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).     

A comparison of Stokes' numerical integration and collocation,and a new combination technique

Summary Basically two different evaluation methods are available to compute geoid heights from residual gravity anomalies in the inner zone: numerical integration and least squares collocation.If collocation is not applied to a global gravity data set, as is usually the case in practice, its result will not be equal to the numerical integration result. However, the cross covariance function between geoid heights and gravity anomalies can be adapted such that the geoid contribution is computed only from a small gravity area up to a certain distance _o from the computation point. Using this modification, identical results are obtained as from numerical integration.Applying this modification makes the results less dependent on the covariance function used. The difference between numerical integration and collocation is mainly caused by the implicitly extrapolated residual gravity anomaly values, outside the original data area. This extrapolated signal depends very much on the covariance function used, while the interpolated values within the original data area depend much less on it.As a sort of by-product, this modified collocation formula also leads to a new combination technique of numerical integration and collocation, in which the optimizing practical properties of both methods are fully exploited.Numerical examples are added as illustration.     

Regional geopotential model improvement for the Iranian geoid determination

Spherical harmonic expansions of the geopotential are frequently used for modelling the earth’s gravity field. Degree and order of recently available models go up to 360, corresponding to a resolution of about50 km. Thus, the high degree potential coefficients can be verified nowadays even by locally distributed sets of terrestrial gravity anomalies. These verifications are important when combining the short wavelength model impact, e.g. for regional geoid determinations by means of collocation solutions. A method based on integral formulae is presented, enabling the improvement of geopotential models with respect to non-global distributed gravity anomalies. To illustrate the foregoing, geoid computations are carried out for the area of Iran, introducing theGPM2 geopotential model in combination with available regional gravity data. The accuracy of the geoid determination is estimated from a comparison with Doppler and levelling data to ±1.4m.     

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.     

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     

A bias-free geodetic boundary value problem approach to height datum unification

A geodetic boundary value problem (GBVP) approach has been formulated which can be used for solving the problem of height datum unification. The developed technique is applied to a test area in Southwest Finland with approximate size of 1.5° × 3° and the bias of the corresponding local height datum (local geoid) with respect to the geoid is computed. For this purpose the bias-free potential difference and gravity difference observations of the test area are used and the offset (bias) of the height datum, i.e., Finnish Height Datum 2000 (N2000) fixed to Normaal Amsterdams Peil (NAP) as origin point, with respect to the geoid is computed. The results of this computation show that potential of the origin point of N2000, i.e., NAP, is (62636857.68 ± 0.5) (m²/s²) and as such is (0.191 ± 0.003) (m) under the geoid defined by W ₀ = 62636855.8 (m²/s²). As the validity test of our methodology, the test area is divided into two parts and the corresponding potential difference and gravity difference observations are introduced into our GBVP separately and the bias of height datums of the two parts are computed with respect to the geoid. Obtaining approximately the same bias values for the height datums of the two parts being part of one height datum with one origin point proves the validity of our approach. Besides, the latter test shows the capability of our methodology for patch-wise application.     

Simulation of regional gravity field recovery from satellite gravity gradiometer data using collocation and FFT

When planning a satellite gravity gradiometer (SGG) mission, it is important to know the quality of the quantities to be recovered at ground level as a function of e.g. satellite altitude, data type and sampling rate, and signal variance and noise. This kind of knowledge may be provided either using the formal error estimates of wanted quantities using least-squares collocation (LSC) or by comparing simulated data at ground level with results computed by methods like LSC or Fast Fourier Transform (FFT). Results of a regional gravity field recovery in a 10^o×20^o area surrounding the Alps using LSC and FFT are reported. Data used as observations in satellite altitude (202 or161 km) and for comparison at ground level were generated using theOSU86F coefficient set, complete to degree 360. These observations are referred to points across simulated orbits. The simulated quantities were computed for a 45 days mission period and 4 s sampling. A covariance function which also included terms above degree 360 was used for prediction and error estimation. This had the effect that the formal error standard deviation for gravity anomalies were considerably larger than the standard deviations of predicted minus simulated quantities. This shows the importance of using data with frequency content above degree 360 in simulation studies. Using data at202 km altitude the standard deviation of the predicted minus simulated data was equal to8.3 mgal for gravity and0.33 m for geoid heights.     

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     

The CARIB97 high-resolution geoid height model for the Caribbean Sea

A 2×2 arc-minute resolution geoid model, CARIB97, has been computed covering the Caribbean Sea. The geoid undulations refer to the GRS-80 ellipsoid, centered at the ITRF94 (1996.0) origin. The geoid level is defined by adopting the gravity potential on the geoid as W ₀=62 636 856.88 m²/s² and a gravity-mass constant of GM=3.986 004 418×10¹⁴ m³/s². The geoid model was computed by applying high-frequency corrections to the Earth Gravity Model 1996 global geopotential model in a remove-compute-restore procedure. The permanent tide system of CARIB97 is non-tidal. Comparison of CARIB97 geoid heights to 31 GPS/tidal (ITRF94/local) benchmarks shows an average offset (h–H–N) of 51 cm, with an Root Mean Square (RMS) of 62 cm about the average. This represents an improvement over the use of a global geoid model for the region. However, because the measured orthometric heights (H) refer to many differing tidal datums, these comparisons are biased by localized permanent ocean dynamic topography (PODT). Therefore, we interpret the 51 cm as partially an estimate of the average PODT in the vicinity of the 31 island benchmarks. On an island-by-island basis, CARIB97 now offers the ability to analyze local datum problems which were previously unrecognized due to a lack of high-resolution geoid information in the area. Received: 2 January 1998 / Accepted: 18 August 1998     

Gibbs sampler for computing and propagating large covariance matrices

The use of sampling-based Monte Carlo methods for the computation and propagation of large covariance matrices in geodetic applications is investigated. In particular, the so-called Gibbs sampler, and its use in deriving covariance matrices by Monte Carlo integration, and in linear and nonlinear error propagation studies, is discussed. Modifications of this technique are given which improve in efficiency in situations where estimated parameters are highly correlated and normal matrices appear as ill-conditioned. This is a situation frequently encountered in satellite gravity field modelling. A synthetic experiment, where covariance matrices for spherical harmonic coefficients are estimated and propagated to geoid height covariance matrices, is described. In this case, the generated samples correspond to random realizations of errors of a gravity field model. AcknowledgementsThe authors are indebted to Pieter Visser and Pavel Ditmar for providing simulation output that was used in the GOCE error generation experiments. Furthermore, the NASA/NIMA/OSU team is acknowledged for providing public ftp access to the EGM96 error covariance matrix. The two anonymous reviewers are thanked for their valuable comments.     

Global height datum unification: a new approach in gravity potential space

The problem of “global height datum unification” is solved in the gravity potential space based on: (1) high-resolution local gravity field modeling, (2) geocentric coordinates of the reference benchmark, and (3) a known value of the geoid’s potential. The high-resolution local gravity field model is derived based on a solution of the fixed-free two-boundary-value problem of the Earth’s gravity field using (a) potential difference values (from precise leveling), (b) modulus of the gravity vector (from gravimetry), (c) astronomical longitude and latitude (from geodetic astronomy and/or combination of (GNSS) Global Navigation Satellite System observations with total station measurements), (d) and satellite altimetry. Knowing the height of the reference benchmark in the national height system and its geocentric GNSS coordinates, and using the derived high-resolution local gravity field model, the gravity potential value of the zero point of the height system is computed. The difference between the derived gravity potential value of the zero point of the height system and the geoid’s potential value is computed. This potential difference gives the offset of the zero point of the height system from geoid in the “potential space”, which is transferred into “geometry space” using the transformation formula derived in this paper. The method was applied to the computation of the offset of the zero point of the Iranian height datum from the geoid’s potential value W ₀=62636855.8 m²/s². According to the geometry space computations, the height datum of Iran is 0.09 m below the geoid.     

The IAG approach to the atmospheric geoid correction in Stokes' formula and a new strategy

The well-known International Association of Geodesy (IAG) approach to the atmospheric geoid correction in connection with Stokes' integral formula leads to a very significant bias, of the order of 3.2 m, if Stokes' integral is truncated to a limited region around the computation point. The derived truncation error can be used to correct old results. For future applications a new strategy is recommended, where the total atmospheric geoid correction is estimated as the sum of the direct and indirect effects. This strategy implies computational gains as it avoids the correction of direct effect for each gravity observation, and it does not suffer from the truncation bias mentioned above. It can also easily be used to add the atmospheric correction to old geoid estimates, where this correction was omitted. In contrast to the terrain correction, it is shown that the atmospheric geoid correction is mainly of order H of terrain elevation, while the term of order H ² is within a few millimetres. Received: 20 May 1998 / Accepted: 19 April 1999     

Downward continuation of the free-air gravity anomalies to the ellipsoid using the gradient solution and terrain correction—An attempt of global numerical computations

The formulas for the determination of the coefficients of the spherical harmonic expansion of the disturbing potential of the earth are defined for data given on a sphere. In order to determine the spherical harmonic coefficients, the gravity anomalies have to be analytically downward continued from the earth's surface to a sphere—at least to the ellipsoid. The goal of this paper is to continue the gravity anomalies from the earth's surface downward to the ellipsoid using recent elevation models. The basic method for the downward continuation is the gradient solution (theg ₁ term). The terrain correction has also been computed because of the role it can play as a correction term when calculating harmonic coefficients from surface gravity data. Theg ₁ term and the terrain correction were expanded into the spherical harmonics up to180 ^th order. The corrections (theg ₁ term and the terrain correction) have the order of about 2% of theRMS value of degree variance of the disturbing potential per degree. The influences of theg ₁ term and the terrain correction on the geoid take the order of 1 meter (RMS value of corrections of the geoid undulation) and on the deflections of the vertical is of the order 0.1″ (RMS value of correction of the deflections of the vertical).     

A synthetic Earth Gravity Model Designed Specifically for Testing Regional Gravimetric Geoid Determination Algorithms

A synthetic [simulated] Earth gravity model (SEGM) of the geoid, gravity and topography has been constructed over Australia specifically for validating regional gravimetric geoid determination theories, techniques and computer software. This regional high-resolution (1-arc-min by 1-arc-min) Australian SEGM (AusSEGM) is a combined source and effect model. The long-wavelength effect part (up to and including spherical harmonic degree and order 360) is taken from an assumed errorless EGM96 global geopotential model. Using forward modelling via numerical Newtonian integration, the short-wavelength source part is computed from a high-resolution (3-arc-sec by 3-arc-sec) synthetic digital elevation model (SDEM), which is a fractal surface based on the GLOBE v1 DEM. All topographic masses are modelled with a constant mass-density of 2,670 kg/m³. Based on these input data, gravity values on the synthetic topography (on a grid and at arbitrarily distributed discrete points) and consistent geoidal heights at regular 1-arc-min geographical grid nodes have been computed. The precision of the synthetic gravity and geoid data (after a first iteration) is estimated to be better than 30 μ Gal and 3 mm, respectively, which reduces to 1 μ Gal and 1 mm after a second iteration. The second iteration accounts for the changes in the geoid due to the superposed synthetic topographic mass distribution. The first iteration of AusSEGM is compared with Australian gravity and GPS-levelling data to verify that it gives a realistic representation of the Earth’s gravity field. As a by-product of this comparison, AusSEGM gives further evidence of the north–south-trending error in the Australian Height Datum. The freely available AusSEGM-derived gravity and SDEM data, included as Electronic Supplementary Material (ESM) with this paper, can be used to compute a geoid model that, if correct, will agree to in 3 mm with the AusSEGM geoidal heights, thus offering independent verification of theories and numerical techniques used for regional geoid modelling.Electronic Supplementary Material Supplementary material is available in the online version of this article at http://dx.doi.org/10.1007/s00190-005-0002-z     

Computation of spherical harmonic coefficients and their error estimates using least-squares collocation

 Equations expressing the covariances between spherical harmonic coefficients and linear functionals applied on the anomalous gravity potential, T, are derived. The functionals are the evaluation functionals, and those associated with first- and second-order derivatives of T. These equations form the basis for the prediction of spherical harmonic coefficients using least-squares collocation (LSC). The equations were implemented in the GRAVSOFT program GEOCOL. Initially, tests using EGM96 were performed using global and regional sets of geoid heights, gravity anomalies and second-order vertical gravity gradients at ground level and at altitude. The global tests confirm that coefficients may be estimated consistently using LSC while the error estimates are much too large for the lower-order coefficients. The validity of an error estimate calculated using LSC with an isotropic covariance function is based on a hypothesis that the coefficients of a specific degree all belong to the same normal distribution. However, the coefficients of lower degree do not fulfil this, and this seems to be the reason for the too-pessimistic error estimates. In order to test this the coefficients of EGM96 were perturbed, so that the pertubations for a specific degree all belonged to a normal distribution with the variance equal to the mean error variance of the coefficients. The pertubations were used to generate residual geoid heights, gravity anomalies and second-order vertical gravity gradients. These data were then used to calculate estimates of the perturbed coefficients as well as error estimates of the quantities, which now have a very good agreement with the errors computed from the simulated observed minus calculated coefficients. Tests with regionally distributed data showed that long-wavelength information is lost, but also that it seems to be recovered for specific coefficients depending on where the data are located. Received: 3 February 2000 / Accepted: 23 October 2000     

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.     

Geoid and high resolution sea surface topography modelling in the mediterranean from gravimetry,altimetry and GOCE data: evaluation by simulation

The determination of local geoid models has traditionally been carried out on land and at sea using gravity anomaly and satellite altimetry data, while it will be aided by the data expected from satellite missions such as those from the Gravity field and steady-state ocean circulation explorer (GOCE). To assess the performance of heterogeneous data combination to local geoid determination, simulated data for the central Mediterranean Sea are analyzed. These data include marine and land gravity anomalies, altimetric sea surface heights, and GOCE observations processed with the space-wise approach. A spectral analysis of the aforementioned data shows their complementary character. GOCE data cover long wavelengths and account for the lack of such information from gravity anomalies. This is exploited for the estimation of local covariance function models, where it is seen that models computed with GOCE data and gravity anomaly empirical covariance functions perform better than models computed without GOCE data. The geoid is estimated by different data combinations and the results show that GOCE data improve the solutions for areas covered poorly with other data types, while also accounting for any long wavelength errors of the adopted reference model that exist even when the ground gravity data are dense. At sea, the altimetric data provide the dominant geoid information. However, the geoid accuracy is sensitive to orbit calibration errors and unmodeled sea surface topography (SST) effects. If such effects are present, the combination of GOCE and gravity anomaly data can improve the geoid accuracy. The present work also presents results from simulations for the recovery of the stationary SST, which show that the combination of geoid heights obtained from a spherical harmonic geopotential model derived from GOCE with satellite altimetry data can provide SST models with some centimeters of error. However, combining data from GOCE with gravity anomalies in a collocation approach can result in the estimation of a higher resolution geoid, more suitable for high resolution mean dynamic SST modeling. Such simulations can be performed toward the development and evaluation of SST recovery methods.     

确定大地水准面的Tikhonov最小二乘配置法

LSC法（最小二乘配置法）因能融合不同种类重力观测数据确定大地水准面的特性而受到广泛关注,但由于协方差矩阵存在病态性,微小的观测误差将被协方差矩阵的小奇异值放大,导致计算的配置结果不稳定且精度偏低。本文提出Tikhonov_LSC法,即在LSC法中引入Tikhonov正则化算法,基于GCV法选择协方差矩阵的正则化参数,利用正则化参数修正协方差矩阵的小奇异值,以抑制其对观测误差的放大影响。基于Tikhonov_LSC法计算大地水准面,能有效提高其稳定性和精度。通过以EGM2008重力场模型分别计算山区、丘陵和海域重力异常作为基础数据确定相应区域大地水准面的实验,验证了该方法的有效性。     

The ellipsoidal correction to the Stokes kernel for precise geoid determination

Solving the geodetic boundary-value problem (GBVP) for the precise determination of the geoid requires proper use of the fundamental equation of physical geodesy as the boundary condition given on the geoid. The Stokes formula and kernel are the result of spherical approximation of this fundamental equation, which is a violation of the proper relation between the observed quantity (gravity anomaly) and the sought function (geoid). The violation is interpreted here as the improper formulation of the boundary condition, which implies the spherical Stokes kernel to be in error compared with the proper kernel of integral transformation. To remedy this error, two correction kernels to the Stokes kernel were derived: the first in both closed and spectral forms and the second only in spectral form. Contributions from the first correction kernel to the geoid across the globe were [−0.867 m, +1.002 m] in the low-frequency domain implied by the GRIM4-S4 purely satellite-derived geopotential model. It is a few centimeters, on average, in the high-frequency domain with some exceptions of a few meters in places of high topographical relief and sizable geological features in accordance with the EGM96 combined geopotential model. The contributions from the second correction kernel to the geoid are [−0.259 m, +0.217 m] and [−0.024 m, +0.023 m] in the low- and high-frequency domains, respectively.     

The GEOID96 high-resolution geoid height model for the United States

The 2 arc-minute × 2 arc-minute geoid model (GEOID96) for the United States supports the conversion between North American Datum 1983 (NAD 83) ellipsoid heights and North American Vertical Datum 1988 (NAVD 88) Helmert heights. GEOID96 includes information from global positioning system (GPS) height measurements at optically leveled benchmarks. A separate geocentric gravimetric geoid, G96SSS, was first calculated, then datum transformations and least-squares collocation were used to convert from G96SSS to GEOID96. Fits of 2951 GPS/level (ITRF94/NAVD 88) benchmarks to G96SSS show a 15.1-cm root mean square (RMS) around a tilted plane (0.06 ppm, 178^∘ azimuth), with a mean value of −31.4 cm (15.6-cm RMS without plane). This mean represents a bias in NAVD 88 from global mean sea level, remaining nearly constant when computed from subsets of benchmarks. Fits of 2951 GPS/level (NAD 83/NAVD 88) benchmarks to GEOID96 show a 5.5-cm RMS (no tilts, zero average), due primarily to GPS error. The correlated error was 2.5 cm, decorrelating at 40 km, and is due to gravity, geoid and GPS errors. Differences between GEOID96 and GEOID93 range from −122 to +374 cm due primarily to the non-geocentricity of NAD 83. Received: 28 July 1997 / Accepted: 2 September 1998