<p>This dissertation places intersection homology and local homology within the framework of persistence, which was originally developed for ordinary homology by Edelsbrunner, Letscher, and Zomorodian. The eventual goal, begun but not completed here, is to provide analytical tools for the study of embedded stratified spaces, as well as for high-dimensional and possibly noisy datasets for which the number of degrees of freedom may vary across the parameter space. Specifically, we create a theory of persistent intersection homology for a filtered stratified space and prove several structural theorems about the pair groups asso- ciated to such a filtration. We prove the correctness of a cubic algorithm which computes these pair groups in a simplicial setting. We also define a series of intersec- tion homology elevation functions for an embedded stratified space and characterize their local maxima in dimension one. In addition, we develop a theory of persistence for a multi-scale analogue of the local homology groups of a stratified space at a point. This takes the form of a series of local homology vineyards which allow one to assess the homological structure within a one-parameter family of neighborhoods of the point. Under the assumption of dense sampling, we prove the correctness of this assessment at a variety of radius scales.</p> / Dissertation

Laboratory experiments were conducted to investigate the propagation of internal solitary waves on a uniform slope in a two-layered free surface fluid system. The laboratory facilities employed in this study is the first in Taiwan, which include a stainless steel wave flume (dimensions: 12 meters long with cross-section 0.5 m wide and 0.7m deep) and experimental apparatus for generating and measuring internal waves. The flume incorporates a movable vertical gate at one end for generating internal solitary waves, and a uniform slope (either £c = 30o, 50o, 60o, 90o, 120o or 130o) at the other end. The upper layer had fresh water with density £l1 (999kg/m3), to a depth H1; the lower layer was saline brine density £l2 (1030 kg/m3), which was slowly filled into the flume to a depth of H2 by gravity through several openings at the bottom of the flume, Boussinesq parameter . A mini pump was used to remove a small quantity of fresh water from one side of the vertical gate to another side. By creating a prescribed difference £bo in the interface levels on either side of the gate beforehand, internal solitary wave was generated by the mechanism of overturning the brine and fresh water behind the movable gate. Five ultrasonic probes at equidistant distance recorded the interface fluctuations, one density probe measured the change of density at the interface, while two electrical capacitance gauges for the free surface displacements likely to occur. Digital cameras were also used to record the motions of internal wave in the flume and on the slope for further analysis. Laboratory test on internal solitary wave were arranged from one of the combinations using different layer thickness ratios H1/H2, interface differences £bo, density ratios £l1/£l2, and bottom slopes £c. In addition to internal solitary wave reflection from a uniform slope, laboratory investigations included internal wave propagation on a rigid impermeable bottom and evolution on a uniform slope. Keeping the total water depth in the flume at H = 40cm, an increase in the depth parameter |H2-H1|/H produced large internal wave amplitude, reduced phase velocity, and enhanced soliton feature. From the experimental result analyzed, it suggests that the Korteweg-de Vries (KdV) theory fits solitary waves of small amplitude, and the modified KdV is suitable for large amplituded waves. Considering wave motion in an inviscid fluid, the dissipation of internal solitary waves propagating in a flume may occur through bottom friction and wave breaking. Subjected to bottom friction alone, the amplitude of most internal solitary waves in the experiments decayed approximate by 10% over a journey of 6 meters. Two types of wave breaking mechanism were found to produce strong mixing and local vortex in the fluid, causing significant energy losses. For internal solitary waves of large amplitudes, reflection coefficient for wave amplitude or energy decreased, as amplitude or energy increased. Under this condition, however, the reflection coefficient due to bottom friction may be assumed as constant. Using the experimental results obtained, empirical equation is now proposed to account for wave dissipation due to for non-breaking internal waves. The equation indicates that decrease in reflection coefficient as wave amplitude or energy increases may be expressed using a second order polynomial. Overall, experimental results suggest that good agreement can be found between experimental data and the empirical equation so derived. Upon assuming the wave reflection coefficient is solely dependent on the incoming wave amplitude or energy, prediction for reflection coefficient can be calculated in a straight forward manner. Either large-scale, high-frequency internal wave motion or internal solitary waves have been observed in natural lakes. The observed rapid decay of internal wave energy after severe breaking events seemed to be mostly due to dissipation on various sloping boundaries in a lake. From the basic laboratory experiments on internal wave reflection from various single slopes, the results many benefit provide researchers to promote further research on practical applications related to limnology.

The threats imposed by the ongoing climate change become successively more clear. In the municipality of Sandviken the possible rise in water level in the lake Storsjön might cause major problems. This study aims to produce an elevation model over one of the threatened areas so that the effect of different water levels in the lake can be studied in the future. The area of study consists of an esker cutting straight through the lake Storsjön. The area is densely populated and the size is about 13 hectares. Measurements were performed, both with GPS, utilizing Network-RTK, and with total station. The formations of the ridge were captured by following the breaklines with a point density of 225 points/ha as an intended goal. The GPS measurements were transformed into the local coordinate system of Sandviken municipality. The measured coordinates for six control points were compared to the true coordinates. The fit was not perfect, so a Helmert transformation was performed on the plane coordinates. The accuracy of the measurements was below 50 mm after the transformation. Net adjustments in plane and height were performed on the total station measurements. The error ellipses for the station points varied from 2 to 44 mm. The over all point density achieved was 274 points/ha. Two elevation models were created, one Triangular Irregular Network (TIN) and one interpolated model using a kriging interpolation. Contour lines from both models were produced and compared with each other. The TIN creates a more angular surface and therefore the contour lines are not as smooth as in the interpolated model. Both models showed good resemblance compared with the original data. / Problem relaterade till klimatförändringar blir alltmer uttalade. I Sandvikens kommun kan problem uppstå om vattennivåerna i Storsjön stiger. Syftet med det här examensarbetet var att skapa en höjdmodell över ett av de utsatta områdena för att effekter av en eventuell vattennivåhöjning skall kunna studeras i framtiden. Området består av en rullstensås som sträcker sig rakt igenom Storsjön. Området är tättbefolkat och ca 13 hektar stort. Mätningarna utfördes med både GPS och totalstation. Vid GPS-mätningarna utnyttjades Nätverks-RTK. Åsens formationer fångades genom att följa brytlinjer med en estimerad punkttäthet på 225 punkter/ha. GPS-mätningarna transformerades till Sandvikens lokala koordinatsystem. De mätta koordinaterna för sex kontrollpunkter jämfördes med de sanna koordinaterna. Eftersom de mätta värdena inte passade in perfekt så utfördes en Helmerttransformation av plankoordinaterna. Noggrannheten på mätta värden var inom 50 mm efter transformationen. Nätutjämningar i plan och höjd gjordes på totalstationsmätningarna. Felellipserna för stationspunkterna varierade mellan 2 och 44 mm. Den sammanlagda punkttätheten uppgick till 274 punkter/ha. Två höjdmodeller framställdes, en TIN-model (Triangular Irregular Network) och en interpolerad model där interpolationsmetoden kriging användes. Höjdkurvor framställda från respektive modell skapades och jämfördes med varandra. Ett TIN ger en mer kantig yta vilket återspeglas i höjdkurvorna som inte är lika jämna som de från den interpolerade modellen. Båda modellerna visade god överensstämmelse med originaldata.

The Province of British Columbia is developing a suite of attributes to assess and monitor forest sustainability. Each attribute is in turn evaluated using a variety of indicators. Recently, digital remote sensing technologies have emerged as both alternative and supplement to traditional monitoring techniques, with light detection and ranging (lidar) in particular showing great promise for estimating a variety of indicators. The goal of this thesis was to review and assess the ability of lidar to estimate selected indicators of forest sustainability. Specifically, digital elevation model (DEM) interpolation (from which indicators are extracted both directly and indirectly) and wildlife tree class distributions were examined. Digital elevation models are a key derivative of lidar data, and their generation is a critical step in the data processing stream. A validation exercise was undertaken to determine which combination of interpolation routine and spatial resolution was the most accurate. Ground returns were randomly subsetted into prediction and validation datasets. Linear, quintic, natural neighbour, spline with tension, regularized spline, inverse distance weighting, and ANUDEM interpolation routines were used to generate surfaces at spatial resolutions of 0.5, 1.0, and 1.5 m. The 0.5 m natural neighbour surface was found to be the most accurate (RMSE=0.17 m). Classification and regression tree analysis indicated that slope and ground return density were the best predictors of interpolation error. The amount and variability of living and dead wood in a forest stand is an important indicator of forest biodiversity. In the second study, the capacity of lidar to estimate the distribution of living and dead trees within forests is investigated. Twenty-two field plots were established in which each stem (DBH>10cm) was assigned to a wildlife tree (WT) class. For each plot, a suite of lidar-derived predictor variables were extracted. Ordinal logistic regression was then employed to predict the cumulative proportions of stems within the WT classes. Results indicated that the coefficient of variation of the lidar height data was the best predictor variable (r = 0.85, p <0.000, RMSE = 4.9%). The derived relationships allowed for the prediction of the proportion of stems within WT classes across the landscape.

In the context of growing tourism and global warming, the fragile landscape of the Himalayas is under immense pressure because of rapid land cover changes in developing countries like Nepal. Remotely sensed data combined with ethnographic knowledge are useful tools for studying such changes. The quantitative change can be measured analyzing satellite images whereas local people’s perceptions provide supportive information. To measure such changes in Sagarmatha National Park of Nepal, Multispectral Scanner (MSS) and Thematic Mapper (TM) images since 1972 were used. Normalized Difference Vegetation Index (NDVI) was calculated for different elevation classes and land cover types. These measurements, along with land cover change (1992- 2006) analysis, shows a significant conversion of the areas covered by ice, shrub and grass to rock and soil. Factors including political conflict due to a Maoist rebellion group, inactive park management, increasing tourist demand, and consequent natural resources exploitation helped to explain the change in the forested areas. This is supported by the information from short, informal, semi-structured interviews with local people. However, the local people are unaware of global warming, which has caused the ice melting and glacial lake expansion. Although global causes are out of the immediate control of land managers, better management practices and managed tourism might help alleviate deteriorating Himalayan ecosystems.

The Province of British Columbia is developing a suite of attributes to assess and monitor forest sustainability. Each attribute is in turn evaluated using a variety of indicators. Recently, digital remote sensing technologies have emerged as both alternative and supplement to traditional monitoring techniques, with light detection and ranging (lidar) in particular showing great promise for estimating a variety of indicators. The goal of this thesis was to review and assess the ability of lidar to estimate selected indicators of forest sustainability. Specifically, digital elevation model (DEM) interpolation (from which indicators are extracted both directly and indirectly) and wildlife tree class distributions were examined. Digital elevation models are a key derivative of lidar data, and their generation is a critical step in the data processing stream. A validation exercise was undertaken to determine which combination of interpolation routine and spatial resolution was the most accurate. Ground returns were randomly subsetted into prediction and validation datasets. Linear, quintic, natural neighbour, spline with tension, regularized spline, inverse distance weighting, and ANUDEM interpolation routines were used to generate surfaces at spatial resolutions of 0.5, 1.0, and 1.5 m. The 0.5 m natural neighbour surface was found to be the most accurate (RMSE=0.17 m). Classification and regression tree analysis indicated that slope and ground return density were the best predictors of interpolation error. The amount and variability of living and dead wood in a forest stand is an important indicator of forest biodiversity. In the second study, the capacity of lidar to estimate the distribution of living and dead trees within forests is investigated. Twenty-two field plots were established in which each stem (DBH>10cm) was assigned to a wildlife tree (WT) class. For each plot, a suite of lidar-derived predictor variables were extracted. Ordinal logistic regression was then employed to predict the cumulative proportions of stems within the WT classes. Results indicated that the coefficient of variation of the lidar height data was the best predictor variable (r = 0.85, p <0.000, RMSE = 4.9%). The derived relationships allowed for the prediction of the proportion of stems within WT classes across the landscape.

Digital Elevation Models (DEMs) are fundamental datasets for environmental modelling. They provide the basic data from which terrain indices that represent or influence environmental phenomena are derived, for example slope gradient and hydrological contributing area, and also the source from which specific morphometric features are quantified and characterised, for example mountains and drainage basins. This thesis focuses on the latter, with the aim being to develop an algorithm to characterise the landscape in terms of five morphometric features (peaks, passes, pits, ridges and valleys) and to assess its validity and effectiveness for characterising landform from DEMs. The research in this thesis is divided into two parts. First, an algorithm of morphometric characterisation of landform from OEMs is developed based on a locally fitted quadratic surface and its positional relationship with the analysis window. Five requirements are taken into account within the algorithm: (1) the ideal cases of different morphometric features are simply and clearly defined; (2) the output is spatially continuous to reflect the inherent fuzziness of landform features; (3) the output is easily combined into a multi-scale index across a range of operational scales; (4) the standard general morphometric parameters can be easily quantified due to the easy calculation of first and second order derivatives from the quadratic surface; and (5) the algorithm is applicable to the different data structures used to represent DEMs. An additional benefit of the quadratic surface is the derivation of the R?? goodness-of-fit statistic, which allows both an assessment of the reliability of the results and the complexity of the terrain. Of the five morphometric features identified using the algorithm, valleys are perhaps the most commonly used. Therefore the second part of this thesis is a more detailed comparison between the Multi-Scale Valleyness (MSV) and three existing algorithms (D8, D∞ and MrVBF). D8 and D∞ are global flow accumulation algorithms, and perform well when characterising valley centre lines. However, they do not identify the valley areas themselves, although this is to be expected given their formulation. MrVBF focuses on characterising valley bottoms and so performs well when characterising valleys in broad and topographically flat areas. It does not identify valleys in the steeper upland parts of a catchment, although this too is something to be expected given its formulation. MSV directly characterises valley areas from a geomorphometric point of view, and performs well for both upland and lowland catchments, irrespective of their width. Overall, the results show that the single- and multi-scale terrain indices developed in this research perform well when characterising the five morphometric features. The approach has considerable potential for use in environmental modelling and terrain analysis.

O presente trabalho propõe uma metodologia para a elaboração de modelos digitais de elevação e ortofotos, em média e pequena escala, de maneira automática, utilizando correlação de imagens digitais, a partir de fotografias aéreas métricas. Aborda conceitos básicos de Fotogrametria e correlação de imagens digitais, e lança o conceito da janela epipolar como ferramenta auxiliar na busca e correlação. Estas janelas permitem obter a máxima correlação simultânea entre duas ou mais imagens, como também a varredura em qualquer direção. Isso torna bastante robusto o método de elaboração dos produtos propostos. Esta metodologia foi implementada em um caso prático onde se utilizou uma cobertura aerofotogramétrica em escala 1:25.000. Os produtos foram validados, a partir de elementos de referência retirados de outra cobertura em escala 1:5.000. As precisões atenderam os requisitos do PEC Classe A para a escala 1:2.500 em planimetia, e intervalos de curvas de nível de 5 metros de eqüidistância, em altimetria. / This work presents a methodology for the production of digital elevation model and orthophotos on medium and small scales, with an automatic manner, using the concepts of digital image correlation. It presents basic fundamentals of photogrammetry and digital image correlation, and launches the concept of epipolar window like an auxiliary search and correlation tool. These windows allow the simultaneous multiple image correlation, in different direction scanning. This makes quite robust the production method of the proposed products. This methodology was implemented in a practical case, where 1:25.000 scale photogrammetric images were used. The products were validated using a 1:5.000 scale coverage as reference. The accuracy and precision achieved are into the Brazilian Cartography Standards as \"Classe A\" for the 1:2.500 scale, in planimmetry, and 5 meter contour lines interval in altimetry.

M.Sc. / Real-time terrain rendering (RTTR) is an exciting eld in computer graphics. The algorithms and techniques developed in this domain allow immersive virtual environments to be created for interactive applications. Many di culties are encountered in this eld of research, including acquiring the data to model virtual worlds, handling huge amounts of geometry, and texturing landscapes that appear to go on forever. RTTR has been widely studied, and powerful methodologies have been developed to overcome many of these obstacles. Complex natural terrain features such as detailed vertical surfaces, overhangs and caves, however, are not easily supported by the majority of existing algorithms. It becomes di cult to add such detail to a landscape. Existing techniques are incredibly e cient at rendering elevation data, where for any given position on a 2D horizontal plane we have exactly 1 altitude value. In this case we have a many-to-1 mapping between 2D position and altitude, as many 2D coordinates may map to 1 altitude value but any single 2D coordinate maps to 1 and only 1 altitude. In order to support the features mentioned above we need to allow for a many-to-many mapping. As an example, with a cave feature for a given 2D coordinate we would have elevation values for the oor, the roof and the outer ground. In this dissertation we build upon established techniques to allow for this manyto- many mapping, and thereby add support for complex terrain features. The many-to-many mapping is made possible by making use of geometry images in place of height-maps. Another common problem with existing RTTR algorithms is texture distortion. Texturing is an inexpensive means of adding detail to rendered terrain. Many existing technique map texture coordinates in 2D, leading to distortion on steep surfaces. Our research attempts to reduce texture distortion in such situations by allowing a more even spread of texture coordinates. Geometry images make this possible as they allow for a more even distribution of sample positions. Additionally we devise a novel means of blending tiled texture that enhances the important features of the individual textures. Fully sampled terrain employs a single global texture that covers the entire landscape. This technique provides great detail, but requires a huge volume of data. Tiled texturing requires comparatively little data, but su ers from disturbing regular patterns. We seek to reduce the gap between tiled textures and fully sampled textures. In particular, we aim at reducing the regularity of tiled textures by changing the blending function. In summary, the goal of this research is twofold. Firstly we aim to support complex natural terrain features|speci cally detailed vertical surfaces, over-hangs and caves. Secondly we wish to improve terrain texturing by reducing texture distortion, and by blending tiled texture together in a manner that appears more natural. We have developed a level of detail algorithm which operates on geometry images, and a new texture blending technique to support these goals.

High-elevation environments above 2500 metres above sea level (m.a.s.l.) were among the planet's last frontiers of human colonization. Research on the speed and tempo of this colonization process is active and holds implications for understanding rates of genetic, physiological and cultural adaptation in our species. Permanent occupation of high-elevation environments in the Andes Mountains of South America tentatively began with hunter-gatherers around 9 ka according to current archaeological estimates, though the timing is currently debated. Recent observations on the archaeological site of Soro Mik'aya Patjxa (8.0-6.5 ka), located at 3800 m.a.s.l. in the Andean Altiplano, offer an opportunity to independently test hypotheses for early permanent use of the region. This study observes low oxygen (delta O-18) and high carbon (delta C-13) isotope values in human bone, long travel distances to low-elevation zones, variable age and sex structure in the human population and an absence of non-local lithic materials. These independent lines of evidence converge to support a model of permanent occupation of high elevations and refute logistical and seasonal use models. The results constitute the strongest empirical support to date for permanent human occupation of the Andean highlands by hunter-gatherers before 7 ka.