Digital Height Model Research Articles

Modeling bird communities using unclassified remote sensing imagery: Effects of the spatial resolution and data period

In this paper we assess the capacity of satellite images to explain and predict bird community patterns in farm-wood landscapes in southwestern France. Our goal is to examine the effect of the images’ acquisition date and spatial resolution on the models’ performance. We also seek to assess whether unclassified images provide results comparable with classified data (i.e. land-cover map). To do that we constructed species richness models (generalized additive models) based on a sample of 573 counting points and on non-classified images made up of NDVI data and digital height model (DHM), making it possible to quantify the spatial and vertical heterogeneity of habitats. To assess the acquisition date effect, we compared the performance of NDVI data acquired on four different dates (February 4th, June 24th, August 19th and October 18th, 2009) by the same sensor (SPOT-5). To assess the spatial resolution effect, we compared five NDVI images acquired over an identical period (September 2010) but by different sensors (WorldView-2, SPOT-5, SPOT-4, Landsat, MODIS) as well as two DHMs obtained from LiDAR (1m) and radar (5m) data. Our results show that for a constant spatial resolution (10m) it is the NDVI data acquired at the beginning of the autumn that provide the best performance. These data better reveal the landscape requirements of birds during the breeding period. For a given period (September 2010), the higher resolution spatial data (2m) are the highest performing. However, in view of the cost of WorldView images, we suggest that 10m data (SPOT-5) provide a good trade-off for studying the distribution of bird communities. For the height data (DHM), the effect of the spatial resolution is not significant. The differences of performance between the spatial resolutions of NDVI data are not as great as those between the data acquisition periods. The performance of unclassified data (NDVI or DHM) is also comparable with that of land-cover maps. This suggests on the one hand that the choice of the NDVI image date is more important than that of the spatial resolution and on the other hand that the NDVI or DHM data are good alternatives to classified data when constructing a bird-habitat predictive model.

Estimation of Above-Ground Forest Biomass Using Landsat ETM+, Aster GDEM and Lidar

The measurement of above-ground biomass is important to understand carbon flow between trees and the atmosphere; remote sensing plays an important role in making this possible for extensive and hard-to-reach areas. This study compared above-ground forest biomass estimation models using data from different sources, including Landsat ETM+, Aster GDEM, ALS (LiDAR) and forest inventories. Two sets of predictors were established: the first included variables extracted from Landsat ETM+ and Aster GDEM, while the second included variables from Landsat in combination with LiDAR products (Digital Terrain Model, Digital Surface Model and Canopy Height Model). The Random Forest algorithm was used to build all models; this method explicitly returns the importance of each predictor and therefore allows the selection of the best set of variables. Estimations were made separately by forest cover for Pinus radiata, Eucalyptus globulus and second-growth Nothofagus glauca. Better results were obtained using the combination Landsat-LiDAR than those using Landsat-Aster GDEM data. Also, the results were better when applying the model to pine cover (pseudo R2 77.22%).

Behaviour of earth’s crust due to topographic loads derived by inverse and direct isostasy

The behaviour of the earth’s crust due to topographic loads can be derived by either inverse or direct approach. As for the inverse approach, it is postulated that the density anomaly is proportional to the earth’s radius vector so that it is linearly related to topography by a convolution of the topography and an isotropic kernel function. Accordingly, one can prove that the attraction of the compensating masses is also a convolution of the topography and an isotropic isostatic response function. Such an isostatic response function can be determined by deconvolution. This paper gives the derivation of such a deconvolution by means of spherical harmonics. A practical determination of the isotropic isostatic response function needs the harmonic analysis of both the topography and the attraction of the compensating masses. Applying the principle of inverse isostasy, by which we aim to achieve zero isostatic anomalies, then the attraction of the compensating masses equals the Bouguer anomalies with an opposite sign. The harmonic analysis of the Bouguer anomalies is thus a combination of the harmonic analysis of the topographic potential and the already existing global reference models. As for the direct approach, consider that the earth’s crust is an infinite thin plate subject to topographic loads. The solution of such a bent plate represents the displacement of the earth’s crust due to topographic loads. The paper illustrates that the exact solution of the bent plate is given by the Kelvin function keix. A practical application has been carried out for both approaches using EGM96 and GPM98CR geopotential earth models as well as TUG87 and TBASE digital height models. The results show that the estimated isotropic isostatic response functions derived by the inverse approach behave similarly as that given by direct approach represented by the Kelvin function keix.

The effect of DHM resolution in computing the topographic-isostatic harmonic coefficients within the window technique

The window technique was suggested earlier to get rid of the double consideration of the topographic-isostatic masses within the data window in the framework of the remove-restore technique. Within the course of the window technique, one needs to compute the harmonic coefficients of the topographic-isostatic masses for the data window. The paper studies the effect of using Digital Height Models (DHMs) with different resolutions of the computed harmonic coefficients of the topographic-isostatic masses for the data window. Two different test areas, one in Austria and one in Egypt, are considered in this investigation. A set of DHMs with different resolutions is available for both test areas. The harmonic coefficients of the topographic-isostatic masses for the data window are computed for both test areas using the available DHMs with different resolutions. A comparison among the potential degree variances of the different DHMs is carried out. The computation of the window topographic-isostatic gravity anomalies for both data sets is performed using the set of the available DHMs with different resolutions. The results show that using a DHM with the grid size of about 5 km for smooth topography and of about 3 km for rough topography gives practically the same topographic-isostatic gravity anomalies for the data window in a significantly much less CPU time compared to that of using the finest DHM.

DEM Generation from High Resolution Satellite Imagery

Digital height models (DHM) covering larger areas can be generated by means of optical or synthetic aperture radar (SAR) images taken from space. An overview of the sensors and the characteristics of generated height models is given. With very high resolution optical satellite stereo pairs a system accuracy of 1.0 ground sampling distance (GSD) standard deviation can be reached. Of course this is not the accuracy of a DEM, which is also influenced by interpolation and includes areas with limited contrast and vegetation, leading to lower quality. In addition, the difference between digital surface models (DSMs), describing the visible surface, and digital terrain models (DTMs), describing the bare ground, has to be respected. The same problem exists for SAR images used by either interferometric SAR (InSAR) or by radargrammetry if no InSAR configuration is available. It has to be ascertained whether existing regional or nearly worldwide DEMs can be used instead of especially produced elevation models. The SRTM DEM and ASTER GDEM are both available, free of charge via the Internet, but their resolution and accuracy are limited. Higher resolution DEMs, such as the SPOT DEM (also named Reference 3D) or NextMap, are not free of charge and they do not cover the whole earth. This will also be the case for the TanDEM-X height model, which will be available in 2014. If more detailed DEMs are required, they can be determined by automatic image matching of very high resolution satellite imagery.

Detection of fallen trees in forested areas using small footprint airborne laser scanning data

Deadwood was identified as an important indicator for habitat condition and biodiversity in forests. The assessment of downed trees is therefore part of sustainable forest management and ecological monitoring. However, manual quantification of deadwood in forests is challenging, time consuming, and considered cost-inefficient. Full-waveform airborne laser scanning (FWF-ALS) can be used to support the assessment process. The amplitude and width of the backscattered pulses contain information on the properties of the surface. We used these observations for the identification of downed trees in a Natura2000 forest site. A high density FWF-ALS data set was acquired under leaf-off conditions. Echo width and type (i.e., first, intermediate, and last) information as well as normalized echo heights were used to filter the point cloud and derive a digital height model (DHM). This DHM depicts downed stems as line-like features. Image processing was applied to derive and refine regions representing fallen trees. Terrestrial reference data consisting of locations and dimensions of downed trees, as well as state of decay were used for evaluation. Direct identification of downed trees in FWF-ALS point clouds was possible (completeness 75%, correctness 90%), but it was influenced by factors such as dimension, state of decay, vegetation density, and penetration of the laser.

Algorithm for improving the co-registration of LiDAR-derived digital canopy height models and field data

Improving trees location under LiDAR-derived digital canopy height models (DCMs) is of great interest as discrepancies between both dataset influence the accuracy of the estimations of forest attributes. A method is proposed for the co-registration of LiDAR-derived DCMs with local field positional measurements under a dense tree canopy. This approach consists of two main stages: (1) the assessment of the match between the LiDAR-derived digital terrain model and topographic surveying measurements when shifting the coordinates around a measured position; and (2) a comparison between the field height of selected trees and the LiDAR-derived DCM. Satisfactory results were obtained from geo-referencing field data and LiDAR models for characterizing the forest structure in heterogeneous Pinus sylvestris stands. Closure error of topographic surveying was 17.7 cm, and GPS accuracy to 95 % probability was below 10 cm, thus considerably lower than the resolution of the LiDAR models (1 m-pixel). The best co-location for field trees and LiDAR models provided a coefficient of determination of 0.56 between field-measured tree heights and LiDAR-derived DCM values.

High‐resolution remote sensing data improves models of species richness

AbstractQuestionCan predictors derived from air‐ and space‐borne high‐resolution remote sensing data improve models of species richness commonly built using coarser‐scaled environmental variables?LocationSwitzerland, covering 41 244 km2 of Central Europe.MethodsWe applied linear regressions to model species richness of woody species, herbs, edaphic bryophytes and epiphytic lichens in Swiss forests. We included high‐resolution predictors derived from digital height models and from satellite spectral images. Coarser‐scaled predictors characterizing climatic and topographic conditions were also included, as were soil properties and geology. We applied hierarchical partitioning to regression models to investigate the independent contribution of each predictor set to species richness models.ResultsPredictors derived from high‐resolution remote sensing data substantially improved the species richness models (increase 14–55% of R2). However, coarse‐scaled climatic and topographic predictors still explained a high proportion of the variance in the species richness data in all models, independently of other predictors commonly used. The importance of the remotely sensed variables was strongly dependent on the biogeographic region considered. The species richness models of smaller organisms of the forest floor (herbs and edaphic bryophytes) benefited greatly from adding high‐resolution topographic predictors, indicating the importance of microtopographic heterogeneity for these groups. Both epiphytic lichens and herbs responded strongly to indicators of structural properties of the forest stand.ConclusionsHigh‐resolution remote sensing data is a proxy for micro‐environmental structures and variation in these structures. Our results show that predictors derived from such data can improve species richness models considerably, especially in regions with low climatic and/or topographic variation. High‐resolution remote sensing variables excellently complement coarser‐scaled predictors, as they are available over large areas at low cost.

Estimation of Forest Carbon Storage Based on Airborne LiDAR Data

The objective of this study was to estimate the carbon storage of forest areas by using airborne light detection and ranging (LiDAR) data. Digital canopy height model (CHM) which generated from point cloud data was combined with accurate geo-referenced true color orthophoto image to produce parameter information of trees (height, canopy diameter, DBH data) in test area by marker controlled watershed segmentation algorithm. The total carbon storage of experimental site could be calculated by adopting binary tree volume table and semi-empirical inversion means. This study suggested that the carbon storage of forest areas can be effectively estimated by using LiDAR data, which will play an important role in forest data updating and process of forget sustainable development.

Flattening Gamma: Radiometric Terrain Correction for SAR Imagery

Enabling intercomparison of synthetic aperture radar (SAR) imagery acquired from different sensors or acquisition modes requires accurate modeling of not only the geometry of each scene, but also of systematic influences on the radiometry of individual scenes. Terrain variations affect not only the position of a given point on the Earth's surface but also the brightness of the radar return as expressed in radar geometry. Without treatment, the hill-slope modulations of the radiometry threaten to overwhelm weaker thematic land cover induced backscatter differences, and comparison of backscatter from multiple satellites, modes, or tracks loses meaning. The ASAR & PALSAR sensors provide state vectors and timing with higher absolute accuracy than was previously available, allowing them to directly support accurate tie-point-free geolocation and radiometric normalization of their imagery. Given accurate knowledge of the acquisition geometry of a SAR image together with a digital height model (DHM) of the area imaged, radiometric image simulation is applied to estimate the local illuminated area for each point in the image. Ellipsoid-based or sigma naught (σ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sup> ) based incident angle approximations that fail to reproduce the effect of topographic variation in their sensor model are contrasted with a new method that integrates terrain variations with the concept of gamma naught (γ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sup> ) backscatter, converting directly from beta naught (β <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sup> ) to a newly introduced terrain-flattened γ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sup> normalization convention. The interpretability of imagery treated in this manner is improved in comparison to processing based on conventional ellipsoid or local incident angle based σ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sup> normalization.

Towards an urban roughness parameterisation using interferometric SAR data taking the Metropolitan Region of Hamburg as an example

Increased surface roughness and the enhanced drag effect of the urban surface are important alterations in urban boundary layer dynamics. An enhanced roughness mapping would be beneficial for several modelling applications since state of the art roughness parameters derived from land use data do not represent the heterogeneity of urban surfaces at higher resolutions. Today most morphometric methods of urban roughness estimation are based on empirically derived functions of obstacle geometry parameters, which have conceptual and practical shortcomings. In this study we propose a new approach based on the topology, statistics and texture of overall roughness elements as represented by a Digital Height Model generated from side-looking Interferometric Synthetic Aperture Radar. These state of the art remotely sensed high resolution data are readily available for Western Europe, the US, as well as parts of Asia and the Caribbean. A set of new angular dependent urban morphology parameters is developed and tested for suitability for the mapping of roughness characteristics by way of an improved methodology. The results are very convincing and consistent over a wide range of surface characteristics.

AUTOMATED CITRUS TREE COUNTING FROM OBLIQUE OR ORTHO IMAGES AND TREE HEIGHT ESTIMATION FROM OBLIQUE IMAGES

The objectives of this research paper were to 1) compare tree counts between oblique and ortho images and 2) determine the accuracy of tree height measurements from oblique images. Accurate determination of tree count and canopy volume can aid growers in planning some orchard operations and estimating yield. The analytical capability of Feature Analyst (FA) to count trees enables the user to quantify the trees in a given area. Eight sample sets of ortho and oblique images were selected from Pictometry aerial imagery software (Pictometry International Corp, 100 Town Center Drive, Suite A, Rochester, NY, 14623) by entering the coordinates of the required area of interest. The images thus obtained were exported to Arc Map in ortho and oblique views. The same training samples randomly selected from the study area in the ortho view were also used for the oblique image in FA to validate/determine the number of trees. The image was represented by a shape file, and an attribute query denoted the quantity of trees. After processing with FA, the majority of the machine counts were below the actual value. Three of the eight ortho images represented over counts, while five represented under count with a total range of -16 to +23%, with a mean error of 8.67%. All the eight oblique images represented under counts with an error range of 0.26 to 27.82% with a mean error of 9.94%. Thus, Oblique images had an error range of 27.56%, while orthogonal images have an error range of 39%, but overall the average error was less for ortho images. An estimate was made of tree height using oblique images in Pictometry. The errors were partially due to the ambiguity in selecting each tree’s base and apex. Assuming there may be 0.15 m error due to resolution in ground area covered and tree height centering contributed an 11% error, overall precision in tree canopy volume measurement was still 85%. INTRODUCTION Feature Analyst (FA) has the ability to count trees in a given acreage. Saraswat et al. (2007) used FA to count the trees in citrus groves in Polk County, Florida. They digitized a few trees in the study area and used these samples to determine the total number of trees in the study area (Saraswat et al., 2008). Attempts were made by previous researchers to determine tree height using Lidar data and digital roof models, but heights could not be determined to 100% accuracy. Imai et al. created a digital height model (DHM) by subtracting Digital Elevation model from a Digital Surface model. Hashiba et al. (2003) generated a digital roof model (DRM) using a typical stereogram concept and basic trigonometric calculations were executed to calculate the tree height. Pictometry is aerial imaging software that has the capabilities to measure the tree height and ground area covered from ortho and oblique images. The aerial images were collected by running flight line mosaics across the study area. These two factors are required to calculate the tree canopy volume. The combination of these factors to determine canopy volume per block then enables estimating the yield with weather factors that determine flowering intensity (Valiente and Albrigo, 2004; Albrigo et al., 2002). Proc. IS on Appl. of Precision Agric. for Fruits and Vegetables Eds.: L.G. Albrigo and R. Ehsani Acta Hort. 824, ISHS 2009 92 MATERIAL The Citrus Research and Education Center groves in Florida were chosen as the study area. The images were randomly selected from four blocks. Eight other sites were identified and ortho and oblique images were obtained for these sites. These were 0.15 m resolution images provided by the Pictometry Visual Intelligence Company. Images were collected by running flight line mosaics across the study area. Each image had three bands (RGB) stacked as a true color composite. Five additional sites were identified and ortho images were extracted from the Pictometry image source. The same site areas (0.3 m resolution) were extracted from a Polk county tax assessor source. All the trees in the study area were citrus.

A comparison of different mass elements for use in gravity gradiometry

Topographic and isostatic mass anomalies affect the external gravity field of the Earth. Therefore, these effects also exist in the gravity gradients observed, e.g., by the satellite gravity gradiometry mission GOCE (Gravity and Steady-State Ocean Circulation Experiment). The downward continuation of the gravitational signals is rather difficult because of the high-frequency behaviour of the combined topographic and isostatic effects. Thus, it is preferable to smooth the gravity field by some topographic-isostatic reduction. In this paper the focus is on the modelling of masses in the space domain, which can be subdivided into different mass elements and evaluated with analytical, semi-analytical and numerical methods. Five alternative mass elements are reviewed and discussed: the tesseroid, the point mass, the prism, the mass layer and the mass line. The formulae for the potential, the attraction components and the Marussi tensor of second-order potential derivatives are provided. The formulae for different mass elements and computation methods are checked by assuming a synthetic topography of constant height over a spherical cap and the position of the computation point on the polar axis. For this special situation an exact analytical solution for the tesseroid exists and a comparison between the analytical solution of a spherical cap and the modelling of different mass elements is possible. A comparison of the computation times shows that modelling by tesseroids with different methods produces the most accurate results in an acceptable computation time. As a numerical example, the Marussi tensor of the topographic effect is computed globally using tesseroids calculated by Gauss–Legendre cubature (3D) on the basis of a digital height model. The order of magnitude in the radial-radial component is about ± 8 E.U.

Production of DTM by using the existing contour data lines

The topographic database in Nepal has been prepared in the national level by making use of the analogue topographic base data. Topographic dynamics in terms of other physical features changes rapidly but changes in the terrain elevation occurs negligibly. As there has been growing demand for digital height model (DHM) of the terrain in spite of the presence of the existing two and half dimensional data such as contours, work was done as to the input, methodology, and quality assessment of the DTM being generated on the national level. Some initiations taken in this respect has been highlighted in this paper.

航空機レーザ計測による広葉樹林の樹高，樹冠面積の抽出

We discuss the possibility of surveying the height and crown area of single tree in broadleaved forest by application of airborne laser scanner system. The process of this study includes : interpose the ground data for preparing accurate DTM (Digital Terrain Model) and DHM (Digital Height Model), extracting top of crown using method of segmentation-based in DHM, create crown area by means of segmentation on defined cross-section which the height can be designated. In result of our study, all of bigger trees, which compose upper crown-layer, had been extracted and the crown area on llm above ground had been computed.

Topography data harmonisation and uncertainties applying SRTM, laser scanner and cartographic elevation models

Abstract. Only a few studies have attempted to quantify topography-depending water fluxes, to evaluate retention and reservoir capacities and surface run-off paths within large river basins because data availability and data quality are critical issues to face this objective. It becomes most relevant if water balance has to be calculated in large or transboundary river basins. The advance of space based earth observation data offers a solution to this information problem. Therefore, this paper mainly focuses on weaknesses and strengths analyzing topography with SRTM (Shuttle Radar Topography Mission) digital height data and thus provides techniques for their improved application in river network derivation, floodplain analysis, watershed hydrology in large as well as in large river basins (&gt;1000 km2). In the analysis different types of digital elevation models (DEM), terrain models (DTM) and land cover classification data (biotope map, Corine Land Cover 1994) have been used. The DHMs are generated from Airborne Laser Scanning (0.5 m), topographic maps (10.0/50.0 m) and SRTM at 30.0 m and 90.0 m spatial resolution. SRTM digital height models are generated by Synthetic Aperture Radar (SAR) and show a high spatial variance in urban areas, regions of dense vegetation canopy, floodplains and water bodies. As study area serve the Elbe basin (Czech Republic, Germany) with its sub-basins and the Saale river basin (Germany, different federal countries Saxony-Anhalt, Saxony and Thuringia).

Analysis of ICESat data using Kalman filter and kriging to study height changes in East Antarctica

We analyze ICESat derived heights collected between Feb.03–Nov.04 using a kriging/Kalman filtering approach to investigate height changes in East Antarctica. The model's parameters are height change to an a priori static digital height model, seasonal signal expressed as an amplitude B and phase θ, and height‐change rate dh/dt for each (100 km)2 block. From the Kalman filter results, dh/dt has a mean of −0.06 m/yr in the flat interior of East Antarctica. Spatially correlated pointing errors in the current data releases give uncertainties in the range 0.06 m/yr, making height change detection unreliable at this time. Our test shows that when using all available data with pointing knowledge equivalent to that of Laser 2a, height change detection with an accuracy level 0.02 m/yr can be achieved over flat terrains in East Antarctica.

Isostatic response of the earth's crust derived by inverse isostasy

Postulate that the density anomaly is proportional to the earth's radius vector so that it is linearly related to the topography by a convolution of the topography and an isotropic kernel function. Accordingly, one can prove that the attraction of the compensating masses is also a convolution of the topography and an isotropic isostatic response function. Such an isostatic response function can be determined by deconvolution. The paper gives the derivation of such a deconvolution by means of spherical harmonics. A practical determination of the isotropic isostatic response function needs the harmonic analysis of both the topography and the attraction of the compensating masses. Applying the principle of inverse isostasy, by which we aim to achieve zero isostatic anomalies, then the attraction of the compensating masses equals the Bouguer anomalies with an opposite sign. The harmonic analysis of the Bouguer anomalies is thus a combination of the harmonic analysis of the topographic potential and the already existed global reference models. A practical application has been carried out using EGM96 and GPM98CR geopotential earth models as well as TUG87 and TBASE digital height models. The results show that the estimated isotropic isostatic response functions behave similarly as that given by the exact bending curve of the earth's crust represented by the Kelvin function kei x. This shows that the isostatic response of the earth's crust computed by direct isostasy using the most realistic isostatic model (modified Vening Meinesz isostatic model with Kelvin function kei x as the exact bending curve) matches the isostatic response of the earth's crust derived by inverse isostasy.

Further development of the urban bioclimate model UBIKLIM, taking local wind systems into account

The urban bioclimate model UBIKLIM is an application model providing an assessment on the influence of urban climate on human beings. On a calm summer's day with fair weather conditions it calculates the meteorological parameters air temperature, humidity, wind velocity and short and long wave radiation based on a digital height model and appropriate land use information. These meteorological parameters determine the heat exchange conditions of human beings and thus can be analyzed with the help of the Klima-Michel model into physiologically significant terms. The result is a bioclimate map on a scale that is appropriate for climate-related planning in an urban environment. In urban areas lying in a distinct relief, a superimposed wind regime, such as a down valley wind, can modify the bioclimate considerably. Up to now it was not possible for UBIKLIM to take this situation into account. In this point UBIKLIM has been developed further, but its easily applicable character has been preserved. This has been realized, firstly by combining an orographically induced wind field with the one calculated in UBIKLIM as a function of land use esp. of urban structure, and secondly by introducing an algorithm by working out conveyance processes in the direction of the superposed wind field. UBIKLIM fills the gap between meso and micro scale models. Results of the further developed model are shown in the example of the city of Freiburg im Breisgau which is influenced by the distinct relief at the southern edge of the Black Poorest.

Detailed gravimetric geoid for the Egyptian south-western desert

Different gravimetric geoid solutions were carried out for the south-western Egyptian desert. A set of 2682 measured gravity stations were available for this investigation. The discrete gravity data are reduced to the geoid through the free-air reduction. Also terrain reduction is performed using a set of fine 30″ × 30″ and coarse 3′ × 3′ Digital Height Models for the test area. Different earth's geopotential models were used for the removing of the reference field. The geoid computations were performed in the space domain using the discrete Stokes integration technique and in the spectral domain using the spherical 2-D FFT technique. For both techniques, the reduced gravity anomalies are gridded on a 3′ × 3′ geographical grid using the least-squares interpolation technique with local covariance functions. A broad comparison among the different geoid solutions is made. The effect of the inner zone densification is in the order of about 2 cm. The effect of the spherical 2-D FFT approximation is mostly less than 1 cm.