lineament − 5% and drainage / streams − 5% and find out the potential zones in terms of good, moderate and poor zones with the area of 49.70 km², 261.61 km² and 46.04 km² respectively. The potential zone wise study area was overlaid with village boundary map and the village wise groundwater potential zones with three categories such as good, moderate and poor zones were obtained. This GIS based output result was validated by conducting field survey by randomly selecting wells in different villages using GPS instruments. The coordinates of each well location were obtained by GPS and plotted in the GIS platform and it was clearly shown that the well coordinates were exactly seated with the classified zones.     

Spatial and Temporal Variability of Net Primary Productivity (NPP) over Terrestrial Biosphere of India Using NOAA-AVHRR Based GloPEM Model

The monitoring of terrestrial carbon dynamics is important in studies related with global climate change. This paper presents results of the inter-annual variability of Net Primary Productivity (NPP) from 1981 to 2000 derived using observations from NOAA-AVHRR data using Global Production Efficiency Model (GloPEM). The GloPEM model is based on physiological principles and uses the production efficiency concept, in which the canopy absorption of photosynthetically active radiation (APAR) is used with a conversion “efficiency” to estimate Gross Primary Production (GPP). NPP derived from GloPEM model over India showed maximum NPP about 3,000 gCm⁻²year⁻¹ in west Bengal and lowest up to 500 gCm⁻²year⁻¹ in Rajasthan. The India averaged NPP varied from 1,084.7 gCm⁻²year⁻¹ to 1,390.8 gCm⁻²year⁻¹ in the corresponding years of 1983 and 1998 respectively. The regression analysis of the 20 year NPP variability showed significant increase in NPP over India (r = 0.7, F = 17.53, p < 0.001). The mean rate of increase was observed as 10.43 gCm⁻²year⁻¹. Carbon fixation ability of terrestrial ecosystem of India is increasing with rate of 34.3 TgC annually (t = 4.18, p < 0.001). The estimated net carbon fixation over Indian landmass ranged from 3.56 PgC (in 1983) to 4.57 PgC (in 1998). Grid level temporal correlation analysis showed that agricultural regions are the source of increase in terrestrial NPP of India. Parts of forest regions (Himalayan in Nepal, north east India) are relatively less influenced over the study period and showed lower or negative correlation (trend). Finding of the study would provide valuable input in understanding the global change associated with vegetation activities as a sink for atmospheric carbon dioxide.     

A non-iterative and non-singular perturbation solution for transforming Cartesian to geodetic coordinates

The Cartesian-to-Geodetic coordinate transformation is re-cast as a minimization algorithm for the height of the Satellite above the reference Earth surface. Optimal necessary conditions are obtained that fix the satellite ground track vector components in the Earth surface. The introduction of an artificial perturbation variable yields a rapidly converging second order power series solution. The initial starting guess for the solution provides 3–4 digits of precision. Convergence of the perturbation series expansion is accelerated by replacing the series solution with a Padé approximation. For satellites with heights < 30,000 km the second-order expansions yields ~mm satellite geodetic height errors and ~10⁻¹² rad errors for the geodetic latitude. No quartic or cubic solutions are required: the algorithm is both non-iterative and non-singular. Only two square root and two arc-tan calculations are required for the entire transformation. The proposed algorithm has been measured to be ~41% faster than the celebrated Bowring method. Several numerical examples are provided to demonstrate the effectiveness of the new algorithm.     

海洋激光雷达反演水体光学参数

研制了一套船载海洋激光雷达,用于探测海水光学参数垂直廓线。2017年8月,该系统在黄海海域进行了实验测量。在准单次散射模型中引入原位测量的光学参数,实现了理想激光雷达回波信号的模拟,并将该理想信号与系统响应函数卷积后精确复现了实验的激光雷达信号。采用Fernald后向迭代积分法(简称Fernald法),比较了不同水体悬浮物激光雷达比下反演的激光雷达衰减系数α与原位漫射衰减系数Kd的差别。基于停航时标定的水体悬浮物激光雷达比,采用Fernald法获得了走航时的激光雷达衰减系数。进一步地,提出一种基于米散射激光雷达数据和原位测量的后向散射数据的融合算法,模拟了高光谱分辨率激光雷达(HSRL)反演α的过程,并将其与Fernald法进行了比较。实验结果表明,自研的海洋激光雷达能够有效探测海水光学参数,基于合适的水体悬浮物激光雷达比的Fernald法可以有效应用于米散射激光雷达的反演,未来无需假设的HSRL在海水光学参数探测领域具有更大的优势。     

On calculating the Earth's C21 and S21 gravity coefficients in the IERS terrestrial reference frame

Modern models of the Earth's gravity field are developed in the IERS (International Earth Rotation Service) terrestrial reference frame. In this frame the mean values for gravity coefficients of the second degree and first order, C ₂₁(IERS) and S ₂₁(IERS), by the current IERS Conventions are recommended to be calculated by using the observed polar motion parameters. Here, it is proved that the formulae presently employed by the IERS Conventions to obtain these coefficients are insufficient to ensure their values as given by the same source. The relevant error of the normalized mean values for C ₂₁(IERS) and S ₂₁(IERS) is 3×10⁻¹², far above the adopted cutoff (10⁻¹³) for variations of these coefficients. Such an error in C ₂₁ and S ₂₁ can produce non-modeled perturbations in motion prediction of certain artificial Earth satellites of a magnitude comparable to the accuracy of current tracking measurements. Received: 14 September 1998 / Accepted: 20 May 1999