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.     

Assessment of crop growth parameters of wheat under stress condition through ground based spectral data

A field experiment was conducted on wheat crop during rabi seasons of 1995–96, 1996–97 and 1997–98 to study the spectral response of wheat crop (between 490 to 1080 nm) under water and nutrient stress condition. An indigenously developed ground truth radiometer having narrow band in visible and near infrared region (490 – 1080 nm) was used. Vegetation indices derived using different band combinations and related to crop growth parameters. The near infrared spectral region of 710 – 1025 nm was found most important for monitoring stress condition. Relationship has been developed between crop growth parameters and vegetation indices. Leaf Area Index (LAI) and chlorophyll could be predicted by knowing different reflectance ratios at milking stage of crop with R² value of 0.78 and 0.89, respectively. Dry biomass (DBM), Plant Water Content (PWC) and grain yield are also significantly related with reflectance ratios at flowering stage of crop with R² value of 0.90, 0.98 and 0.74, respectively.     

Spectral models for estimation of chlorophyll content, growth and yield of wheat crop

Field experiments were conducted during 1998–99 and 1999–2000 at research farm of the Department of Agricultural Meteorology, CCS Haryana Agricultural University, Hisar. Five wheat cultivars: WH 542, PBW 343, UP 2338, Raj 3765 and Sonak were sown on 25^th November, 10^th and 25^th December with four nitrogen levels viz., no nitrogen. 50, 100 and 150% of recommended dose. Leaf area index, dry matter at anthesis, final dry biomass and grain yield were recorded in all the treatments. Chlorophyll and wax contents of wheat leaves were estimated at different growth stages. Multiband spectral reflectance was measured using hand-held radiometer. Spectral indices such as simple ratio, normalized difference, transformed vegetation index, perpendicular vegetation index and greenness index were computed using the multiband spectral data. Values of all the spectral indices were maximum in 25 November sown crop with maximum dose of nitrogen (180 kg N ha^-1). PBW 343 showed higher values of all the spectral indices in comparison with other cultivars. The spectral indices recorded during maximum leaf area index stage were correlated with crop parameters. Using stepwise regression, empirical models for chlorophyll, leaf area index, dry biomass and yield prediction were developed. The ’R²’ values of these models ranged between 0.87 and 0.95.     

Cotton yield prediction through spectral parameters

To predict the crop yield from spectral parameters, a field experiment was conducted on cotton crop during 1997-98 Kharif season on a sandy loam soil at the Punjab Agricultural Unjversity, Ludhiana. India. Spectral reflectance and agronomic measurements were made for cotton species (American and Desi cotton), sown on two dates (May 1 and May 29) under five nitrogen levels (0, 40, 80, 120 and 160 kg/ha). Regression analysis showed that growth variables had poor correlation with seed cotton yield for all three models, however, yield attributes were significantly and highly correlated for second degree model with seed cotton yield. The integrated Radiance Ratio (RR) and Normalized Difference Vegetation Index (NDVI) measured over time were significantly correlated quadratically with seed cotton yield on three time segment periods viz., 81–110, 111–140 and 141–200 DAS, but highest correlation values were obtained during 81–110 DAS, In American cotton, the highest correlation coefficient for RR and NDVI were 0.91 and 0.81, respectively; whereas for Desi cotton these values were 0.88 and 0.84, respectively.     

The geopotential value <Emphasis Type="Italic">W</Emphasis><Subscript>0</Subscript> for specifying the relativistic atomic time scale and a global vertical reference system

The TOPEX/Poseidon (T/P) satellite alti- meter mission marked a new era in determining the geopotential constant W ₀. On the basis of T/P data during 1993–2003 (cycles 11–414), long-term variations in W ₀ have been investigated. The rounded value W ₀ = 62636856.0 ± 0.5) m ² s ⁻² has already been adopted by the International Astronomical Union for the definition of the constant L _G = W ₀/c ² = 6.969290134 × 10⁻¹⁰ (where c is the speed of light), which is required for the realization of the relativistic atomic time scale. The constant L _G , based on the above value of W ₀, is also included in the 2003 International Earth Rotation and Reference Frames Service conventions. It has also been suggested that W ₀ is used to specify a global vertical reference system (GVRS). W ₀ ensures the consistency with the International Terrestrial Reference System, i.e. after adopting W ₀, along with the geocentric gravitational constant (GM), the Earth’s rotational velocity (ω) and the second zonal geopotential coefficient (J ₂) as primary constants (parameters), then the ellipsoidal parameters (a,α) can be computed and adopted as derived parameters. The scale of the International Terrestrial Reference Frame 2000 (ITRF2000) has also been specified with the use of W ₀ to be consistent with the geocentric coordinate time. As an example of using W ₀ for a GVRS realization, the geopotential difference between the adopted W ₀ and the geopotential at the Rimouski tide-gauge point, specifying the North American Vertical Datum 1988 (NAVD88), has been estimated.     

Statistics of GPS ionospheric scintillation and irregularities over polar regions at solar minimum

A statistical study of the occurrence characteristic of GPS ionospheric scintillation and irregularity in the polar latitude is presented. These measurements were made at Ny-Alesund, Svalbard [78.9°N, 11.9°E; 75.8°N corrected geomagnetic latitude (CGMLat)] and Larsemann Hills, East Antarctica (69.4°S, 76.4°E; 74.6°S CGMLat) during 2007–2008. It is found that the GPS phase scintillation and irregularity activity mainly takes place in the months 10, 11 and 12 at Ny-Alesund, and in the months 5, 6 at Larsemann Hills. The seasonal pattern of phase scintillation with respect to the station indicates that the GPS phase scintillation occurrence is a local winter phenomenon, which shows consistent results with past studies of 250 MHz satellite beacon measurements. The occurrence rates of GPS amplitude scintillation at the two stations are below 1%. A comparison with the interplanetary magnetic field (IMF) B _y and B _z components shows that the phase scintillation occurrence level is higher during the period from later afternoon to sunset (16–19 h) at Ny-Alesund, and from sunset to pre-midnight (18–23 h) at Larsemann Hills for negative IMF components. The findings seem to indicate that the dependence of scintillation and irregularity occurrence on geomagnetic activity appears to be associated with the magnetic local time (MLT).     

Studies on spectral reflectance under normal and nitrogen,phosphorus and pest and disease stress condition in soybean (<Emphasis Type="Italic">Glycine max</Emphasis> L.)

The results emerged out of the studies on spectral reflectance under normal and nitrogen and phosphorus stress condition in soybean (Glycine max L.) conducted at Marathwada Agricultural University experimental farm, Parbhani duringkharif 2004–05 showed that crop growth and bio-physiological parameters viz., Height, chlorophyll, leaf area index and total biomass influenced by pest and disease and nutrient stress resulted in detectable spectral reflectance variation. Poor crop growth, reduced canopy cover, chlorophyll content and biomass production are the effects observed in nutrient deficient crops. These above changes in soybean crop were related to spectral indices (RVI and NDVI) that are resulted in discrimination of stressed and normal (non-stressed) soybean crop.     

Can suspended sediment concentrations be estimated from multispectral imagery using only image-derived information?

Sampling for suspended sediment concentrations (SSC) in inland waters is traditionally based on collecting samples at sparse locations and in limited intervals. A number of investigators explored the utility of earth-observing satellites and air-borne sensors for monitoring of SSC over vast areas. Two approaches are commonly deployed: (1) empirical relationships between a chosen remotely sensed quantity and the actual in-situ SSC; and (2) bio-optical models founded on radiative transfer modeling. Unfortunately, in-situ measurements are often unavailable for direct image calibration, and inherent optical properties of optically active constituents (specific scattering and absorption coefficients) are usually unknown. This paper examines the possibility to retrieve SSC from multispectral satellite imagery without any in-situ data, i.e. using only image-derived information. The fundamental principle of image selfcalibration relies on the fact that in the visual domain of wavelengths (∼400–700 nm) the at-sensor reflectance becomes “saturated“ at high SSC, whereas the near-infrared domain (∼700–900 nm) remains almost perfectly linearly related to sediment concentrations. The core idea of the self-calibrating procedure is rather simple and is based on fitting an exponential function between reflectance and SSC, with SSC replaced by a linear relationship between SSC and reflectance in the near-infrared domain. As a first approximation of the non-linearity between reflectance and SSC levels in the 400–700 nm range, we used the equation proposed by Schiebe et al. (1992), although other equations, especially those arising from optical theory could be used as well. The technique is illustrated on a moderately sediment-laden reservoir and two scenes acquired from Landsat ETM+. The standard error of the estimated SSC was below 15 mg/L (i.e. ∼25 % relative error for the observed range of SSC). Although the proposed algorithm does not yield better results than other models mentioned in the literature, the primary advantage of the outlined methodology is that no in-situ measurements (water sampling nor spectral profiling) are needed — i.e. only image-derived information is used.     

Approximating the Bayesian estimate of the standard deviation in a linear model

The Bayesian estimates _b of the standard deviation σ in a linear model—as needed for the evaluation of reliability—is well known to be proportional to the square root of the Bayesian estimate (s ²) _b of the variance component σ² by a proportionality factor involving the ratio of Gamma functions. However, in analogy to the case of the respective unbiased estimates, the troublesome exact computation ofa _b may be avoided by a simple approximation which turns out to be good enough for most applications even if the degree of freedom ν is rather small. Paper presented to the Int. Conf. on “Practical Bayesian Statistics”, Cambridge (U.K.), 8.–11. July 1986.     

Utility of Hyperspectral Data for Potato Late Blight Disease Detection

The study was carried out to investigate the utility of hyperspectral reflectance data for potato late blight disease detection. The hyperspectral data was collected for potato crop at different level of disease infestation using hand-held spectroradiometer over the spectral range of 325–1075 nm. The data was averaged into 10-nm wide wavebands, resulting in 75 narrowbands. The reflectance curve was partitioned into five regions, viz. 400–500 nm, 520–590 nm, 620–680 nm, 770–860 nm and 920–1050 nm. The notable differences in healthy and diseased potato plants were noticed in 770–860 nm and 920–1050 nm range. Vegetation indices, namely NDVI, SR, SAVI and red edge were calculated using reflectance values. The differences between the vegetation indices for plants at different levels of disease infestation were found highly significant. The optimal hyperspectral wavebands to discriminate the healthy plants from disease infested plants were 540, 610, 620, 700, 710, 730, 780 and 1040 nm whereas upto 25% infestation could be discriminated using reflectance at 710, 720 and 750 nm.     

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     

A Critical Analysis of a Recent Test of the Lense–Thirring Effect with the LAGEOS Satellites

In this paper, we quantitatively discuss the impact of the current uncertainties in the even zonal harmonic coefficients J _l of the Newtonian part of the terrestrial gravitational potential on the measurement of the general relativistic Lense–Thirring effect. We use a suitable linear combination of the nodes Ω of the laser-ranged LAGEOS and LAGEOS-II satellites. The one-sigma systematic error due to mismodelling of the J _l coefficients ranges from ~ 4% for the EIGEN−GRACE02S gravity field model to ~ 9% for the GGM02S model. Another important source of systematic error of gravitational origin is represented by the secular variations j _l of the even zonal harmonics. While the relativistic and J _l signals are linear in time, the shift due to j _l is quadratic. We quantitatively assess their impact on the measurement of the Lense–Thirring effect with numerical simulations obtaining a 10−20% one-sigma total error over 11 years for EIGEN-GRACE02S. Ciufolini and Pavlis (Nature 431:958–960, 2004) claim a total error of 5% at the one-sigma level.     

Impact of Pastoral Practices on Forest Cover and Regeneration in the Outer Fringes of Kedarnath Wildlife Sanctuary,Western Himalaya

The study reports the impact of migratory livestock in the buffer area of Kedarnath Wildlife Sanctuary. The cumulative impacts of livestock grazing, fuel wood extraction and other anthropogenic pressures on forest cover in and around camping sites have been analyzed using satellite data and field observations. Multi-date satellite data were used to map the forest cover around 16 seasonal livestock camps. An annual loss of about 7.25 ha of forest covers during 1976–1990, 13.57 ha during 1990–1999 and 11.46 ha during 1999–2005 are recorded. A loss of 108.53 ha of forestland has been estimated during 1976–1990, about 122.16 ha during 1990–1999 and 68.78 ha during 1999–2005. It is also observed that many of the isolated camping sites are now connected due to loss of vegetation. Phytosociological analysis indicates that the number of seedlings and saplings in three species of Quercus (Q. leucotrichophora, Q. floribunda and Q. semecarpifolia) as compared to mature tress is very less around cattle camping sites. The ratio of number of sapling with seedling and trees indicates that the status of sapling is nil to very poor, an indication of loss of saplings due to either trampling or grazing.     

Intuitive derivation of loop inverses and array algebra

Array algebra forms the general base of fast transforms and multilinear algebra making rigorous solutions of a large number (millions) of parameters computationally feasible. Loop inverses are operators solving the problem of general matrix inverses. Their derivation starts from the inconsistent linear equations by a parameter exchangeX→L ₀, where X is a set of unknown observables,A ₀ forming a basis of the so called “problem space”. The resulting full rank design matrix of parameters L₀ and its ℓ-inverse reveal properties speeding the computational least squares solution expressed in observed values . The loop inverses are found by the back substitution expressing ∧X in terms ofL through . Ifp=rank (A) ≤n, this chain operator creates the pseudoinverseA ⁺. The idea of loop inverses and array algebra started in the late60's from the further specialized case,p=n=rank (A), where the loop inverse A ₀ ⁻¹ (AA ₀ ⁻¹ )^ℓ reduces into the ℓ-inverse A^ℓ=(A^TA)⁻¹A^T. The physical interpretation of the design matrixA A ₀ ⁻¹ as an interpolator, associated with the parametersL ₀, and the consideration of its multidimensional version has resulted in extended rules of matrix and tensor calculus and mathematical statistics called array algebra.     

Development and implementation of a multiscale biomass model using hyperspectral vegetation indices for winter wheat in the North China Plain

Crop monitoring during the growing season is important for regional management decisions and biomass prediction. The objectives of this study were to develop, improve and validate a scale independent biomass model. Field studies were conducted in Huimin County, Shandong Province of China, during the 2006–2007 growing season of winter wheat (Triticum aestivum L.). The field design had a multiscale set-up with four levels which differed in their management, such as nitrogen fertilizer inputs and cultivars, to create different biomass conditions: small experimental fields (L1), large experimental fields (L2), small farm fields (L3), and large farm fields (L4). L4, planted with different winter wheat varieties, was managed according to farmers’ practice while L1 through L3 represented controlled field experiments. Multitemporal spectral measurements were taken in the fields, and biomass was sampled for each spectral campaign. In addition, multitemporal Hyperion data were obtained in 2006 and 2007. L1 field data were used to develop biomass models based on the relation between the winter wheat spectra and biomass: several published vegetation indices, including NRI, REP, OSAVI, TCI, and NDVI, were investigated. A new hyperspectral vegetation index, which uses a four-band combination in the NIR and SWIR domains, named GnyLi, was developed. Following the multiscale concept, the data of higher levels (L2 through L4) were used stepwise to validate and improve the models of the lower levels, and to transfer the improved models to the next level. Lastly, the models were transferred and validated at the regional scale using Hyperion images of 2006 and 2007. The results showed that the GnyLi and NRI models, which were based on the NIR and SWIR domains, performed best with R² > 0.74. All the other indices explained less than 60% model variability. Using the Hyperion data for regionalization, GnyLi and NRI explained 81–89% of the biomass variability. These results highlighted that GnyLi and NRI can be used together with hyperspectral images for both plot and regional level biomass estimation. Nevertheless, additional studies and analyses are needed to test its replicability in other environmental conditions.     

Monitoring agricultural cropping patterns across the Laurentian Great Lakes Basin using MODIS-NDVI data

The Moderate Resolution Imaging Spectrometer (MODIS) Normalized Difference Vegetation Index (NDVI) 16-day composite data product (MOD12Q) was used to develop annual cropland and crop-specific map products (corn, soybeans, and wheat) for the Laurentian Great Lakes Basin (GLB). The crop area distributions and changes in crop rotations were characterized by comparing annual crop map products for 2005, 2006, and 2007. The total acreages for corn and soybeans were relatively balanced for calendar years 2005 (31,462 km² and 31,283 km², respectively) and 2006 (30,766 km² and 30,972 km², respectively). Conversely, corn acreage increased approximately 21% from 2006 to 2007, while soybean and wheat acreage decreased approximately 9% and 21%, respectively. Two-year crop rotational change analyses were conducted for the 2005–2006 and 2006–2007 time periods. The large increase in corn acreages for 2007 introduced crop rotation changes across the GLB. Compared to 2005–2006, crop rotation patterns for 2006–2007 resulted in increased corn–corn, soybean–corn, and wheat–corn rotations. The increased corn acreages could have potential negative impacts on nutrient loadings, pesticide exposures, and sediment-mediated habitat degradation. Increased in US corn acreages in 2007 were related to new biofuel mandates, while Canadian increases were attributed to higher world-wide corn prices. Additional study is needed to determine the potential impacts of increases in corn-based ethanol agricultural production on watershed ecosystems and receiving waters.     

Potato varietal discrimination using ground based multiband radiometer

The potential usefulness of spectral properties and vegetation indices in varietal discrimination of potato genotypes was studied in the field experiment. Spectral measurements were recorded in different bands in blue (450–520 nm), green (520–590 nm), red (620–680 nm) and infrared (770–860 nm) of the electromagnetic spectrum at different stages during crop growth period. A ground based hand held multiband radiometer (Model/041) was used for the purpose. The mean per cent green reflectance value among different genotypes was lowest in genotype MS/86-89, while it was observed highest in genotype JX-216. Significant difference among these genotypes was found at all growth stages except 6 week after planting. Consequent to variation in spectral reflectance the vegetation indices like, NDVI, RVI, TVI and DVI showed significant difference among genotypes at all growth stages except at 8^th week after planting. The vegetation indices are good indicators of crop growth and condition. Similarly, fresh weight, dry weight, and leaf area index were also highest in MS/86-89, followed by KUFRI Bahar and KUFRI Sutlej while in case of leaf area index it was followed by Kufri Sutlej and Kufri Bahar. JX-23 was highest in chlorophyll content and tuber yield followed by MS/86-89 and JW-160, while lowest chlorophyll content was seen in MS/89-1095 and poorest tuber yield in MS/89-60. Most of the genotypes exhibited considerable variation in their spectral response and vegetation indices thereby indicating the possibility of their discrimination through remote sensing technique.     

Yield Estimation Model and Water Productivity of Wheat Crop (Triticum aestivum) in an Irrigation Command Using Remote Sensing and GIS

Crop yield estimation has an important role on economy development and its accuracy and speed influence yield price and helps in deciding the excess or deficit production conditions. The water productivity evaluates the irrigation command through water use efficiency (WUE). Remote sensing (RS) and geographical information system (GIS) techniques were used for crop yield and water productivity estimation of wheat crop (Triticum aestivum) grown in Tarafeni South Main Canal (TSMC) irrigation command of West Bengal State in India. One IRS P6 image and four wide field sensor (WiFS) images for different months of winter season were used to determine the Normalized Difference Vegetation Index (NDVI) and Soil Adjusted Vegetation Index (SAVI) for area under wheat crop. The temporally and spatially distributed spectral growth profile and AREASUM of NDVI (A_NDVI) and SAVI (A_SAVI) with time after sowing of wheat crop were developed and correlated with actual crop yield of wheat (Y_act). The developed relationships between A_SAVI and Y_act resulted high correlation in comparison to that of A_NDVI. Using the developed model the RS based wheat yield (Y_RS) predicted from A_SAVI varied on entire TSMC irrigation command from 22.67 to 33.13 q ha⁻¹ respectively, which gave an average yield of 26.50 q ha⁻¹. The RS generated yield based water use efficiency (WUE_YRS) for water supplied from canal of TSMC irrigation command was found to be 6.69 kg ha⁻¹ mm⁻¹.     

Inversion of PROSAIL Model for Retrieval of Plant Biophysical Parameters

The current development of satellite technology particularly in the sensors like POLDER and MISR, has emphasized more on directional reflectance measurements (i.e. spectral reflectance of the target measured from different view zenith and azimuth angles) of the earth surface features mainly the vegetation for retrieval of biophysical parameters at regional scale using radiative transfer models. This approach being physical process based and uses directional reflectance measurement has been found to better and more reliable compared to the conventional statistical approach used till date and takes care of anisotropic nature (i.e. reflectance from the target is different if measured from different view angles) of the target. Keeping this in view a field experiment was conducted in mustard crop to evaluate the radiative transfer model for biophysical parameter retrieval through its inversion with the objectives set as (i) to relate canopy biophysical parameters and geometry to its bidirectional reflectance, (ii) to evaluate a canopy reflectance model to best represent the radiative transfer within the canopy for its inversion and (iii) to retrieve crop biophysical parameters through inversion of the model. Two varieties of the mustard crop (Brassica juncea L) were grown with two nitrogen treatments. The bidirectional reflectance data obtained at 5 nm interval for a range of 400–1100 nm were integrated to IRS LISS–II sensor’s four band values using Newton Cotes Integration technique. Biophysical parameters like leaf area index, leaf chlorophyll content, leaf length, plant height and average leaf inclination angle, biomass etc were estimated synchronizing with the bi-directional reflectance measurements. Radiative transfer model PROSAIL model was validated and its inversion was done to retrieve LAI and ALA. Look Up Table (LUT) of Bidirectional reflectance distribution function (BRDF) was prepared simulating through PROSAIL model varying only LAI (0.2 interval from 1.2 to 5.4 ) and ALA (5° interval from 40° to 55°) parameters and inversion was done using a merit function and numerical optimization technique given by Press et al. (1986). The derived LAI and ALA values from inversion were well matched with observed one with RMSE 0.521 and 5.57, respectively.     

Green's function solution to spherical gradiometric boundary-value problems

 Three independent gradiometric boundary-value problems (BVPs) with three types of gradiometric data, {Γ _rr }, {Γ _{r θ},Γ _{r λ}} and {Γ_θθ−Γ_λλ,Γ_θλ}, prescribed on a sphere are solved to determine the gravitational potential on and outside the sphere. The existence and uniqueness conditions on the solutions are formulated showing that the zero- and the first-degree spherical harmonics are to be removed from {Γ _{r θ},Γ _{r λ}} and {Γ_θθ−Γ_λλ,Γ_θλ}, respectively. The solutions to the gradiometric BVPs are presented in terms of Green's functions, which are expressed in both spectral and closed spatial forms. The logarithmic singularity of the Green's function at the point ψ=0 is investigated for the component Γ _rr . The other two Green's functions are finite at this point. Comparisons to the paper by van Gelderen and Rummel [Journal of Geodesy (2001) 75: 1–11] show that the presented solution refines the former solution. Received: 3 October 2001 / Accepted: 4 October 2002