Derivation of topographic variables from a digital elevation model given by a spheroidal trapezoidal grid

Digital elevation models (DEMs) given by spheroidal trapezoidal grids are more appropriate for large regional, sub-continental, continental and global geological and soil studies than square-spaced DEMs. Here we develop a method for derivation of topographic variables, specifically horizontal (k) and vertical h (k) landsurface curvatures, from spheroidal trapezoidal-spaced DEMs. First, we v derive equations for calculation of partial derivatives of elevation with DEMs of this sort. Second, we produce formulae for estimation of the method accuracy in terms of root mean square errors of partial derivatives of elevation, as well as k h and k (m and m respectively). We design the method for the case that the v kh k v Earth's shape can be ignored, that is, for DEM grid sizes of no more than 225 km. We test the method by the example of fault recognition using a DEM of a part of Central Eurasia. A comparative analysis of test results and factual geological data demonstrates that the method actually works in regions marked by complicated topographic and tectonic conditions. Upon increasing DEM grid size, one can produce generalised maps of k and k. Spatial distributions of m and m h v kh k v depend directly on the distribution of elevation RMSE. Areas with high values of m are marked by low values of m, and vice versa, areas with high values kh k v of m are marked by low values of m. Data on m and m should be utilised k v kh kh k v to control and improve applications of k and k to geological studies. The method h v developed opens up new avenues for carrying out some 'conventional' raster operations directly on geographical co-ordinates.     

A discrete square global grid system based on the parallels plane projection

We developed a direct partitioning method to construct a seamless discrete global grid system (DGGS) with any resolution based on a two‐dimensional projected plane and the earth ellipsoid. This DGGS is composed of congruent square grids over the projected plane and irregular ellipsoidal quadrilaterals on the ellipsoidal surface. A new equal area projection named the parallels plane (PP) projection derived from the expansion of the central meridian and parallels has been employed to perform the transformation between the planar squares and the corresponding ellipsoidal grids. The horizontal sides of the grids are parts of the parallel circles and the vertical sides are complex ellipsoidal curves, which can be obtained by the inverse expression of the PP projection. The partition strategies, transformation equations, geometric characteristics and distortions for this DGGS have been discussed. Our analysis proves that the DGGS is area‐preserving while length distortions only occur on the vertical sides off the central meridian. Angular and length distortions positively correlate to the increase in latitudes and the spanning of longitudes away from a chosen central meridian. This direct partition only generates a small number of broken grids that can be treated individually.     

Finite cube elements method for calculating gravity anomaly and structural index of solid and fractal bodies with defined boundaries

In this paper, we present the finite cube elements method (FCEM); a novel numerical tool for calculating the gravity anomaly g and structural index SI of solid models with defined boundaries and variable density distributions, tilted or in normal position (e.g. blocks, faulted blocks, cylinders, spheres, hemispheres, triaxial ellipsoids). Extending the calculation to fractal objects, such as Menger sponges of different orders and bodies defined by polyhedrons, demonstrates the robustness of FCEM. In addition, approximating the cube element by a sphere of equal volume makes the calculation of gravitation and related derivatives much simpler. In gravity modelling of a sphere, cubes with edges of 100 m and 200 m achieve a good compromise between running time and overall error.
Displaying the distribution of SI of the studied models on contour maps and profiles will have a strong impact on the forward and inverse modelling of potential field data, especially for Euler deconvolution.
For Menger sponges, plots of gravity elements g and its derivatives show similar patterns independent of fractal order. Moreover, both the pattern and magnitude of SI are independent of fractal order, allowing the use of SI as a new invariant measure for fractal objects. However, SI pattern and magnitude strongly depend on the depth to the buried bodies as do other elements
In this study, we also present a new type of plot; the structural index against distance variation diagrams from which we extract the three critical SI ( CSI ) values, one per axis. The inversion of gravity anomaly data at CSI values gives the optimal mean location of the buried body.     

Morphometric characterisation of landform from DEMs

We describe a method of morphometric characterisation of landform from digital elevation models (DEMs). The method is implemented first by classifying every location into morphometric classes based on the mathematical shape of a locally fitted quadratic surface and its positional relationship with the analysis window. Single‐scale fuzzy terrain indices of peakness, pitness, passness, ridgeness, and valleyness are then calculated based on the distance of the analysis location from the ideal cases. These can then be combined into multi‐scale terrain indices to summarise terrain information across different operational scales. The algorithm has four characteristics: (1) the ideal cases of different geomorphometric features are simply and clearly defined; (2) the output is spatially continuous to reflect the inherent fuzziness of geomorphometric features; (3) the output is easily combined into a multi‐scale index across a range of operational scales; and (4) the standard general morphometric parameters are quantified as the first and second order derivatives of the quadratic surface. An additional benefit of the quadratic surface is the derivation of the R ² goodness of fit statistic, which allows an assessment of both the reliability of the results and the complexity of the terrain. An application of the method using a test DEM indicates that the single‐ and multi‐scale terrain indices perform well when characterising the different geomorphometric features.     

Dealing with double vagueness in DEM morphometric analysis

A method is presented to explicitly incorporate spatial and scale vagueness – double vagueness – into geomorphometric analyses. Known limitations of usual practices include using a single fixed set of crisp thresholds for morphometric classification and the imposition of a single arbitrary number of scales of analysis to the entire digital elevation model (DEM). Among the advantages of the proposed method are: fuzzification of morphometric classification rules, scale-dependent adaptive fuzzy set parametrization and an objective definition of maximum scale of analysis on a cell-by-cell basis. The method was applied to several DEMs ranging from the ocean floor to surface landscapes of both Earth and Mars. The result was evaluated with respect to modal morphometric features and to characteristic scales, suggesting a more robust method for deriving both morphometric classifications and terrain attributes. We argue that the method would be preferable to any single-scale crisp approach, at least in the context of preliminary hands-off morphometric analyses of DEMs.     

Third-order geomorphometric variables (derivatives): definition,computation and utilization of changes of curvatures

Third-order geomorphometric variables (based on third derivatives of the altitudinal field) have been neglected in geomorphometry, but their application to the delimitation of surface objects will lead to their increasing significance in future. New techniques of computation, presented and evaluated here, facilitate their use. This paper summarizes recent knowledge concerning definition, computation and geomorphologic interpretation of these variables. Formulae defining various third-order variables are unified based on the physical definition of slope gradient. Methods for their computation are compared from the point of view of method error and error generated by digital elevation model (DEM) inaccuracy. For exact mathematical test surfaces, the most natural and simple variant of the method of central differences (CD2) shows a method error 2–3 times smaller than the other methods used recently in geomorphometry. However, success in coping with DEM inaccuracy depends (for a given grid mesh) on the number and weighting of points from which the derivative is computed. This was tested for surfaces with varying degrees of random error. Here least squares-based methods are the most effective for mixed derivatives (especially for finer grids and less accurate DEMs), while a variant of the CD method, that repeats numerical evaluation of first derivatives (CD1), is the most successful for derivatives in cardinal directions. The CD2 method is generally the most successful for coarser grids where the method error is dominant.

Utilization of third-order variables is documented from examples of terrain feature (ridge, valley and edge) extraction and from a first statistical test of the hypothesis that real segments of the land surface have a tendency to a constant value of some morphometric variable. For detection of (sharp) ridges and valleys, it is shown that the rate of change of tangential curvature is inadequate: rate of change of normal curvature is also required. A basic confirmation of the constant-value tendency is provided.     

Geomorphometric feature analysis using morphometric parameterization and artificial neural networks

This paper presents a semi-automatic method using an unsupervised neural network to analyze geomorphometric features as landform elements. The Shuttle Radar Topography Mission (SRTM) provided detailed digital elevation models (DEMs) for all land masses between 60°N and 57°S. Exploiting these data for recognition and extraction of geomorphometric features is a challenging task. Results obtained with two methods, Wood's morphometric parameterization and the Self Organizing Map (SOM), are presented in this paper.Four morphometric parameters (slope, minimum curvature, maximum curvature and cross-sectional curvature) were derived by fitting a bivariate quadratic surface with a window size of 5 by 5 to the SRTM DEM. These parameters were then used as input to the two methods. Wood's morphometric parameterization provides point-based features (peak, pit and pass), line-based features (channel and ridge) and area-based features (planar). Since point-based features are defined as having a very small slope when their neighbors are considered, two tolerance values (slope tolerance and curvature tolerance) are introduced. Selection of suitable values for the tolerance parameters is crucial for obtaining useful results.The SOM as an unsupervised neural network algorithm is employed for the classification of the same morphometric parameters into ten classes characterized by morphometric position (crest, channel, ridge and plan area) subdivided by slope ranges. These terrain features are generic landform element and can be used to improve mapping and modeling of soils, vegetation, and land use, as well as ecological, hydrological and geomorphological features. These landform elements are the smallest homogeneous divisions of the land surface at the given resolution. The result showed that the SOM is an efficient scalable tool for analyzing geomorphometric features as meaningful landform elements, and uses the full potential of morphometric characteristics.     

Numerical inversion of deformation caused by pressure sources: application to Mount Etna (Italy)

The interpretation of geodetic data in volcanic areas is usually based on analytical deformation models. Although numerical finite element (FE) modelling allows realistic features such as topography and crustal heterogeneities to be included, the technique is not computationally convenient for solving inverse problems using classical methods. In this paper, we develop a general tool to perform inversions of geodetic data by means of 3-D FE models. The forward model is a library of numerical displacement solutions, where each entry of the library is the surface displacement due to a single stress component applied to an element of the grid. The final solution is a weighted combination of the six stress components applied to a single element-source. The pre-computed forward models are implemented in a global search algorithm, followed by an appraisal of the sampled solutions. After providing extended testing, we apply the method to model the 1993–1997 inflation phase at Mt Etna, documented by GPS and EDM measurements. We consider four different forward libraries, computed in models characterized by homogeneous/heterogeneous medium and flat/topographic free surface. Our results suggest that the elastic heterogeneities of the medium can significantly alter the position of the inferred source, while the topography has minor effect.     

Evaluating the Potential of an Airborne Laser-scanning System for Measuring Volume Changes of Glaciers

Airborne laser scanning (ALS) is well suited for the production of digital elevation models (DEM), and can, in contrast to photographic methods, be used to acquire a DEM independently of surface texture and external light sources. ALS thus serves as a tool to generate DEMs of firn areas where photogram- metric methods often fail.
The potential of an integrated ALS system – comprising a laser scanner, precise differential global positioning system, and a gyro platform – for DEM generation of firn areas is currently being assessed. The Unteraargletscher, Bernese Alps, Switzerland, has been chosen as a test site. As part of a pilot project aimed at determining the mass balance distribution of that glacier without the use of in situ information, ALS measurements were conducted in autumn 1997. The DEM derived from laser measurements is extremely sensitive to the position and attitude of the aircraft. Currently the main work focuses on assessing and improving the system's accuracy by error modelling and by the development of error-correction algorithms.
Preliminary results from Unteraargletscher are presented, and the potential of this method for the generation of DEMs of firn areas is discussed.     

Modelling surface drainage patterns in altered landscapes using LiDAR

The detailed topographic information contained in light detection and ranging (LiDAR) digital elevation models (DEMs) can present significant challenges for modelling surface drainage patterns. These data frequently represent anthropogenic infrastructure, such as road embankments and drainage ditches. While LiDAR DEMs can improve estimates of catchment boundaries and surface flow paths, modelling efforts are often confounded difficulties associated with incomplete representation of infrastructure. The inability of DEMs to represent embankment underpasses (e.g. bridges, culverts) and the problems with existing automated techniques for dealing with these problems can lead to unsatisfactory results. This is often dealt with by manually modifying LiDAR DEMs to incorporate the effects of embankment underpasses. This paper presents a new DEM pre-processing algorithm for removing the artefact dams created by infrastructure in sites of embankment underpasses as well as enforcing flow along drainage ditches. The application of the new algorithm to a large LiDAR DEM of a site in Southwestern Ontario, Canada, demonstrated that the least-cost breaching method used by the algorithm could reliably enforce drainage pathways while minimizing the impact to the original DEM.     

A Bayesian approach to inverse modelling of stratigraphy, part 2: Validation tests

A novel inverse modelling method is applied to the problem of constraining the environmental parameters (e.g. relative sea level, sediment supply) that control stratigraphic architecture. This technique links forward modelling of shallow-marine wave/storm-dominated stratigraphy to a combination of inverse methods formulated in a Bayesian framework. We present a number of examples in which relative sea-level and sediment-supply curves were inferred from synthetic vertical successions of grain size (e.g. wells) and synthetic thickness curves (e.g. seismically derived isopachs) extracted from a forward model simulation. These examples represent different scenarios that are designed to test the impact of data distribution, quantity and quality on the uncertainty of the inferred parameters. The inverse modelling approach successfully reproduces the gross stratigraphic architectures and relative sea level and sediment-supply histories of the synthetic forward model simulation, within the constraints of the modelled data quality. The relative importance of the forcing parameters can be evaluated by their sensitivity and impact on the inverted data. Of equal importance, the inverse results allow complete characterisation of the uncertainties inherent to the stratigraphic modelling tool and to the data quality, quantity and distribution. The numerical scheme also successfully deals with the problem of non-uniqueness of the solution of the inverse problem. These preliminary results suggest that the inverse method is a powerful tool in constraining stratigraphic architecture for hydrocarbon reservoir characterisation and modelling, and it may ultimately provide a process-based geological complement to standard geostatistical tools.     

The fine structure of the normal incidence synthetic seismogram

Summary . The spectral function of a perfectly elastic, horizontally stratified medium has been demonstrated previously to provide an attractive formulation to describe the properties of the one-dimensional synthetic seismogram (Robinson & Treitel). Here we examine the mathematical framework of the Model in still greater detail. Knowledge of this fine structure of the synthetic seismogram leads to the solution of two particular seismic inverse problems. First, we consider a layered medium with an arbitrary surface reflection coefficient c _o, where | c _o|<1, and which contains an impulsive source immediately above the surface. Given the corresponding synthetic seismogram, we develop an inverse, or backward recursion formalism which recovers the entire series of original reflection coefficients. Second, we consider a similar problem for an impulsive source located just below the surface. Both inversion procedures constitute a continuation of the work of Goupillaud and of Sherwood & Trorey, and represent a generalization of the classical technique originally proposed by Kunetz which, however, only holds for the marine case, c_o =±1. The present approach is not so constrained and thereby becomes applicable to land seismograms as well.
If products of third or higher order in the reflection coefficients can be neglected, significant simplifications arise in the theory. In that event the usual representation of the synthetic seismogram as a ratio of two polynomials in the complex variable z becomes particularly revealing. The numerator polynomial is then approximately equal to the z transform of the reflection coefficient series, while the denominator polynomial is approximately equal to the z transform of the autocorrelation of these reflection coefficients. The resulting simplified theory affords important computational savings in the appropriate backward recursion algorithms.     

Ellipsoidal quadtrees for indexing of global geographical data

Systems for landscape visualization and geographical data handling require methods for efficient data access. Retrieval of data from large geographical databases, ten to thousands of Gbytes, is usually optimized with spatial indexing mechanisms. The simplest form of spatial indexing is achieved by dividing the database into congruent grid cells. The subsequent subdivision of the grid cells can be based on so-called quadtrees. Quadtrees for two-dimensional division and subdivision are appropriate for cartographical data. A geographical database, with objects stored in geocentric or geodetic (geographical) co-ordinates, requires indexing mechanisms that take into account the shape of the Earth. In this paper, we present a method for indexing of geographical data, named Ellipsoidal Quadtrees (EQT). In contrast to other global indexing methods, EQT is based on the Earth ellipsoid and not a spherical approximation. EQT division and subdivision make it possible to divide the Earth surface into a mesh of quadrangles with equal areas. We will demonstrate that EQT is flexible. It can be used for indexing databases of various sizes, including national and global databases. Tests on real data show that the performance of EQT is good.     

Geodetic singularities in the gravity field of a non-homogeneous planet

Summary. An inverse geodetic singularity problem is considered for a non-homogeneous spherical planet. The singularity condition is expressed in terms of the density distribution and of the geometrical parameters of the configuration; the condition for the density distribution is deduced which gives rise to singularities of parabolic type in the external gravity field of the planet. The structure of the gravity field in the neighbourhood of the singularities is investigated in detail together with the behaviour of the gradients of the disturbances in the geodetic coordinates.     

基于地图方里网的全球剖分系统

针对现有剖分模型的不足,提出了一种基于地图方里网的全球剖分系统,有效避免了传统经纬度格网模型在高纬度地区的形状退化和正多面体格网模型的面片形状不规则问题。制定了相应的编码,实现了面片编码与传统地理坐标之间的转换和邻接关系的计算,最后对方里网在地球椭球面上的变形规律进行了研究。研究结果表明:基于地图方里网的剖分系统不仅具有科学的数据组织形式,而且面片单元的变形面积小、变形规律稳定。     

Gravitational viscoelastic relaxation of eccentrically nested spheres

We present a semi-analytical solution to the 2-D forward modelling of viscoelastic relaxation in a heterogeneous model consisting of eccentrically nested spheres. Several numerical methods for 2-D and 3-D viscoelastic relaxation modelling have been applied recently, including finite-element and spectral-finite-difference schemes. The present semi-analytical approach provides a model response against which more general numerical algorithms can be validated. The eccentrically nested sphere solution has been tested by comparing it with the analytical solutions for viscoelastic relaxation in a homogeneous sphere and in two concentrically nested spheres, and good agreement was obtained.     

Error assessment of grid-based flow routing algorithms used in hydrological models

This paper reports an investigation on the accuracy of grid-based routing algorithms used in hydrological models. A quantitative methodology has been developed for objective and data-independent assessment of errors generated from the algorithms that extract hydrological parameters from gridded DEM. The generic approach is to use artificial surfaces that can be described by a mathematical model, thus the ‘true’ output value can be pre-determined to avoid uncertainty caused by uncontrollable data errors. Four mathematical surfaces based on an ellipsoid (representing convex slopes), an inverse ellipsoid (representing concave slopes), saddle and plane were generated and the theoretical ‘true’ value of the Specific Catchment Area (SCA) at any given point on the surfaces could be computed using mathematical inference. Based on these models, tests were made on a number of algorithms for SCA computation. The actual output values from these algorithms on the convex, concave, saddle and plane surfaces were compared with the theoretical ‘true’ values, and the errors were then analysed statistically. The strengths and weaknesses of the selected algorithms are also discussed.     

中国数字地形学研究进展（英文）

Geomorphometry,the science of digital terrain analysis(DTA),is an important focus of research in both geomorphology and geographical information science(GIS).Given that 70% of China is mountainous,geomorphological research is popular among Chinese scholars,and the development of GIS over the last 30 years has led to significant advances in geomorphometric research.In this paper,we review Chinese progress in geomorphometry based on the published literature.There are three major areas of progress:digital terrain modelling methods,DTA methods,and applications of digital terrain models(DTMs).First,traditional vector-and raster-based terrain modelling methods,including the assessment of uncertainty,have received widespread attention.New terrain modelling methods such as unified raster and vector,high-fidelity,and real-time dynamic geographical scene modelling have also attracted research attention and are now a major focus of digital terrain modelling research.Second,in addition to the popular DTA methods based on topographical derivatives,geomorphological features,and hydrological factors extracted from DTMs,DTA methods have been extended to include analyses of the structure of underlying strata,ocean surface features and even socioeconomic spatial structures.Third,DTMs have been applied to fields including global climate change,analysis of various typical regions,lunar surface and other related fields.Clearly,Chinese scholars have made significant progress in geomorphometry.Chinese scholars have had the greatest international impact in areas including high-fidelity digital terrain modelling and DTM-based regional geomorphological analysis,particularly in the Loess Plateau and the Tibetan Plateau regions.     

The multibody space-time geodetic boundary value problem and the Honkasalo term

Summary. The geodetic boundary value problem is solved by taking temporal changes of geometry and gravity into account. It is aimed to get complete information about the mechanical response of the Earth to an external tidal force which is only imperfectly known. The information is used to solve the inverse Love-Shida problem, e.g. by computing the Love -Shida operators. The general solution (4.25), (4.26) is specialized for a perfectly known external gravity field and for a free fixed boundary of the Earth.     

An automated approach to the classification of the slope units using digital data

Digital elevation and remote sensing data sets contain different, yet complementary, information related to geomorphological features. Digital elevation models (DEMs) represent the topography, or land form, whereas remote sensing data record the reflectance/emittance, or spectral, characteristics of surfaces. Computer analysis of integrated digital data sets can be exploited for geomorphological classification using automated methods developed in the remote sensing community. In the present study, geomorphological classification in a moderate- to high-relief area dominated by slope processes in southwest Yukon Territory, Canada, is performed with a combined set of geomorphometric and spectral variables in a linear discriminant analysis. An automated method was developed to find the boundaries of geomorphological objects and to extract the objects as groups of aggregated pixels. The geomorphological objects selected are slope units, with the boundaries being breaks of slope on two-dimensional downslope profiles. Each slope unit is described by variables summarizing the shape, topographic, and spectral characteristics of the aggregated group of pixels. Overall discrimination accuracy of 90% is achieved for the aggregated slope units in ten classes.