Nonlinear filtering of continuous systems: foundational problems and new results

The previous work of Xu on discrete nonlinear filtering is extended to continuous systems. The new results are summarized as follows: (1) a second-order unbiased prediction of the true state governed by a vector stochastic differential equation is worked out; (2) a set of coupled differential equations for a new truncated second-order nonlinear filter and its variance–covariance matrix are derived from the frequentist point of view. The new filter is proved to be unbiased to the second-order approximation; and, most importantly, (3) comparison of the new filtering and accuracy results with the literature on nonlinear filtering has indicated that more than 40 years of nonlinear filtering of continuous systems may have foundational problems.Acknowledgments.This work is supported by a Grant-in-Aid for Scientific Research (C13640422). The author thanks Prof J.A.R. Blais, Prof A. Dermanis and Prof B. Schaffrin for their constructive comments.     

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

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

On a four-dimensional integrated geodesy

In contrast to continuous global considerations of time dependent boundary value problems an attempt is made to define4D-linear observation equations in the framework of integrated geodesy for discrete, more or less regional and local applications (deformation analysis) where time variations in position and in the gravity field have to be considered. The derivation is a strict analogue and extension of the3D integrated approach. In addition the construction of time dependent covariance functions is discussed, which are necessary to solve for unknown displacements and changes in the gravity potential in the generalized least squares collocation model.     

A WORKING METHOD FOR THE CALIBRATION OF PLOTTING INSTRUMENTS USING COMPUTERS

Methods of testing and adjusting photogrammetric plotting instruments are reviewed and the precision of traditional simple methods is investigated. Practical tests confirm the theoretical results and suggest the most efficient methods for determining the co-ordinates of the perspective centre, the principal distance and the adjustments necessary to the interior orientation. These methods are incorporated in a procedure for testing instruments using either a large computer or a desk top computer. It is concluded that an improvement in accuracy and efficiency can be obtained by using redundant observations and a computer to handle the mathematical adjustment.     

利用神经网络实现动态大系统稳定性分析与镇定

本文给出了利用人工神经网络（ＡＮＮ）理论和技术实现线性定常离散和连续时间动态大系统稳定性分析与镇定的方法（算法）；并在计算机上进行了一系列的数值实验，验证了该方法的正确性、有效性和可实现性。     

Towards an integral strategy for modelling uncertain geospatial data

This paper endeavours to put the discussion on errors and uncertainties in geographical information systems (GISs) in a more systematic way by examining the strength and weakness of discrete objects and continuous fields, the two distinct schools of spatial data modelling. In doing so, it argues that neither discrete objects nor continuous fields alone provide objective and complete representations of highly complex geographical phenomena, though there are good reasons for asserting that continuous fields are better suited to modelling spatial dependence, heterogeneity and fuzziness significant in geographical reality than discrete objects. Thus, there seems to be merit in adopting an integrated model incorporating analytical capabilities of fields and generalization functions of objects, for which extended TIN(triangulated irregular network) models along with their duals (Voronoi diagrams) provide a pragmatical solution.     

Geoid determination using one-step integration

A residual (high-frequency) gravimetric geoid is usually computed from geographically limited ground, sea and/or airborne gravimetric data. The mathematical model for its determination from ground gravity is based on the transformation of observed discrete values of gravity into gravity potential related to either the international ellipsoid or the geoid. The two reference surfaces are used depending on height information that accompanies ground gravity data: traditionally orthometric heights determined by geodetic levelling were used while GPS positioning nowadays allows for estimation of geodetic (ellipsoidal) heights. This transformation is usually performed in two steps: (1) observed values of gravity are downward continued to the ellipsoid or the geoid, and (2) gravity at the ellipsoid or the geoid is transformed into the corresponding potential. Each of these two steps represents the solution of one geodetic boundary-value problem of potential theory, namely the first and second or third problem. Thus two different geodetic boundary-value problems must be formulated and solved, which requires numerical evaluation of two surface integrals. In this contribution, a mathematical model in the form of a single Fredholm integral equation of the first kind is presented and numerically investigated. This model combines the solution of the first and second/third boundary-value problems and transforms ground gravity disturbances or anomalies into the harmonically downward continued disturbing potential at the ellipsoid or the geoid directly. Numerical tests show that the new approach offers an efficient and stable solution for the determination of the residual geoid from ground gravity data.     

L1-norm pre-analysis measures for geodetic networks

 Several pre-analysis measures which help to expose the behavior of L ₁ -norm minimization solutions are described. The pre-analysis measures are primarily based on familiar elements of the linear programming solution to L ₁-norm minimization, such as slack variables and the reduced-cost vector. By examining certain elements of the linear programming solution in a probabilistic light, it is possible to derive the cumulative distribution function (CDF) associated with univariate L ₁-norm residuals. Unlike traditional least squares (LS) residual CDFs, it is found that L ₁-norm residual CDFs fail to follow the normal distribution in general, and instead are characterized by both discrete and continuous (i.e. piecewise) segments. It is also found that an L ₁ equivalent to LS redundancy numbers exists and that these L ₁ equivalents are a byproduct of the univariate L ₁ univariate residual CDF. Probing deeper into the linear programming solution, it is found that certain combinations of observations which are capable of tolerating large-magnitude gross errors can be predicted by comprehensively tabulating the signs of slack variables associated with the L ₁ residuals. The developed techniques are illustrated on a two-dimensional trilateration network. Received: 6 July 2001 / Accepted: 21 February 2002     

Synthetic RapidEye data used for the detection of area-based spruce tree mortality induced by bark beetles

Tree mortality caused by outbreaks of the bark beetle Ips typographus (L.) plays an important role in the natural dynamics of Norway spruce (Picea abies L.) stands, which could cause far-reaching changes in the occurrence and duration of vegetation phenology. Field-based early detection of tree disturbances is hampered by logistic, terrain, and technical shortcomings, and by the inability to continuously monitor disturbances over large areas. Despite achievements in remote mapping of bark-beetle-induced tree mortalities, early warning has been mostly unsuccessful mainly because of the lack of spectral sensitivity and discrepancies in definitions of field- and image-based disturbance classes. Here we applied a method based on inter-annual phenology of Norway spruce stands derived from synthetic multispectral data to part of the Bavarian Forest National Park in Germany. We fused temporally continuous Moderate Resolution Imaging Spectroradiometer and discrete RapidEye data using a flexible spatiotemporal data fusion method to achieve validated 8-day RapidEye-like composites of normalized difference vegetation index for 2011. We assumed that the dead trees delineated on 2012 aerial photographs were those in which bark beetle infestations were initiated in 2011. Samples were drawn with variable-sized buffering to represent the areas prone to infestations and their surroundings. We applied a conditional inference random forest to select the best image date among the entire 46 synthetic datasets to best discriminate between the core infestation patches and their surroundings from the subsequent year. Of the discrete time points identified, day 281 of the year represented the highest discrepancy between aerial image-based dead trees and their surroundings. Classification results were significantly correlated with beetle count data obtained using pheromone traps. Our method provided valuable information for management purposes and enabled wall-to-wall mapping of stands prone to infestation and its uncertainty. The results offer potential implications for rapid and cost-effective monitoring of bark beetle outbreaks using satellite data, which would be of great benefit for both management and research tasks.     

A comprehensive framework for exploratory spatial data analysis: Moran location and variance scatterplots

Abstract

A significant Geographic Information Science (GIS) issue is closely related to spatial autocorrelation, a burning question in the phase of information extraction from the statistical analysis of georeferenced data. At present, spatial autocorrelation presents two types of measures: continuous and discrete. Is it possible to use Moran's I and the Moran scatterplot with continuous data? Is it possible to use the same methodology with discrete data? A particular and cumbersome problem is the choice of the spatial-neighborhood matrix (W) for points data. This paper addresses these issues by introducing the concept of covariogram contiguity, where each weight is based on the variogram model for that particular dataset: (1) the variogram, whose range equals the distance with the highest Moran I value, defines the weights for points separated by less than the estimated range and (2) weights equal zero for points widely separated from the variogram range considered. After the W matrix is computed, the Moran location scatterplot is created in an iterative process. In accordance with various lag distances, Moran's I is presented as a good search factor for the optimal neighborhood area. Uncertainty/transition regions are also emphasized. At the same time, a new Exploratory Spatial Data Analysis (ESDA) tool is developed, the Moran variance scatterplot, since the conventional Moran scatterplot is not sensitive to neighbor variance. This computer-mapping framework allows the study of spatial patterns, outliers, changeover areas, and trends in an ESDA process. All these tools were implemented in a free web e-Learning program for quantitative geographers called SAKWeb© (or, in the near future, myGeooffice.org).     

A STATISTICAL APPROACH TO FEATURE CLASSIFICATION ON REMOTE SENSING IMAGERY

The mathematical basis for a feature classification algorithm is described which combines elements of game theory with Bayesian and suboptimal [feature classification] decision rules. Comparison of reflectance values with training area parameters, according to a sequence of diminishing a priori probabilities that the values will be assigned to that particular class results in reductions in computer time during classification. Results of the procedure are demonstrated through a pair of “before” and “after” images. Translated from: Metody kompleksnykh aerokosmicheskikh issledovaniy Sibiri, L. K. Zyat'kova, ed. Novosibirsk: Nauka, 1985, pp. 75–79.     

Variance-covariance estimation of GPS Networks

Summary It is quite easy to estimate the variance-covariance (VCV) matrix for single session surveys or local networks, but difficult where these local networks are combined together to form a regional network. Our main aim is to develop an appropriate VCV model to combine all the different types of networks, either global, regional or local. By careful estimation and combination of the individual VCVs of the local networks, we can form a unique VCV for local, regional and global networks. Different techniques are used to derive appropriate models for the variancecovariance components of the Global Positioning System (GPS) networks. The VCV models were estimated using homogeneous and heterogeneous data. The variance-covariance components are empirically derived using (a) the covariance of the observations of homogeneous data, (b) a combination of the covariance of the observationsP ^–1 and the covariance of the signal componentsC _ss (for either homogeneous and/or heterogeneous data), (c) only the variances are used to determine the variancecovariance, their covariances being zeros. We compare the solutions of the VCV developed for homogeneous and/or heterogeneous data with other developed VCVs. It was observed that the derived VCV model for the combined homogeneous and/or heterogeneous data of case (b) gives the best estimates in all cases.     

Bayesian dynamic modeling for large space-time datasets using Gaussian predictive processes

In this paper, we extend the applicability of a previously proposed class of dynamic space-time models by enabling them to accommodate large datasets. We focus on the common setting where space is viewed as continuous but time is taken to be discrete. Scalability is achieved by using a low-rank predictive process to reduce the dimensionality of the data and ease the computational burden of estimating the spatio-temporal process of interest. The proposed models are illustrated using weather station data collected over the northeastern United States between 2000 and 2005. Here our interest is to use readily available predictors, association among measurements at a given station, as well as dependence across space and time to improve prediction for incomplete station records and locations where station data does not exist.     

LOCAL ATTRACTIONS AND THEIR INFLUENCE ON THE DETERMINATION OF HEIGHTS ABOVE MEAN SEA LEVEL

Abstract

The Mythical Spheroid.—The preceding article dealt with the fact that the spheroid of reference is a myth and that, even if it were not, we could not get hold of it at any given place. In order to apply corrections to observed quantities or, more generally, to operate upon them mathematically, we must make some assumption such as that of the spheroidal level surface. Probably a lot of harm has been done by attaching the notion of too concrete a thing to the spheroid. Disputes and misconceptions have arisen. People talk of“putting the spheroid down at a point” and imagine that the obedient thing is still at their feet when they get to another point, perhaps distant, in their system of triangulation or what not. Actually the spheroid may be disobedient not only as regards the direction of the vertical but also because it is above their heads or below their feet. What happens is that at each point afresh the computer treats the observations as if they were made there on the surface of a spheroid. In the same way, but travelling still farther along the road of hypothesis, he may treat observations for astronomical positions as if the compensation for visible elevations were uniformly distributed as a deficiency of density down to a depth of 122·2 kilometres. That was the depth which happened to give the smallest sum of squares of residuals in a certain restricted area, but nobody imagines that it corresponds with a physical reality, especially the ·2! It was a convenient mathematical instrument which, once the theory was to be given a trial, had to be fashioned out of some assumption or another. All this has little to do with geodetic levelling but is meant to try to banish the spheroid out of the reader's mind or at least to the back of his mind. In what follows we shall be compelled to make a certain amount of use of the family of spheroids but always with the above strictures in view.     

Disconnected Cartographies and Immappancy

Abstract

Manual shading, traditionally produced manually by specifically trained cartographers, is still considered superior to automatic methods, particularly for mountainous landscapes. However, manual shading is time-consuming and its results depend on the cartographer and as such difficult to replicate consistently. For this reason there is a need to create an automatic method to standardize its results. A crucial aspect of manual shading is the continuous change of light direction (azimuth) and angle (zenith) in order to better highlight discrete landforms. Automatic hillshading algorithms, widely available in many geographic information systems (GIS) applications, do not provide this feature. This may cause the resulting shaded relief to appear flat in some areas, particularly in areas where the light source is parallel to the mountain ridge. In this work we present a GIS tool to enhance the visual quality of hillshading. We developed a technique based on clustering aspect to provide a seamless change of lighting throughout the scene. We also provide tools to change the light zenith according to either elevation or slope. This way the cartographer has more room for customizing the shaded relief representation. Moreover, the method is completely automatic and this guarantees consistent and reproducible results. This method has been embedded into an ArcGIS toolbox.     

Finding least-cost paths across a continuous raster surface with discrete vector networks

The problem of finding the least-cost path from a source point to a destination point can be dealt with by routing across a continuous surface or routing along a discrete network. The solutions within these two contexts are linked to the use of a raster- or a vector-based least-cost path algorithm. This study presents a technique which integrates raster- and vector-based least-cost path algorithms for determining the least-cost path across a continuous raster surface with discrete vector networks. The technique incorporates ancillary vector data sets that are required to examine the travel cost at each link, connections between nodes, and the representation of intersecting links in the discrete vector network into raster-based least-cost path analysis. The integrated technique presented here is applicable to all-terrain vehicle navigation where a continuous raster surface and discrete vector networks need to be considered simultaneously in order to find least-cost paths. This paper describes the concept behind, and details of, the integrated technique. Applications of the technique with synthetic and real-world data sets are also presented. They provide proof that the technique is effective in finding least-cost paths across a continuous raster surface with discrete vector networks.     

Integrating Discrete and Continuous Data in an OpenGeospatial‐Compliant Specification

The integrated management of heterogeneous spatial data, such as continuous fields and discrete data, is an important issue for the Geographic Information (GI) community. Indeed, GI users are forced to navigate among and operate with several tools in order to solve their spatial problems, due to the lack of systems capable of integrating different components, each meant to provide a specific solution. The aim of this article is to propose an OpenGeospatial‐compliant solution which supports expert users in handling problems involving heterogeneous data by means of a seamless approach. A class hierarchy modeling spatial discrete objects, continuous data, relationships, and operations, is described, whereby data are organized in agreement with the binary representation. A running example is illustrated to support readers' understanding of the proposed solution. Finally, some guidelines about an implementation modality are given, to demonstrate the applicability of the proposal to an existing DBMS.     

Towards an integral strategy for modelling uncertain geospatial data

This paper endeavours to put the discussion on errors and uncertainties in geographical information systems (GISs) in a more systematic way by examining the strength and weakness of discrete objects and continous fields, the two distinct schools of spatial data modelling. In doing so, it argues that neither discrete objects nor continous fields alone provide objective and complete representations of highly complex geographical phenomena, though there are good reasons for asserting that continuous fields are better suited to modelling spatial dependence, heterogeneity and fuzzines significant in geographical reality than discrete objects. Thus, there seems to be merit in adopting an integrated model incorporating analytical capabilities of fields and generalization functions of objects, for which extended TIN (triangulated irregular network) models along with their duals (Voronoi diagrams) provide a pragmatical solution.     

Partial iterates for symmetrizing non-parametric color correction

Mosaic generation is a central tool in various fields ranging way beyond the scope of photogrammetry and requires the radiometry and color of the images to be corrected. Correction can either be done by a global parametric approach (looking for an optimal gain or gamma for each image of the mosaic), or by iteratively correcting image pairs with a non-parametric approach. Such non-parametric approaches allow for much finer correction but are asymmetric, i.e. they require the choice of a source image that will be corrected to match a target image. Thus the result on the whole mosaic will be very dependant on the order in which images are corrected. In this paper, we propose to use partial iterates to symmetrize non-parametric correction in order to solve this problem. Partial iterates formalize what partially applying a bijective function means and we explain how this can be done in both the continuous and discrete domain. This mechanism is applied to a simple non-parametric approach (histogram transfer of the luminance) to show its potential.     

Fields as a framework for integrating GIS and environmental process models. Part 1: Representing spatial continuity

Linking a GIS to a spatially distributed, physically-based environmental model offers many advantages. However, the implementation of such linkages is generally problematic. Many problems arise because the relationship between the reality being represented by the mathematical model and the data model used to organize the spatial data in the GIS has not been rigorously defined. In particular, while many environmental models are based on theories that assume continuity and incorporate physical fields as independent variables, current GISs can only represent continuous phenomena in a variety of discrete data models. This paper outlines a strategy in which field variables are used to enable modellers to work directly with the spatial data as spatially continuous phenomena. This allows the manner in which the spatial data has been discretized and the ways in which it can be manipulated to be treated independently from the conceptual modelling of physical processes. Modellers can express their spatial data needs as representations of reality, rather than as elements of a GIS database, and a GIS-independent language for model development may result. By providing a formal linkage between the various models of spatial phenomena, a mechanism is created for the explicit expression of transformation rules between the different spatial data models stored and manipulated by a GIS.