Truncation error formulas for the geoidal height and the deflection of the vertical

A general formula giving Molodenskii coefficientsQ _n of the truncation errors for the geoidal height is introduced in this paper. A relation betweenQ _n andq _n, Cook’s truncation function, is also obtained. Cook (1951) has treated the truncation errors for the deflection of the vertical in the Vening Meinesz integration. Molodenskii et al. (1962) have also derived the truncation error formulas for the deflection of the vertical. It is proved in this paper that these two formulas are equivalent.     

Construction of anisotropic covariance functions using Riesz-representers

A reproducing-kernel Hilbert space (RKHS) of functions harmonic in the set outside a sphere with radius R ₀, having a reproducing kernel K ₀(P,Q) is considered (P, Q, and later P _n being points in the set of harmonicity). The degree variances of this kernel will be denoted σ_{0 n} . The set of Riesz representers associated with the evaluation functionals (or gravity functionals) related to distinct points P _n ,n = 1,…,N, on a two-dimensional surface surrounding the bounding sphere, will be linearly independent. These functions are used to define a new N-dimensional RKHS with kernel (a _n >0)

A recurrence relation for the βn-function

A recurrence relation is presented for the smoothing function, β_n, which is used in geodesy to relate spherical harmonics to their mean values over circular areas (caps). The proposed formula does not require the computation of the Legendre's polynomials. Moreover, it is numerically more stable than the formulas ofPellinen (1969) andMeissl (1971).     

Noise-signal index threshold: a new noise-reduction technique for generation of reference spectra and efficient hyperspectral image classification

Reference spectra of terrestrial targets are usually collected using field spectro-radiometers for mineral abundance mapping and target detection. These spectra often have noise that masks characteristic absorption and reflection features and affects the efficiency of material mapping. This work aims at obtaining an empirical technique for reduction of high-frequency noise from field spectra. The proposed noise correction technique uses a ‘normalized’ measure R_n , where R_n  = (L_n  ? F_n )/L_n for each band (n) calculated from field and laboratory spectra of test material, with F_n and L_n being the depth of the absorption feature in field and laboratory spectra, respectively. On the basis of the assumption of the constancy of this ratio in neighbouring bands, an empirical algorithm that approximates the ratio R_n of a noisy band to the corrected ratio of an adjacent band is used to obtain the noise-corrected field spectra. The classification accuracy increases significantly when noise reduced field spectra are used as reference spectra.     

On the approximation of external gravitational potential with closed systems of (trial) functions

Summary Let S be the (regular) boundary-surface of an exterior regionE _e in Euclidean space ℜ³ (for instance: sphere, ellipsoid, geoid, earth's surface). Denote by {φ_n} a countable, linearly independent system of trial functions (e.g., solid spherical harmonics or certain singularity functions) which are harmonic in some domain containingE _e ∪ S. It is the purpose of this paper to show that the restrictions {ϕ_n} of the functions {φ_n} onS form a closed system in the spaceC (S), i.e. any functionf, defined and continuous onS, can be approximated uniformly by a linear combination of the functions ϕ_n. Consequences of this result are versions of Runge and Keldysh-Lavrentiev theorems adapted to the chosen system {φ_n} and the mathematical justification of the use of trial functions in numerical (especially: collocational) procedures.     

New analytical and numerical approaches for geopotential modeling

 The standard analytical approach which is applied for constructing geopotential models OSU86 and earlier ones, is based on reducing the boundary value equation to a sphere enveloping the Earth and then solving it directly with respect to the potential coefficients _n,m . In an alternative procedure, developed by Jekeli and used for constructing the models OSU91 and EGM96, at first an ellipsoidal harmonic series is developed for the geopotential and then its coefficients _n,m ^e are transformed to the unknown _n,m . The second solution is more exact, but much more complicated. The standard procedure is modified and a new simple integral formula is derived for evaluating the potential coefficients. The efficiency of the standard and new procedures is studied numerically. In these solutions the same input data are used as for constructing high-degree parts of the EGM96 models. From two sets of _n,m (n≤360,|m|≤n), derived by the standard and new approaches, different spectral characteristics of the gravity anomaly and the geoid undulation are estimated and then compared with similar characteristics evaluated by Jekeli's approach (`etalon' solution). The new solution appears to be very close to Jekeli's, as opposed to the standard solution. The discrepancies between all the characteristics of the new and `etalon' solutions are smaller than the corresponding discrepancies between two versions of the final geopotential model EGM96, one of them (HDM190) constructed by the block-diagonal least squares (LS) adjustment and the other one (V068) by using Jekeli's approach. On the basis of the derived analytical solution a new simple mathematical model is developed to apply the LS technique for evaluating geopotential coefficients. Received: 12 December 2000 / Accepted: 21 June 2001     

Accurate formula expressing the difference between the normal gravity and its radial component

A simple formula is presented giving the value of γ−γ _r to better than 0.001 mgal associated with an arbitrary reference ellipsoid, where γ is the normal gravity and γ _r is its radial component. Further simplifications of this formula are possible, depending on the desired accuracy. Since in the actual field g−g_r equals γ−γ _r to a good approximation, this formula makes it possible to work in terms of g_r rather than in terms of the measured quantity g. Such a choice is attractive mainly because the spherical harmonic expansion of g_r is very simple.     

NOTE ON THE REDUCTION OF CIRCUMMERIDIAN ALTITUDES TO THE MERIDIAN

Abstract

The method of reducing circummeridian altitudes or zenith distances to the meridian, using the factors m and n as tabulated by Chauvenet, is well known. The following method, which does not use these factars, has been faund both more convenient and more accurate in practice. The formula can be easily obtained by expanding m and n in powers of t, but far the sake af campleteness the derivatian is here given from the beginning.     

GPS radio occultation measurements on ionospheric electron density from low Earth orbit

Since the proof-of-concept GPS/Meteorology (GPS/MET) experiment successfully demonstrated active limb sounding of the Earth’s neutral atmosphere and ionosphere via GPS radio occultation (RO) from low Earth orbit, the developments of electron density (n _e) retrieval techniques and powerful processing systems have made a significant progress in recent years. In this study, the researches of n _e profiling from space-based GPS RO observations are briefly reviewed. Applying to the Formosat-3/Constellation Observing System for Meteorology, Ionosphere and Climate (FS3/COSMIC) data, we also present a compensatory Abel inversion technique including the effects of large-scale horizontal gradients and/or inhomogeneous ionospheric n _e obtained from an improved near real-time phenomenological model of the TaiWan Ionospheric Model. The results were evaluated by the ionosonde foF2 and foE data and showed improvements of rms foF2 difference from 29.2 to 16.5% in relative percentage and rms foE difference from 54.2 to 32.7% over the standard Abel inversion.     

Determination of complexity factor and its relationship with accuracy of representation for DEM terrain

Based on the estimating rule of the normal vector angles between two adjacent terrain units, we use the concept of terrain complexity factor to quantify the terrain complexity of DEM, and then the formula of terrain complexity factor in Raster DEM and TIN DEM is deduced theoretically. In order to make clear how the terrain complexity factor E _CF and the average elevation h affect the accuracy of DEM terrain representation RMSE _Et , the formula of Gauss synthetical surface is applied to simulate several real terrain surfaces, each of which has different terrain complexity. Through the statistical analysis of linear regression in simulation data, the linear equation between accuracy of DEM terrain representation RMSE _Et , terrain complexity factor E _CF and the average elevation h is achieved. A new method is provided to estimate the accuracy of DEM terrain representation RMSE _Et with a certain terrain complexity and it gives convincing theoretical evidence for DEM production and the corresponding error research in the future.     

Digitizing the thermal and hydrological parameters of land surface in subtropical China using AMSR-E brightness temperatures

ABSTRACT

Digitizing the land surface temperature (T_s) and surface soil moisture (m_v) is essential for developing the intelligent Digital Earth. Here, we developed a two parameter physical-based passive microwave remote sensing model for jointly retrieving T_s and m_v using the dual-polarized T_b of Aqua satellite advanced microwave scanning radiometer (AMSR-E) C-band (6.9 GHz) based on the simplified radiative transfer equation. Validation using in situ T_s and m_v in southern China showed the average root mean square errors (RMSE) of T_s and m_v retrievals reach 2.42 K (R² = 0.61, n = 351) and 0.025 g cm^?3 (R² = 0.68, n = 663), respectively. The results were also validated using global in situ T_s (n = 2362) and m_v (n = 1657) of International Soil Moisture Network. The corresponding RMSE are 3.44 k (R² = 0.86) and 0.039 g cm^?3 (R² = 0.83), respectively. The monthly variations of model-derived Ts and mv are highly consistent with those of the Moderate Resolution Imaging Spectroradiometer T_s (R² = 0.57; RMSE = 2.91 k) and ECV_SM m_v (R² = 0.51; RMSE = 0.045 g cm^?3), respectively. Overall, this paper indicates an effective way to jointly modeling T_s and m_v using passive microwave remote sensing.     

A new method for computing the ellipsoidal correction for Stokes's formula

 This paper generalizes the Stokes formula from the spherical boundary surface to the ellipsoidal boundary surface. The resulting solution (ellipsoidal geoidal height), consisting of two parts, i.e. the spherical geoidal height N ₀ evaluated from Stokes's formula and the ellipsoidal correction N ₁, makes the relative geoidal height error decrease from O(e ²) to O(e ⁴), which can be neglected for most practical purposes. The ellipsoidal correction N ₁ is expressed as a sum of an integral about the spherical geoidal height N ₀ and a simple analytical function of N ₀ and the first three geopotential coefficients. The kernel function in the integral has the same degree of singularity at the origin as the original Stokes function. A brief comparison among this and other solutions shows that this solution is more effective than the solutions of Molodensky et al. and Moritz and, when the evaluation of the ellipsoidal correction N ₁ is done in an area where the spherical geoidal height N ₀ has already been evaluated, it is also more effective than the solution of Martinec and Grafarend. Received: 27 January 1999 / Accepted: 4 October 1999     

Accuracy of GPS-derived relative positions as a function of interstation distance and observing-session duration

 Ten days of GPS data from 1998 were processed to determine how the accuracy of a derived three-dimensional relative position vector between GPS antennas depends on the chord distance (denoted L) between these antennas and on the duration of the GPS observing session (denoted T). It was found that the dependence of accuracy on L is negligibly small when (a) using the `final' GPS satellite orbits disseminated by the International GPS Service, (b) fixing integer ambiguities, (c) estimating appropriate neutral-atmosphere-delay parameters, (d) 26 km ≤ L ≤ 300 km, and (e) 4 h ≤T ≤ 24 h. Under these same conditions, the standard error for the relative position in the north–south dimension (denoted S _n and expressed in mm) is adequately approximated by the equation S _n =k _n /T ^0.5 with k _n =9.5 ± 2.1 mm · h^0.5 and T expressed in hours. Similarly, the standard errors for the relative position in the east–west and in the up-down dimensions are adequately approximated by the equations S _e =k _e /T ^0.5 and S _u =k _u /T ^0.5, respectively, with k _e =9.9 ± 3.1 mm · h^0.5 and k _u =36.5 ± 9.1 mm · h^0.5. Received: 5 February 2001 / Accepted: 14 May 2001     

Satellite detection of increases in global land surface evapotranspiration during 1984–2007

Abstract

As a key component of digital earth, remotely sensed data provides the compelling evidence that the amount of water vapour transferred from the entire global surface to the atmosphere increased from 1984 to 2007. The validation results from the earlier evapotranspiration (ET) estimation algorithm based on net radiation (R _n ), Normalised Difference Vegetation Index (NDVI), air temperature and diurnal air temperature range (DTaR) showed good agreement between estimated monthly ET and ground-measured ET from 20 flux towers. Our analysis indicates that the estimated actual ET has increased on average over the entire global land surface except for Antarctica during 1984–2007. However, this increasing trend disappears after 2000 and the reason may be that the decline in net radiation and NDVI during this period depleted surface soil moisture. Moreover, the good correspondence between the precipitation trend and the change in ET in arid and semi-arid regions indicated that surface moisture linked to precipitation affects ET. The input parameters R _n , T _air, NDVI and DTaR show substantial spatio-temporal variability that is almost consistent with that of actual ET from 1984 to 2007 and contribute most significantly to the variation in actual ET.     

Harmonic analysis of total electron contents time series: methodology and results

In an attempt to model regular variations of the ionosphere, the least-squares harmonic estimation is applied to the time series of the total electron contents (TEC) provided by the JPL analysis center. Multivariate and modulated harmonic estimation spectra are introduced and estimated for the series to detect the regular and modulated dominant frequencies of the periodic patterns. Two significant periodic patterns are the diurnal and annual signals with periods of 24/n hours and 365.25/n days (n = 1, 2, …), which are the Fourier series decomposition of the regular daily and yearly periodic variations of the ionosphere. The spectrum shows a cluster of periods near 27 days, thereby indicating irregularities at this solar cycle period. A series of peaks, with periods close to the diurnal signal and its harmonics, are evident in the spectrum. In fact, the daily signal harmonics of ω _i  = 2πi are modulated with the annual signal harmonics of ω _j  = 2πj/365.25 as ω _ijM  = 2πi(1 ± j/365.25i). Among them, at low and midlatitudes, the largest variations belong to the diurnal signal modulated to the semiannual signal. Some preliminary results on the modulated part are presented. The maximum ranges of the modulated daily signal are ±15 TECU and ±6 TECU at high and low solar periods, respectively. A model consisting of purely harmonic functions plus modulated ones is capable of studying known regular anomalies of the ionosphere, which is currently in progress.     

Determination of the free core nutation period from tidal gravity observations of the GGP superconducting gravimeter network

This study is based on 25 long time-series of tidal gravity observations recorded with superconducting gravimeters at 20 stations belonging to the Global Geodynamic Project (GGP). We investigate the diurnal waves around the liquid core resonance, i.e., K ₁, ψ₁ and φ₁, to determine the free core nutation (FCN) period, and compare these experimental results with models of the Earth response to the tidal forces. For this purpose, it is necessary to compute corrected amplitude factors and phase differences by subtracting the ocean tide loading (OTL) effect. To determine this loading effect for each wave, it was thus necessary to interpolate the contribution of the smaller oceanic constituents from the four well determined diurnal waves, i.e., Q ₁, O ₁, P ₁, K ₁. It was done for 11 different ocean tide models: SCW80, CSR3.0, CSR4.0, FES95.2, FES99, FES02, TPXO2, ORI96, AG95, NAO99 and GOT00. The numerical results show that no model is decisively better than the others and that a mean tidal loading vector gives the most stable solution for a study of the liquid core resonance. We compared solutions based on the mean of the 11 ocean models to subsets of six models used in a previous study and five more recent ones. The calibration errors put a limit on the accuracy of our global results at the level of ± 0.1%, although the tidal factors of O ₁ and K ₁ are determined with an internal precision of close to 0.05%. The results for O ₁ more closely fit the DDW99 non-hydrostatic anelastic model than the elastic one. However, the observed tidal factors of K ₁ and ψ₁ correspond to a shift of the observed resonance with respect to this model. The MAT01 model better fits this resonance shape. From our tidal gravity data set, we computed the FCN eigenperiod. Our best estimation is 429.7 sidereal days (SD), with a 95% confidence interval of (427.3, 432.1).     

Application of geographical information system for mapping/monitoring seismological hazards in Pakistan

This study attempts to use the geographic information system (GIS) technique to map and understand the tectonics and crustal structures of Pakistan. Maps of surficial tectonic features and seismological parameters including Moho depth, P_n velocity and P_g velocity are complied. Based on the seismological data-set of the country the earthquake hazard map of Pakistan is also presented by applying regression technique on seismological, geological and topographical parameters. A case study of 8 October 2005 earthquake is used to validate the hazard map. It is envisaged that the developed GIS database would help policy-makers and scientists in natural hazard evaluation, seismic risk assessment and understanding of earthquake occurrences in Pakistan.     

Harmonic maps

Harmonic maps are generated as a certain class of optimal map projections. For instance, if the distortion energy over a meridian strip of the International Reference Ellipsoid is minimized, we are led to the Laplace–Beltrami vector-valued partial differential equation. Harmonic functions x(L,B), y(L,B) given as functions of ellipsoidal surface parameters of Gauss ellipsoidal longitude L and Gauss ellipsoidal latitude B, as well as x(,q), y(,q) given as functions of relative isometric longitude =L–L₀ and relative isometric latitude q=Q–Q₀ gauged to a vector-valued boundary condition of special symmetry are constructed. The easting and northing {x(b,),y(b,)} of the new harmonic map is then given. Distortion energy analysis of the new harmonic map is presented, as well as case studies for (1) B[–40°,+40°], L[–31°,+49°], B₀= ±30°, L₀=9° and (2) B[46°,56°], L{[4.5°, 7.5°]; [7.5°, 10.5°]; [10.5°,13.5°]; [13.5°,16.5°]}, B₀= 51°, L₀ {6°,9°,12°,15°}.     

Upward and downward continuations in the geopotential determination

Summary The geopotential on and outside the earth is represented as a series in surface harmonics. The principal terms in it correspond to the solid harmonics of the external potential expansion with the coefficients being Stokes’ constantsC _nm andS _nm . The additional terms which occur near the earth’s surface due to its non-sphericity and topography are expressed in terms of Stokes’ constants too. This allows performing downward continuation of the potential derived from satellite observations. In the boundary condition which correlates Stokes’ constants and the surface gravity anomalies there occur additional terms due to the earth’s non-sphericity and topography. They are expressed in terms of Stokes’ constants as well. This improved boundary condition can be used for upward and downward continuations of the gravity field. Simple expressions are found representingC _nm andS _nm as explicit functions of the surface anomalies and its derivatives. The formula for the disturbing potential on the surface is derived in terms of the surface anomalies. All the formulas do not involve the earth’s surface in clinations.