Soil is extremely important for providing food, biomass and raw materials, water and nutrient storage; supporting biodiversity and providing foundations for man-made structures. However, its health is threatened by human activities, which can greatly affect the potential of soils to fulfil their functions and, consequently, result in environmental, economic and social damage. These issues require the characterisation of the impact and spatial extent of the problems. This can be achieved through the creation of detailed and comprehensive soil maps that describe both the spatial and vertical variability of key soil properties. Detailed three-dimensional (3D) digital soil maps can be readily used and embedded into environmental models. Three-dimensional soil mapping is not a new concept. However, only with the recent development of more powerful computers has it become feasible to undertake such data processing. Common techniques to estimate soil properties in the three-dimensional space include geostatistical interpolation, or a combination of depth functions and geostatistics. However, these two methods are both partially flawed. Geostatistical interpolation and kriging in particular, estimate soil properties in unsampled locations using a weighted average of the nearby observations. In order to produce the best possible estimate, this form of interpolation minimises the variance of each weighted average, thus decreasing the standard deviation of the estimates, compared to the soil observations. This appears as a smoothing effect on the data and, as a consequence, kriging interpolation is not reliable when the dataset is not sampled with a sampling designs optimised for geostatistics. Depth function approaches, as they are generally applied in literature, implement a spline regression of the soil profile data that aims to better describe the changes of the soil properties with depth. Subsequently, the spline is resampled at determined depths and, for each of these depths, a bi-dimensional (2D) geostatistical interpolation is performed. Consequently, the 3D soil model is a combination of a series of bi-dimensional slices. This approach can effectively decrease or eliminate any smoothing issues, but the way in which the model is created, by combining several 2D horizontal slices, can potentially lead to erroneous estimations. The fact that the geostatistical interpolation is performed in 2D implies that an unsampled location is estimated only by considering values at the same depth, thus excluding the vertical variability from the mapping, and potentially undermining the accuracy of the method. For these reasons, the literature review identified a clear need for developing, a new method for accurately estimating soil properties in 3D – the target of this research, The method studied in this thesis explores the concept of soil specific depth functions, which are simple mathematical equations, chosen for their ability to describe the general profile pattern of a soil dataset. This way, fitting the depth function to a particular sample becomes a diagnostic tool. If the pattern shown in a particular soil profile is dissimilar to the average pattern described by the depth function, it means that in that region there are localised changes in the soil profiles, and these can be identified from the goodness of fit of the function. This way, areas where soil properties have a homogeneous profile pattern can be easily identified and the depth function can be changed accordingly. The application of this new mapping technique is based on the geostatistical interpolation of the depth function coefficients across the study area. Subsequently, the equation is solved for each interpolated location to create a 3D lattice of soil properties estimations. For this way of mapping, this new methodology was denoted as top-down mapping method. The methodology was assessed through three case studies, where the top-down mapping method was developed, tested, and validated. Three datasets of diverse soil properties and at different spatial extents were selected. The results were validated primarily using cross-validation and, when possible, by comparing the estimates with independently sampled datasets (independent validation). In addition, the results were compared with estimates obtained using established literature methods, such as 3D kriging interpolation and the spline approach, in order to define some basic rule of application. The results indicate that the top-down mapping method can be used in circumstances where the soil profiles present a pattern that can be described by a function with maximum three coefficients. If this condition is met, as it was with key soil properties during the research, the top-down mapping method can be used for obtaining reliable estimates at different spatial extents.

Tato práce se zabývá metodou úplně nejmenších čtverc·, která slouží pro odhad parametr· v lineárních modelech. V práci je uveden základní popis metody a její asymptotické vlastnosti. Je vysvětleno, jakým zp·sobem lze v konceptu metody využít neparametrický bootstrap pro hledání odhadu. Vlastnosti bootstrap od- had· jsou pak simulovány na pseudo náhodně vygenerovaných datech. Simulace jsou prováděny pro dvourozměrný parametr v r·zných nastaveních základního modelu. Jednotlivé bootstrap odhady jsou v rovině řazeny pomocí Mahalanobis a Tukey statistical depth function. Simulace potvrzují, že bootstrap odhad dává dostatečně dobré výsledky, aby se dal využít pro reálné situace.

Models are important tools for understanding the hydrological processes that govern water transport in the landscape and for prediction at times and places where no observations are available. The degree of trust placed on models, however, should not exceed the quality of the data they are fed with. The overall aim of this thesis was to tune the modeling process to account for the uncertainty in the data, by identifying robust parameter values using methods from computational geometry. The methods were developed and tested on data from the Choluteca River basin in Honduras. Quality control of precipitation and discharge data resulted in a rejection of 22% percent of daily raingage data and the complete removal of one out of the seven discharge stations analyzed. The raingage network was not found sufficient to capture the spatial and temporal variability of precipitation in the Choluteca River basin. The temporal variability of discharge was evaluated through a Monte Carlo assessment of the rating-equation parameter values over a moving time window of stage-discharge measurements. Al hydrometric stations showed considerable temporal variability in the stage-discharge relationship, which was largest for low flows, albeit with no common trend. The problem with limited data quality was addressed by identifying robust model parameter values within the set of well-performing (behavioral) parameter-value vectors with computational-geometry methods. The hypothesis that geometrically deep parameter-value vectors within the behavioral set were hydrologically robust was tested, and verified, using two depth functions. Deep parameter-value vectors tended to perform better than shallow ones, were less sensitive to small changes in their values, and were better suited to temporal transfer. Depth functions rank multidimensional data. Methods to visualize the multivariate distribution of behavioral parameters based on the ranked values were developed. It was shown that, by projecting along a common dimension, the multivariate distribution of behavioral parameters for models of varying complexity could be compared using the proposed visualization tools. This has a potential to aid in the selection of an adequate model structure considering the uncertainty in the data. These methods allowed to quantify observational uncertainties. Geometric methods have only recently begun to be used in hydrology. It was shown that they can be used to identify robust parameter values, and some of their potential uses were highlighted. / Modeller är viktiga verktyg för att förstå de hydrologiska processer som bestämmer vattnets transport i landskapet och för prognoser för tider och platser där det saknas mätdata. Graden av tillit till modeller bör emellertid inte överstiga kvaliteten på de data som de matas med. Det övergripande syftet med denna avhandling var att anpassa modelleringsprocessen så att den tar hänsyn till osäkerheten i data och identifierar robusta parametervärden med hjälp av metoder från beräkningsgeometrin. Metoderna var utvecklade och testades på data från Cholutecaflodens avrinningsområde i Honduras. Kvalitetskontrollen i nederbörds- och vattenföringsdata resulterade i att 22 % av de dagliga nederbördsobservationerna måste kasseras liksom alla data från en av sju analyserade vattenföringsstationer. Observationsnätet för nederbörd befanns otillräckligt för att fånga upp den rumsliga och tidsmässiga variabiliteten i den övre delen av Cholutecaflodens avrinningsområde. Vattenföringens tidsvariation utvärderades med en Monte Carlo-skattning av värdet på parametrarna i avbördningskurvan i ett rörligt tidsfönster av vattenföringsmätningar. Alla vattenföringsstationer uppvisade stor tidsvariation i avbördningskurvan som var störst för låga flöden, dock inte med någon gemensam trend. Problemet med den måttliga datakvaliteten bedömdes med hjälp av robusta modellparametervärden som identifierades med hjälp av beräkningsgeometriska metoder. Hypotesen att djupa parametervärdesuppsättningar var robusta testades och verifierades genom två djupfunktioner. Geometriskt djupa parametervärdesuppsättningar verkade ge bättre hydrologiska resultat än ytliga, var mindre känsliga för små ändringar i parametervärden och var bättre lämpade för förflyttning i tiden. Metoder utvecklades för att visualisera multivariata fördelningar av välpresterande parametrar baserade på de rangordnade värdena. Genom att projicera längs en gemensam dimension, kunde multivariata fördelningar av välpresterande parametrar hos modeller med varierande komplexitet jämföras med hjälp av det föreslagna visualiseringsverktyget. Det har alltså potentialen att bistå vid valet av en adekvat modellstruktur som tar hänsyn till osäkerheten i data. Dessa metoder möjliggjorde kvantifiering av observationsosäkerheter. Geometriska metoder har helt nyligen börjat användas inom hydrologin. I studien demonstrerades att de kan användas för att identifiera robusta parametervärdesuppsättningar och några av metodernas potentiella användningsområden belystes.