Basic equations for constructing geopotential models from the gravitational potential derivatives of the first and second orders in the terrestrial reference frame

This research represents a continuation of the investigation carried out in the paper of Petrovskaya and Vershkov (J Geod 84(3):165–178, 2010) where conventional spherical harmonic series are constructed for arbitrary order derivatives of the Earth gravitational potential in the terrestrial reference frame. The problem of converting the potential derivatives of the first and second orders into geopotential models is studied. Two kinds of basic equations for solving this problem are derived. The equations of the first kind represent new non-singular non-orthogonal series for the geopotential derivatives, which are constructed by means of transforming the intermediate expressions for these derivatives from the above-mentioned paper. In contrast to the spherical harmonic expansions, these alternative series directly depend on the geopotential coefficients ${\bar{{C}}_{n,m}}$ and ${\bar{{S}}_{n,m}}$ . Each term of the series for the first-order derivatives is represented by a sum of these coefficients, which are multiplied by linear combinations of at most two spherical harmonics. For the second-order derivatives, the geopotential coefficients are multiplied by linear combinations of at most three spherical harmonics. As compared to existing non-singular expressions for the geopotential derivatives, the new expressions have a more simple structure. They depend only on the conventional spherical harmonics and do not depend on the first- and second-order derivatives of the associated Legendre functions. The basic equations of the second kind are inferred from the linear equations, constructed in the cited paper, which express the coefficients of the spherical harmonic series for the first- and second-order derivatives in terms of the geopotential coefficients. These equations are converted into recurrent relations from which the coefficients ${\bar{{C}}_{n,m}}$ and ${\bar{{S}}_{n,m}}$ are determined on the basis of the spherical harmonic coefficients of each derivative. The latter coefficients can be estimated from the values of the geopotential derivatives by the quadrature formulas or the least-squares approach. The new expressions of two kinds can be applied for spherical harmonic synthesis and analysis. In particular, they might be incorporated in geopotential modeling on the basis of the orbit data from the CHAMP, GRACE and GOCE missions, and the gradiometry data from the GOCE mission.     

Validation of Chlorophyll-a Algorithms in the Coastal Waters of Bay of Bengal Initial Validation Results from OCM-2

In this paper we report chlorophyll measurements made during an ocean colour validation cruise in April 2011 of the research vessel, Sagar Paschimi in the coastal waters of Northern Bay of Bengal. The chlorophyll-a concentration in these waters range from 0.2 to 4.0 mg/m³. Chlorophyll-a concentration from OCM-2 was estimated using the global ocean colour algorithms namely, OC2, OC3, OC4 and Chl-a algorithms respectively. OCM data was processed using the global SeaWiFS Data Analysis System (SeaDAS) in which all the above mentioned algorithms are embedded for estimating the chlorophyll-a concentration. A comparative study was made between and in-situ and satellite derived chlorophyll-a concentration. Although the matchups between in-situ and satellite data from OCM-2 were sparse, it indicates that direct application of the standard SeaWiFS algorithm-the OC4-V4 algorithm—in the coastal waters of the Bay of Bengal will underestimate chlorophyll-a by up to 30%. The results show a good correlation with an R value of 0.61 using OC2 algorithm. However, all the other global algorithms over estimate the chlorophyll-a concentration even in low chlorophyll concentration range. The comparison between in-situ and all the existing chlorophyll algorithms shows the efficiency of these algorithms for quantification of chlorophyll in coastal waters and hence the need to develop regional algorithms and fluorescence based algorithms for better quantification.     

Canadian gravimetric geoid model 2010

A new gravimetric geoid model, Canadian Gravimetric Geoid 2010 (CGG2010), has been developed to upgrade the previous geoid model CGG2005. CGG2010 represents the separation between the reference ellipsoid of GRS80 and the Earth’s equipotential surface of $W_0=62{,}636{,}855.69~\mathrm{m}^2\mathrm{s}^{-2}$ W 0 = 62 , 636 , 855.69 m 2 s ? 2 . The Stokes–Helmert method has been re-formulated for the determination of CGG2010 by a new Stokes kernel modification. It reduces the effect of the systematic error in the Canadian terrestrial gravity data on the geoid to the level below 2 cm from about 20 cm using other existing modification techniques, and renders a smooth spectral combination of the satellite and terrestrial gravity data. The long wavelength components of CGG2010 include the GOCE contribution contained in a combined GRACE and GOCE geopotential model: GOCO01S, which ranges from $-20.1$ ? 20.1 to 16.7 cm with an RMS of 2.9 cm. Improvement has been also achieved through the refinement of geoid modelling procedure and the use of new data. (1) The downward continuation effect has been accounted accurately ranging from $-22.1$ ? 22.1 to 16.5 cm with an RMS of 0.9 cm. (2) The geoid residual from the Stokes integral is reduced to 4 cm in RMS by the use of an ultra-high degree spherical harmonic representation of global elevation model for deriving the reference Helmert field in conjunction with a derived global geopotential model. (3) The Canadian gravimetric geoid model is published for the first time with associated error estimates. In addition, CGG2010 includes the new marine gravity data, ArcGP gravity grids, and the new Canadian Digital Elevation Data (CDED) 1:50K. CGG2010 is compared to GPS-levelling data in Canada. The standard deviations are estimated to vary from 2 to 10 cm with the largest error in the mountainous areas of western Canada. We demonstrate its improvement over the previous models CGG2005 and EGM2008.     

Geospatial Technologies for National Geomorphology and Lineament Mapping Project – A Case Study of Goa State

A three level classification system, based on the genesis of landforms, was used to map the geomorphology of the Goa state. The first level corresponds to the process that was responsible for landform generation, the second level or the intermediate level was assigned based on the morphography, and the third level corresponds to the individual landforms units identified based on the morphostructure. The mapping was carried out using IRS-P6 LISS-III (23.5 m) satellite image as the primary data source. Ancillary data such as geological map, topographic map, digital elevation model (DEM), field data collected by global positioning system (GPS) and web portals for image visualisation, were also used for the mapping purpose. A new software designed for mapping landforms based on the genesis, was used in this study to create a seamless geomorphology and lineament database of the Goa state in a GIS environment. A total of 58 landforms within six types of genetic classes were mapped in this area. Similarly, structural and geomorphic lineaments were also delineated using the satellite data. The database created has multi-purpose usability such as environmental studies, mining activity assessment, coastal zone management and wasteland development, since the classification system used is focused on processes, not theme specific.     

Understanding the Spatial Variability of Chlorophyll a and Total Suspended Matter Distribution Along the Southwest Bay of Bengal Using In-Situ and OCM-2 & MODIS-Aqua Measurements

Spatial and temporal distribution of chlorophyll a (chl a) and Total Suspended Matter (TSM) and inter comparison of Ocean Color Monitor-2 (OCM-2) and Moderate Resolution Imaging Spectro-radiometer (MODIS-Aqua) derived chlorophyll a and TSM was made along the southwest Bay of Bengal (BoB). The in-situ chl a and TSM concentration measured during different seasons were ranged from 0.09 to 10.63 μgl^?1 and 11.04–43.75 mgl^?1 respectively. OCM-2 and MODIS derived chl a showed the maximum (6–8 μgl^?1) at nearshore waters and the minimum (0–1 μgl^?1) along the offshore waters. OCM-2 derived TSM imageries showed the maximum (50–60 mgl^?1) along the nearshore waters of Palk Strait and the moderate concentration (2–5 mgl^?1) was observed in the offshore waters. MODIS derived minimum TSM concentration (13.244 mgl^?1) was recorded along the offshore waters, while the maximum concentration of 15.78 mgl^?1 was found along the Kodiakarai region. The inter-comparison of OCM-2 and MODIS chl a data (R ² ?=?0.549, n?=?49, p?<?0.001, SEE?=?±0.117) indicate that MODIS data overestimates chl a concentration in the nearshore waters of the southern BoB compared to the OCM-2. The correlation between OCM-2 and MODIS-Aqua TSM data (R ² ?=?0.508, N?=?53, P?<?0.001 and SEE?=?±0.024) confirms that variation in the range of values measured by OCM-2 (2–60 mgl^?1) and the MODIS (13–16 mgl^?1) derived TSM values. Despite problems in range of measurements, persistent cloud cover etc., the launch of satellites like OCM-2 with relatively high spatial resolutions makes job easier and possible to monitor chl a distribution and sediment discharges on day to day basis in the southwest BoB.     

An Analysis of Texture Measures in PCA-Based Unsupervised Classification of SAR Images

In single-band single-polarized SAR images, intensity and texture are the information source available for unsupervised land cover classification. Every textural feature measure identifies texture patterns by different approaches. For efficient land cover classification, textural measures have to be chosen suitably. Therefore, in this letter, the role of various intensity and textural measures is analyzed for their discriminative ability for unsupervised SAR image classification into various land cover types like water, urban, and vegetation areas. To make the algorithm adaptable, these textural features are fused using principal component analysis (PCA), and principal components are used for classification purposes. To highlight the effectiveness of PCA, the difference between PCA- and non-PCA-based classifications is also analyzed. Analysis of the role of texture measures for unsupervised classification of real-world SAR data with application of PCA is presented in this letter. The analysis of how every individual feature measure contributes for classification process is presented, and then, textural measures for a feature set are chosen according to their role in improving classification accuracy. By analysis, it is observed that the feature set comprising mean, variance, wavelet components, semivariogram, lacunarity, and weighted rank fill ratio provides good classification accuracy of up to 90.4% than by using individual textural measures, and this increased accuracy justifies the complexity involved in the process.     

Singular value decomposition and cluster analysis as regression diagnostics tools for geodetic applications

It is well known that high-leverage observations significantly affect the estimation of parameters. In geodetic literature, mainly redundancy numbers are used for the detection of single high-leverage observations or of single redundant observations. In this paper a further objective method for the detection of groups of important and less important (and thus redundant) observations is developed. In addition, the parameters which are predominantly affected by these groups of observations are identified. This method thus complements other diagnostics tools, such as, e.g., multiple row diagnostics methods as described in statistical literature (see, e.g., Belsley et al. in Regression diagnostics: identifying influential data and sources of collinearity. Wiley, New York, 1980). The method proposed in this paper is based on geometric aspects of adjustment theory and uses the singular value decomposition of the design matrix of an adjustment problem together with cluster analysis methods for regression diagnostics. It can be applied to any geodetic adjustment problem and can be used for the detection of (groups of) observations that significantly affect the estimated parameters or that are of negligible impact. One of the advantages of the proposed method is the improvement of the reliability of observation plans and thus the reduction of the impact of individual observations (and outliers) on the estimated parameters. This is of particular importance for the very long baseline interferometry technique which serves as an application example of the regression diagnostics tool.     

The determination of potential difference by the joint application of measured and synthetical gravity data: a case study in Hungary

In an elementary approach every geometrical height difference between the staff points of a levelling line should have a corresponding average g value for the determination of potential difference in the Earth’s gravity field. In practice this condition requires as many gravity data as the number of staff points if linear variation of g is assumed between them. Because of the expensive fieldwork, the necessary data should be supplied from different sources. This study proposes an alternative solution, which is proved at a test bed located in the Mecsek Mountains, Southwest Hungary, where a detailed gravity survey, as dense as the staff point density (~1 point/34 m), is available along a 4.3-km-long levelling line. In the first part of the paper the effect of point density of gravity data on the accuracy of potential difference is investigated. The average g value is simply derived from two neighbouring g measurements along the levelling line, which are incrementally decimated in the consecutive turns of processing. The results show that the error of the potential difference between the endpoints of the line exceeds 0.1 mm in terms of length unit if the sampling distance is greater than 2 km. Thereafter, a suitable method for the densification of the decimated g measurements is provided. It is based on forward gravity modelling utilising a high-resolution digital terrain model, the normal gravity and the complete Bouguer anomalies. The test shows that the error is only in the order of 10⁻³mm even if the sampling distance of g measurements is 4 km. As a component of the error sources of levelling, the ambiguity of the levelled height difference which is the Euclidean distance between the inclined equipotential surfaces is also investigated. Although its effect accumulated along the test line is almost zero, it reaches 0.15 mm in a 1-km-long intermediate section of the line.     

Characterization of diurnal cycles in ZTD from a decade of global GPS observations

The diurnal cycle of the tropospheric zenith total delay (ZTD) is one of the most obvious signals for the various physical processes relating to climate change on a short time scale. However, the observation of such ZTD oscillations on a global scale with traditional techniques (e.g. radiosondes) is restricted due to limitations in spatial and temporal resolution. Nowadays, the International GNSS Service (IGS) provides an important data source for investigating the diurnal and semidiurnal cycles of ZTD and related climatic signals. In this paper, 10 years of ZTD data from 1997 to 2007 with a 2-hour temporal resolution are derived from global positioning system (GPS) observations taken at 151 globally distributed IGS reference stations. These time series are used to investigate diurnal and semidiurnal oscillations. Significant diurnal and semidiurnal oscillations of ZTD are found for all GPS stations used in this study. The diurnal cycles (24 hours period) have amplitudes between 0.2 and 10.9 mm with an uncertainty of about 0.5 mm and the semidiurnal cycles (12 h period) have amplitudes between 0.1 and 4.3 mm with an uncertainty of about 0.2 mm. The larger amplitudes of the diurnal and semidiurnal ZTD cycles are observed in the low-latitude equatorial areas. The peak times of the diurnal cycles spread over the whole day, while the peak value of the semidiurnal cycles occurs typically about local noon. These GPS-derived diurnal and semidiurnal ZTD signals are similar with the surface pressure tides derived from surface synoptic pressure observations, indicating that atmospheric tides are the main driver of the diurnal and semidiurnal ZTD variations.