Reljefo atvaizdavimas naudojamas geodezijoje, geoinformacinėse sistemose, kompiuteriniuose žaidimuose ir kitur. Dažniausiai išmatuoti reljefo duomenys nėra pakankamai tikslūs, ir su jais atvaizduojami paviršiai atrodo nenatūraliai. Šiame darbe aptariami trys pagrindiniai, viešai prieinami viso Žemės paviršiaus aukščių duomenų šaltiniai: GTOP30, ASTER GDEM ir SRTM. Nagrinėjami populiariausi reljefo interpoliavimo metodai tokie kaip bitiesinis, bikubinis, atvirkščiai proporcingo atstumo ir natūralių kaimynų. SRTM duomenų interaktyviai peržiūrai buvo sukurtas ArcGrid Viewer įrankis, leidžiantis peržiūrėti ArcGrid formatu išsaugotus reljefus. Interpoliavimo metodų tyrimui buvo sukurtas atskiras GeoSG įrankis. Su juo vienu metu galima stebėti kelis paviršius ir atlikti paviršių interpoliavimą bitiesiniu, bikubiniu, atvirkščiai proporcingo atstumo ir natūralių kaimynų metodais. Su šiuo įrankiu atliktas interpoliavimo metodų tikslumo įvertinimas. Nustatyta, kad geriausiai interpoliuoja bikubinis metodas, o blogiausiai – atvirkščiai proporcingo atstumo metodas. / Terrain rendering is used in geodesy, geographic information systems, computer games and other fields. Usually measured terrain data isn’t sufficiently precise, and if used for rendering, the result looks unnatural. This paper analyzes three main topographic data sources of planet Earth, that are publicly available: GTOP30, ASTER GDEM and SRTM. It compares the most often used methods for terrain data: bilinear, bicubic, inverse distance weighted and natural neighbor interpolation. To monitor SRTM data interactively an ArcGrid Viewer tool was created. It can be used to see terrain files in ArcGrid format. To study interpolation methods, a separate GeoSG tool was created. It can display several terrains at once and interpolate terrain data using bilinear, bicubic, inverse distance weighted and natural neighbor methods. This tool is used to evaluate interpolation methods. Results showed, that the best method for interpolation is the bicubic interpolation method, the worst – inverse distance weighted interpolation method.

This diploma thesis deals with the comparison of the development of nitrates and phosphates in various regions of the Czech Republic. The thesis contains of the description of the circulation of groundwater and its importance to the human society, the circulation of nitrogen and phosphorus and their occurrence in waters. Further, this thesis incorporates the description of the development and the way of observation of the concentration of nitrates and phosphates in groundwater. The selection of observational objects and used methods for the evaluation of the concentrations are described in the thesis. The aim of the thesis is the determination of trends of the concentrations of nitrates and phosphates during time periods and the statistical evaluation of the progress of the concentrations in regional cities of the Czech Republic. The next part of the thesis deals with the spatial analysis of the concentrations of nitrates and phosphates in the region of České Budějovice. This area was chosen for its great amount of the observational objects with a long time period of measuring.

Předmětem této práce je polohově orientovaná analýza v kontextu optimalizace mobilních sítí. Popisuje nástroj pro odhadování základních parametrů sítě na místech s neznámými parametry sítě na základě databáze RTR NetTest. Je zde stručně představena oblast velkých dat, strojového učení a shrnutí o konceptu a funkcionalitě aplikace NetTest. Práce ukazuje a porovnává skupinu regresních metod na základě jejich komplexnosti a vhodnosti pro vytvoření map odhadovaných parametrů sítě. Po jejich důkladné 1D analýze je IDW a GPR analyzováno ve 2D a využito pro vytvoření skupiny map odhadu parametrů sítě. Je posouzena i jejich přesnost na základě referenčního měření aplikací NetTest.

This thesis engages various methods of calculation of the spatial distribution of precipitation and the evaluation of differences in totals of rainfall in a chosen catchment. The aim of this work was to choose suitable methods for the calculation of areal rainfall and subsequently to create those methods. Most methods were developed in the GIS program. The next objective was to explore differences in monthly precipitation from hydrological years 2006 and 2007 calculated in the catchment by chosen methods. Chosen methods for the calculation of the spatial distribution of precipitation in the catchment were: arithmetic mean, Thiessen polygons, isohyetal method and inverse distance weighting method. (IDW). As a suitable catchment was selected the catchment of the Kopaninský stream where relatively dense rainfall measurement network is installed. During the solution of this thesis the area of interest had to be extended by the area between the used rain gauges which was delimited in the GIS program. The reason for an enlargement was a fact that a decomposition of rain gauges in the catchment of the Kopaninský stream did not permit the creation of isohyetal lines in this whole catchment. During the calculation and the comparison of results of individual methods was found that results of used methods were surprisingly nearly the same. It was also stated that not even sufficient dense rain gauges network does not guarantee 100% exactness for calculating the amount of the precipitation in the selected area because of their extraordinary variability.

This thesis engages a comparison of the development of atmospheric deposition of nitrogen and phosphorus in different regions of the Czech Republic. The work explains the formation of atmospheric deposition and related terms such as the source of pollution, emissions, air pollution and nitrogen oxides. There are stated types of atmospheric deposition, sampling procedure and the process of determining the size of atmospheric deposition. This work describes the methods that were used in the calculations and evaluations. It also deals with the selection of suitable measuring stations for processing the development of atmospheric nitrogen deposition. The atmospheric deposition of phosphorus were not measured in the territory of the Czech Republic in the period from 2000 to 2010. The aim of this work is to find trends in time series of concentrations of NOx and statistical evaluation of the development of atmospheric nitrogen deposition. For these chapters were selected concentration of nitrogen oxides in the regional towns of the Czech Republic. In the chapter of spatial analyzes of the concentration of nitrogen oxides is selected Ústecký kraj, mainly due to the fact that this area is characterized by significantly high atmospheric deposition of nitrogen and total air pollution. A therm the Black Triangle is often used for this location.

Ostlänken är en planerad höghastighetsjärnväg mellan Järna och Linköping med stegvis byggstart från 2017, ett omfattande projekt som inkluderar totalt cirka 20 kilometer tunnel, varav en tunnel planeras att korsa Getåravinen. Undermarksbyggande är komplicerat och kontakt med svaghetszoner i berg innebär ökade risker för anläggning, miljö och omgivande fastigheter. För att åstadkomma en riskbedömning, dimensionering och ekonomisk kalkyl utförs undersökningar och ingenjörsgeologiska prognoser av bergkvalité och geologisk struktur. Studier har visat ett samband mellan naturliga lineament och svaghetszoner i berg. Därför var uppsatsens syfte att bedöma geologiska byggbarhetsfaktorer genom en strukturgeologisk karaktärisering av lineamentet Getåravinen. Arbetet fokuserade på ett avgränsat område inom Getåravinen där Sweco, på uppdrag av Trafikverket, har utfört ingenjörsgeologiska undersökningar inför den planerade anläggningen av Ostlänken.Arbetet utgick från en studie av lineament genom geologiska kartor, kärnborrprov och hällkarteringar kring Getåravinen, dessa analyserades sedan för att identifiera strukturgeologiska samband. Metoderna Inverse Distance Weighted och kriging användes för att prediktera berget i borrkärnornas omgivning utifrån kvalitéklassningarna Rock Mass Rating, Q-systemet och Rock Quality Designation. För att visa resultatet pedagogiskt och förbättra underlaget inför byggteknisk diskussion visualiserades resultatet i en Building Information Model i mjukvaruprogrammet AutoCAD Civil 3D. Resultatet visade cirka 20° avvikelse mellan lineamentet och huvudsprickriktningen i Getåravinen, det bekräftade ett sprickmönster som då generellt kan antas gälla för lineament vilket är användbar information vid undermarksbyggande. Prediktionerna visade en rumslig beskrivning av bergets klassificeringar och samband med lineamentet. Nyttan av prediktiva metoder var därmed god då även effektiv kvantifiering av bergklasserna möjliggjordes vilket kan implementeras vid undermarksbyggande. / The East Link is a planned high-speed rail link between Järna and Linköping with a gradual construction start in 2017, a comprehensive project covering a total of approximately 20 kilometers of tunnel, of which one tunnel is planned to cross the Getå ravine. Subsurface construction is complicated, and contact with weak zones in the rock implies increased risks for the tunnel construction, the environment and the surrounding real estates. In order to accomplish a risk assessment, dimensioning and economic calculation, geological engineering investigations of the rock’s quality and geological structure are carried out. Studies have shown a relationship between natural lineaments and weaknesses in the rock. Therefore, the purpose of this thesis was to assess geological constructability factors in the Getå ravine by using a characterization of structural geology. The focus was within a demarcated area where the consulting firm Sweco, on behalf of the Swedish Transport Administration, had performed engineering geological surveys for the planned construction of the East link.The thesis was based on a study of lineament in geological maps, core drill samples and bedrock mapping in the surroundings of the Getå ravine, which were later analysed to identify structural geological relations. The methods Inverse Distance Weighted and Kriging were used to predict the rock ambient to the core kernels based on the quality ratings Rock Mass Rating, Q System and Rock Quality Designation. In order to visualize the results pedagogically and enhance the basis for technical discussion regarding civil engineering, the results were visualized in a Building Information Model in the software AutoCAD Civil 3D. The result showed approximately 20° deviation between the lineament and the primary joint strike in the Getå ravine, it confirmed a joint pattern that can be generally assumed for natural lineaments which is useful information in subsurface construction. The predictions showed a spatial distribution of the rock's different quality ratings and relationship with the lineament. The utility of the predictive methods was therefore certain because of efficient quantification of the rock quality was made possible, which can be implemented when characterizing the rock into quality domains prior subsurface construction.

Dioxins are highly toxic persistent environmental pollutants that occur in marine harbour sediments as the results of industrial practices around the world and pose a significant risk to human health. To adequately remediate contaminated sediments, the spatial extent of contamination must first be determined by spatial interpolation. The ability to lower sampling frequency and perform laboratory analysis on fewer samples, yet still produce an adequate pollutant distribution map, would reduce the initial cost of new remediation projects. Fuzzy Set Theory has been shown as a way to reduce uncertainty due to data sparsity and provides an advantageous way to quantify gradational changes like those of pollutant concentrations through fuzzing clustering based approaches; Fuzzy modelling has the ability to utilize these advantages for making spatial predictions. To assess the ability of fuzzy modeling to make spatial predictions using fewer sample points, its predictive ability was compared to Ordinary Kriging (OK) and Inverse Distance Weighting (IDW) under increasingly sparse data conditions. This research used a Takagi-Sugeno (T-S) fuzzy modelling approach with fuzzy c-means clustering to make spatial predictions of lead concentrations in soil to determine the efficacy of the fuzzy model for applications of modeling dioxins in marine sediment. The spatial density of the data used to make the predictions was incrementally reduced to simulate increasingly sparse spatial data conditions. To determine model performance, the data at each increment not used for making the spatial predictions was used as validation data, which the model attempted to predict and the performance was analyzed. Initially, the parameters associated with the T-S fuzzy model were determined by the optimum observed performance, where the combination of parameters that produced the most accurate prediction of the validation data were retained as optimal for each increment of the data reduction. To determine performance Mean Absolute Error, the Coefficient of Determination, and Root Mean Squared Error were selected as metrics. To give each metric equal weighting a binned scoring system was developed where each metric received a score from 1 to 10, the average represented that methods score. The Akaike Information Criterion (AIC) was also employed to determine the effect of the varied validation set lengths on performance. For the T-S fuzzy model as the amount of data used to solve the respective validation set points was reduced the number of clusters was lower and the cluster centres were more spread out, the fuzzy overlap between clusters was larger, and the widths of the membership function in the T-S fuzzy model were wider. Although it was possible to determine an optimal number of clusters, fuzzy overlap, and membership function width that yielded an optimal prediction of the validation data, gain in performance was minor compared to many other combinations of parameters. Therefore, for the data used in this study the T-S fuzzy model was insensitive to parameter choice. For OK, as the data was reduced, the range of spatial dependence in the data from variography became lower, and for IDW the power parameters optimal value became lower to give a greater weighting to more widely spread points. For the TS fuzzy model, OK, and IDW the increasingly sparse data conditions resulted in an increasingly poor model performance for all metrics. This was supported by AIC values for each method at each increment of the data reduction that were within 1 point of each other. The ability of the methods to predict outlier points and reproduce the variance in the validation sets was very similar and overall quite poor. Based on the scoring system IDW did exhibit a slight outperformance of the T-S fuzzy model, which slightly outperformed OK. However, the scoring system employed in this research was overly sensitive and so was only useful for assessing relative performance. The performance of the T-S model was very dependent on the number of outliers in the respective validation set. For modeling under sparse data conditions, the T-S fuzzy modeling approach using FCM clustering and constant width Gaussian shaped membership functions used in this research did not show any advantages over IDW and OK for the type of data tested. Therefore, it was not possible to speculate on a possible reduction in sampling frequency for delineating the extent of contamination for new remediation projects. / Graduate

Spatial interpolation techniques play an important role in hydrology as many point observations need to be interpolated to create continuous surfaces. Despite the availability of several tools and methods for interpolating data, <a>not all of them work consistently for hydrologic applications</a><a>. One of the techniques, Laplace Equation, which is used in hydrology for creating flownets, has rarely been used for interpolating hydrology data</a>. The objective of this study is to examine the efficiency of Laplace formulation (LF) in interpolating hydrologic data and compare it wih other widely used methods such as the inverse distance weighting (IDW), natural neighbor, and kriging. Comparison is performed quantitatively for using root mean square error (RMSE), visually for creating reasonable surfaces and computationally for ease of operation and speed. Data related to surface elevation, river bathymetry, precipitation, temperature, and soil moisture data are used for different areas in the United States. RMSE results show that LF performs better than IDW and is comparable to other methods for accuracy. LF is easy to use as it requires fewer input parameters compared to IDW and Kriging. Computationally, LF is comparable to other methods in terms of speed when the datasets are not large. Overall, LF offers an robust alternative to existing methods for interpolating various hydrology data. Further work is required to improve its computational efficiency with large data size and find out the effects of different cell size.

Ce mémoire de thèse concerne l'interpolation sur les variétés de Grassmann et ses applications à la réduction de modèles en mécanique et plus généralement aux systèmes d'équations aux dérivées partielles d'évolution. Après une description de la méthode POD, nous introduisons les fondements théoriques en géométrie des variétés de Grassmann, qui seront utilisés dans le reste de la thèse. Ce chapitre donne à ce mémoire à la fois une rigueur mathématique au niveau des algorithmes mis au point, leur domaine de validité ainsi qu'une estimation de l'erreur en distance grassmannienne, mais également un caractère auto-contenu "self-contained" du manuscrit. Ensuite, on présente la méthode d'interpolation sur les variétés de Grassmann introduite par David Amsallem et Charbel Farhat. Cette méthode sera le point de départ des méthodes d'interpolation que nous développerons dans les chapitres suivants. La méthode de Amsallem-Farhat consiste à choisir un point d'interpolation de référence, envoyer l'ensemble des points d'interpolation sur l'espace tangent en ce point de référence via l'application logarithme géodésique, effectuer une interpolation classique sur cet espace tangent, puis revenir à la variété de Grassmann via l'application exponentielle géodésique. On met en évidence par des essais numériques l'influence du point de référence sur la qualité des résultats. Dans notre premier travail, nous présentons une version grassmannienne d'un algorithme connu dans la littérature sous le nom de Pondération par Distance Inverse (IDW). Dans cette méthode, l'interpolé en un point donné est considéré comme le barycentre des points d'interpolation où les coefficients de pondération utilisés sont inversement "proportionnels" à la distance entre le point considéré et les points d'interpolation. Dans notre méthode, notée IDW-G, la distance géodésique sur la variété de Grassmann remplace la distance euclidienne dans le cadre standard des espaces euclidiens. L'avantage de notre algorithme, dont on a montré la convergence sous certaines conditions assez générales, est qu'il ne requiert pas de point de référence contrairement à la méthode de Amsallem-Farhat. Pour remédier au caractère itératif (point fixe) de notre première méthode, nous proposons une version directe via la notion de barycentre généralisé. Notons enfin que notre algorithme IDW-G dépend nécessairement du choix des coefficients de pondération utilisés. Dans notre second travail, nous proposons une méthode qui permet un choix optimal des coefficients de pondération, tenant compte de l'auto-corrélation spatiale de l'ensemble des points d'interpolation. Ainsi, chaque coefficient de pondération dépend de tous les points d'interpolation et non pas seulement de la distance entre le point considéré et un point d'interpolation. Il s'agit d'une version grassmannienne de la méthode de Krigeage, très utilisée en géostatique. La méthode de Krigeage grassmannienne utilise également le point de référence. Dans notre dernier travail, nous proposons une version grassmannienne de l'algorithme de Neville qui permet de calculer le polynôme d'interpolation de Lagrange de manière récursive via l'interpolation linéaire entre deux points. La généralisation de cet algorithme sur une variété grassmannienne est basée sur l'extension de l'interpolation entre deux points (géodésique/droite) que l'on sait faire de manière explicite. Cet algorithme ne requiert pas le choix d'un point de référence, il est facile d'implémentation et très rapide. De plus, les résultats numériques obtenus sont remarquables et nettement meilleurs que tous les algorithmes décrits dans ce mémoire. / This dissertation deals with interpolation on Grassmann manifolds and its applications to reduced order methods in mechanics and more generally for systems of evolution partial differential systems. After a description of the POD method, we introduce the theoretical tools of grassmannian geometry which will be used in the rest of the thesis. This chapter gives this dissertation a mathematical rigor in the performed algorithms, their validity domain, the error estimate with respect to the grassmannian distance on one hand and also a self-contained character to the manuscript. The interpolation on Grassmann manifolds method introduced by David Amsallem and Charbel Farhat is afterward presented. This method is the starting point of the interpolation methods that we will develop in this thesis. The method of Amsallem-Farhat consists in chosing a reference interpolation point, mapping forward all interpolation points on the tangent space of this reference point via the geodesic logarithm, performing a classical interpolation on this tangent space and mapping backward the interpolated point to the Grassmann manifold by the geodesic exponential function. We carry out the influence of the reference point on the quality of the results through numerical simulations. In our first work, we present a grassmannian version of the well-known Inverse Distance Weighting (IDW) algorithm. In this method, the interpolation on a point can be considered as the barycenter of the interpolation points where the used weights are inversely proportional to the distance between the considered point and the given interpolation points. In our method, denoted by IDW-G, the geodesic distance on the Grassmann manifold replaces the euclidean distance in the standard framework of euclidean spaces. The advantage of our algorithm that we show the convergence undersome general assumptions, does not require a reference point unlike the method of Amsallem-Farhat. Moreover, to carry out this, we finally proposed a direct method, thanks to the notion of generalized barycenter instead of an earlier iterative method. However, our IDW-G algorithm depends on the choice of the used weighting coefficients. The second work deals with an optimal choice of the weighting coefficients, which take into account of the spatial autocorrelation of all interpolation points. Thus, each weighting coefficient depends of all interpolation points an not only on the distance between the considered point and the interpolation point. It is a grassmannian version of the Kriging method, widely used in Geographic Information System (GIS). Our grassmannian Kriging method require also the choice of a reference point. In our last work, we develop a grassmannian version of Neville's method which allow the computation of the Lagrange interpolation polynomial in a recursive way via the linear interpolation of two points. The generalization of this algorithm to grassmannian manifolds is based on the extension of interpolation of two points (geodesic/straightline) that we can do explicitly. This algorithm does not require the choice of a reference point, it is easy to implement and very quick. Furthermore, the obtained numerical results are notable and better than all the algorithms described in this dissertation.

This thesis presents a comparison of the development of the concentrations of nitrogen oxides and sulfur dioxide in the air in the Usti region. The work described generally air pollution and related terms such as the sources of pollution emission, immission and atmospheric deposition. The study also evaluated pollution nitrogen oxides and sutur dioxide in the Usti region. This work describes the methods that were used in the calculations and evaluations. It also deals with the selecting appropriate measuring stations to handle the development of the concentrations of these substances in the air. The aim of this work is to find trends in time series of concentrations of NOx and SO2 and statistical evaluation of development levels in selected cities Ústí Region. The next section is evaluated spatial analysis of the concentration of nitrogen oxides and sulfur dioxide in the air in the Ústí Region. This area was chosen primarily because it is characterized by significant air pollution.