General Circulation Models (GCMs) are tools designed to simulate time series of climate variables globally, accounting for effects of greenhouse gases in the atmosphere. They attempt to represent the physical processes in the atmosphere, ocean, cryosphere and land surface. They are currently the most credible tools available for simulating the response of the global climate system to increasing greenhouse gas concentrations, and to provide estimates of climate variables (e.g. air temperature, precipitation, wind speed, pressure etc.) on a global scale. GCMs demonstrate a significant skill at the continental and hemispheric spatial scales and incorporate a large proportion of the complexity of the global system; they are, however, inherently unable to represent local subgrid-scale features and dynamics. The spatial scale on which a GCM can operate (e.g., 3.75° longitude x 3.75° latitude for Coupled Global Climate Model, CGCM2) is very coarse compared to that of a hydrologic process (e.g., precipitation in a region, streamflow in a river etc.) of interest in the climate change impact assessment studies. Moreover, accuracy of GCMs, in general, decreases from climate related variables, such as wind, temperature, humidity and air pressure to hydrologic variables such as precipitation, evapotranspiration, runoff and soil moisture, which are also simulated by GCMs. These limitations of the GCMs restrict the direct use of their output in hydrology. This thesis deals with developing statistical downscaling models to assess climate change impacts and methodologies to address GCM and scenario uncertainties in assessing climate change impacts on hydrology. Downscaling, in the context of hydrology, is a method to project the hydrologic variables (e.g., rainfall and streamflow) at a smaller scale based on large scale climatological variables (e.g., mean sea level pressure) simulated by a GCM. A statistical downscaling model is first developed in the thesis to predict the rainfall over Orissa meteorological subdivision from GCM output of large scale Mean Sea Level Pressure (MSLP). Gridded monthly MSLP data for the period 1948 to 2002, are obtained from the National Center for Environmental Prediction/ National Center for Atmospheric Research (NCEP/NCAR) reanalysis project for a region spanning 150 N -250 N in latitude and 800 E -900 E in longitude that encapsulates the study region. The downscaling model comprises of Principal Component Analysis (PCA), Fuzzy Clustering and Linear Regression. PCA is carried out to reduce the dimensionality of the larger scale MSLP and also to convert the correlated variables to uncorrelated variables. Fuzzy clustering is performed to derive the membership of the principal components in each of the clusters and the memberships obtained are used in regression to statistically relate MSLP and rainfall. The statistical relationship thus obtained is used to predict the rainfall from GCM output. The rainfall predicted with the GCM developed by CCSR/NIES with B2 scenario presents a decreasing trend for non-monsoon period, for the case study. Climate change impact assessment models developed based on downscaled GCM output are subjected to a range of uncertainties due to both ‘incomplete knowledge’ and ‘unknowable future scenario’ (New and Hulme, 2000). ‘Incomplete knowledge’ mainly arises from inadequate information and understanding about the underlying geophysical process of global change, leading to limitations in the accuracy of GCMs. This is also termed as GCM uncertainty. Uncertainty due to ‘unknowable future scenario’ is associated with the unpredictability in the forecast of socio-economic and human behavior resulting in future Green House Gas (GHG) emission scenarios, and can also be termed as scenario uncertainty. Downscaled outputs of a single GCM with a single climate change scenario represent a single trajectory among a number of realizations derived using various GCMs and scenarios. Such a single trajectory alone can not represent a future hydrologic scenario, and will not be useful in assessing hydrologic impacts due to climate change. Nonparametric methods are developed in the thesis to model GCM and scenario uncertainty for prediction of drought scenario with Orissa meteorological subdivision as a case study. Using the downscaling technique described in the previous paragraph, future rainfall scenarios are obtained for all available GCMs and scenarios. After correcting for bias, equiprobability transformation is used to convert the precipitation into Standardized Precipitation Index-12 (SPI-12), an annual drought indicator, based on which a drought may be classified as a severe drought, mild drought etc. Disagreements are observed between different predictions of SPI-12, resulting from different GCMs and scenarios. Assuming SPI-12 to be a random variable at every time step, nonparametric methods based on kernel density estimation and orthonormal series are used to determine the nonparametric probability density function (pdf) of SPI-12. Probabilities for different categories of drought are computed from the estimated pdf. It is observed that there is an increasing trend in the probability of extreme drought and a decreasing trend in the probability of near normal conditions, in the Orissa meteorological subdivision. The single valued Cumulative Distribution Functions (CDFs) obtained from nonparametric methods suffer from limitations due to the following: (a) simulations for all scenarios are not available for all the GCMs, thus leading to a possibility that incorporation of these missing climate experiments may result in a different CDF, (b) the method may simply overfit to a multimodal distribution from a relatively small sample of GCMs with a limited number of scenarios, and (c) the set of all scenarios may not fully compose the universal sample space, and thus, the precise single valued probability distribution may not be representative enough for applications. To overcome these limitations, an interval regression is performed to fit an imprecise normal distribution to the SPI-12 to provide a band of CDFs instead of a single valued CDF. Such a band of CDFs represents the incomplete nature of knowledge, thus reflecting the extent of what is ignored in the climate change impact assessment. From imprecise CDFs, the imprecise probabilities of different categories of drought are computed. These results also show an increasing trend of the bounds of the probability of extreme drought and decreasing trend of the bounds of the probability of near normal conditions, in the Orissa meteorological subdivision. Water resources planning requires the information about future streamflow scenarios in a river basin to combat hydrologic extremes resulting from climate change. It is therefore necessary to downscale GCM projections for streamflow prediction at river basin scales. A statistical downscaling model based on PCA, fuzzy clustering and Relevance Vector Machine (RVM) is developed to predict the monsoon streamflow of Mahanadi river at Hirakud reservoir, from GCM projections of large scale climatological data. Surface air temperature at 2m, Mean Sea Level Pressure (MSLP), geopotential height at a pressure level of 500 hecto Pascal (hPa) and surface specific humidity are considered as the predictors for modeling Mahanadi streamflow in monsoon season. PCA is used to reduce the dimensionality of the predictor dataset and also to convert the correlated variables to uncorrelated variables. Fuzzy clustering is carried out to derive the membership of the principal components in each of the clusters and the memberships thus obtained are used in RVM regression model. RVM involves fewer number of relevant vectors and the chance of overfitting is less than that of Support Vector Machine (SVM). Different kernel functions are used for comparison purpose and it is concluded that heavy tailed Radial Basis Function (RBF) performs best for streamflow prediction with GCM output for the case considered. The GCM CCSR/NIES with B2 scenario projects a decreasing trend in future monsoon streamflow of Mahanadi which is likely to be due to high surface warming. A possibilistic approach is developed next, for modeling GCM and scenario uncertainty in projection of monsoon streamflow of Mahanadi river. Three GCMs, Center for Climate System Research/ National Institute for Environmental Studies (CCSR/NIES), Hadley Climate Model 3 (HadCM3) and Coupled Global Climate Model 2 (CGCM2) with two scenarios A2 and B2 are used for the purpose. Possibilities are assigned to GCMs and scenarios based on their system performance measure in predicting the streamflow during years 1991-2005, when signals of climate forcing are visible. The possibilities are used as weights for deriving the possibilistic mean CDF for the three standard time slices, 2020s, 2050s and 2080s. It is observed that the value of streamflow at which the possibilistic mean CDF reaches the value of 1 reduces with time, which shows reduction in probability of occurrence of extreme high flow events in future and therefore there is likely to be a decreasing trend in the monthly peak flow. One possible reason for such a decreasing trend may be the significant increase in temperature due to climate warming. Simultaneous occurrence of reduction in Mahandai streamflow and increase in extreme drought in Orissa meteorological subdivision is likely to pose a challenge for water resources engineers in meeting water demands in future.

This thesis examines the structure of the Tropical Easterly Jet (TEJ) in a General Circulation Model (GCM). The TEJ is observed only during the Indian summer monsoon period and is strongest during July and August. The jet structure simulated by an atmospheric GCM (CAM-3.1) in July has been compared with reanalysis data. The simulated TEJ was displaced westward by ~ 25◦ when compared to observations. The removal of orography had no impact on the jet structure. This demonstrated that the Tibetan Plateau did not play an important role in the location and structure of the jet. The changes in cumulus scheme in the GCM had a large influence on the location of the jet maxima. To examine the factors which control the location and structure of the jet, a series of experiments were conducted using an aqua-planet version of the model. The impact of different sea surface temperature (SST) profiles was studied. The rainfall in the GCM was primarily in the regions where the SST attained a maximum. By altering the location of SST maximum (and hence the rainfall maximum), the impact of location of rainfall maximum on the location and structure of the jet was studied. When the rainfall maximum was located close to the equator, it did not generate a strong jet but had an influence on the vertical structure of the jet. A large number of simulations were conducted with multiple rainfall maxima and the need for these was demonstrated since only then was the observed jet structure well simulated. Based on the simulations, it was concluded that the simulation of the TEJ by CAM-3.1 was unrealistic because of large unrealistic rainfall over Saudi Arabia in this GCM. Equatorial heating has been shown to be important to simulate proper jet structure. The zonal structure of the jet was also influenced by rainfall in the Pacific Ocean. Although the aqua-planet configuration of the CAM-3.1 GCM provided several useful insights, the simulation was not perfect on account of errors in the simulation of the temperature profile in the lower troposphere. An ideal-physics configuration of the GCM was used. This removed the cumulus physics and instead imposed the observed heating pro-files. Both upper tropospheric friction and radiative-convective atmospheric temperatures were required to simulate the TEJ. The problems with the simulation of structure in the jet exit region was corrected by using radiative-convective atmospheric temperatures that were qualitatively similar to those observed in northern hemisphere summer time. The ideal-physics configuration reconfirmed that the Saudi Arabian rainfall was responsible for the westward shift of the TEJ in the simulations. The ideal-physics simulations showed that the simple analytical model proposed by Gillin1980 was not suitable for the simulation of TEJ. The above the simulations indicate that a shift in the location of the jet is related to a shift in the rainfall pattern. Based on this insight one would expect that the jet location will be different in good and bad monsoon periods. This is indeed the case. In July 2002 the Indian monsoon failed after beginning well in June. In June the TEJ is consequently located west ward compared to July. The same situation prevails even in good and poor monsoon years. In a good monsoon year (July 1988) the jet maximum is located westward when compared to a bad monsoon year (July 2002). In this thesis we have clearly demonstrated the role of anomalous rainfall on the location of the TEJ. This thesis has shown that an accurate simulation of the TEJ depends upon the accurate simulation of various rainfall centers that act as multiple heat sources in the atmosphere. The rainfall in the equatorial region does not influence the strength of the TEJ but alters the vertical structure of the jet. The strength the jet is dependent on the intensity of rainfall and the latitudinal distance from the equator. The complex vertical structure of the jet is not simulated by simple analytical models of the jet.

Submitted by Renata Lopes (renatasil82@gmail.com) on 2016-05-10T13:29:13Z No. of bitstreams: 1 leticiavasconcellosdesouza.pdf: 334599 bytes, checksum: ecaf1358f31b66f2a2e8740f4db33535 (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2016-06-15T13:12:10Z (GMT) No. of bitstreams: 1 leticiavasconcellosdesouza.pdf: 334599 bytes, checksum: ecaf1358f31b66f2a2e8740f4db33535 (MD5) / Made available in DSpace on 2016-06-15T13:12:10Z (GMT). No. of bitstreams: 1 leticiavasconcellosdesouza.pdf: 334599 bytes, checksum: ecaf1358f31b66f2a2e8740f4db33535 (MD5) Previous issue date: 2015-08-14 / Este trabalho é voltado para professores que atuam nas séries finais do Ensino Fundamental. Tem como objetivo mostrar que é possível introduzir o estudo de Congruência Modular nesse segmento de ensino, buscando facilitar a resolução de diversas situações-problema. A motivação para escolha desse tema é que há a possibilidade de tornar mais simples a resolução de muitos exercícios trabalhados nessa etapa de ensino e que são inclusive cobrados em provas de admissão à escolas militares e em olimpíadas de Matemática para esse nível de escolaridade. Inicialmente é feita uma breve síntese do conjunto dos Números Inteiros, com suas operações básicas, relembrando também o conceito de números primos, onde é apresentado o crivo de Eratóstenes; o mmc (mínimo múltiplo comum) e o mdc (máximo divisor comum), juntamente com o Algoritmo de Euclides. Apresenta-se alguns exemplos de situações-problema e exercícios resolvidos envolvendo restos deixados por uma divisão para então, em seguida, ser dada a definição de congruência modular. Finalmente, são apresentadas sugestões de exercícios para serem trabalhados em sala de aula, com uma breve resolução. / The aims of this work is teachers working in the final grades of elementary school. It aspires to show that it is possible to introduce the study of Modular congruence this educational segment, seeking to facilitate the resolution of numerous problem situations. The motivation for choosing this theme is that there is the possibility to make it simpler to solve many problems worked at this stage of education and are even requested for admittance exams to military schools and mathematical Olympiads for that level of education. We begin with a brief summary about integer numbers, their basic operations, also recalling the concept of prime numbers, where the sieve of Eratosthenes is presented; the lcm (least common multiple) and the gcd (greatest common divisor), along with the Euclidean algorithm. We present some examples of problem situations and solved exercises involving debris left by a division and then, we give the definition of modular congruence . Finally , we present suggestions for exercises to be worked in the classroom, with a short resolution.

In this thesis, we investigated modes of intraseasonal variability (ISV) observed in the Indian monsoon rainfall and how these modes modulate rainfall over India. We identified a decreasing trend in the intensity of low-frequency intraseasonal mode with increasing strength in synoptic variability over India. We also made an attempt to understand the reason for these observed trends using numerical simulations. In the first part of the thesis, satellite rainfall estimates are used to understand the spatiotem-poral structures of convection in the intraseasonal timescale and their intensity during boreal sum-mer over south Asia. Two dominant modes of variability with periodicities of 10–20-days (high-frequency) and 20–60-days (low-frequency) are found, with the latter strongly modulated by sea surface temperature. The 20–60-day mode shows northward propagation from the equatorial In-dian Ocean linked with eastward propagating modes of convective systems over the tropics. The 10–20-day mode shows a complex space-time structure with a northwestward propagating anoma-lous pattern emanating from the Indonesian coast. This pattern is found to be interacting with a structure emerging from higher latitudes propagating southeastwards. This could be related to ver-tical shear of zonal wind over northern India. The two modes exhibit variability in their intensity on the interannual time scale and contribute a significant amount to the daily rainfall variability in a season. The intensities of the 20–60-day and 10–20-day modes show significantly strong inverse and direct relationship, respectively, with the all-India June–September rainfall. This study also establishes that the probability of occurrence of substantial rainfall over central India increases significantly if the two intraseasonal modes simultaneously exhibit positive anomalies over the region. There also exists a phase-locking between the two modes. In the second part of the thesis, we investigated the changing nature of these intraseasonal modes over Indian region, and their association with extreme rainfall events using ground based observed rainfall. We found that the relative strength of the northward propagating 20–60-day mode has a significant decreasing trend during the past six decades, possibly attributed to the weakening of large-scale circulation in the region during monsoon. This reduction is compensated by a gain in synoptic-scale (3–9 days) variability. The decrease in the low-frequency ISV is associated with a significant decreasing trend in the percentage of extreme events during the active phase of the monsoon. However, this decrease is balanced by a significant increasing trend in the percentage of extreme events in break phase. We also find a significant rise in occurrence of extremes during early- and late-monsoon months, mainly over the eastern coastal regions of India. We do not observe any significant trend in the high-frequency ISV. In the last part of the thesis, we used numerical simulations to understand the observed changes in the ISV features. Using the atmospheric component of a global climate model (GCM), we have performed two experiments: control experiment (CE) and heating experiment (HE). The CE is the default simulation for 10 years. In HE, we prescribed heating in the atmosphere in such a way that it mimics the conditions for extreme rainfall events as observed over central India during June– September. Heating is prescribed primarily during the break phase of the 20–60-day mode. This basically increases the number of extremes, majority of which are in break phase. The design of the experiment reflects the observed current scenario of increased extreme events during breaks. We found that the increased extreme events in the HE decreased the intensity of the 20–60-day mode over the Indian region. This reduction is associated with a reduction of rainfall in active phase and increase in the length of break phase. A reduction in the seasonal mean over India is also observed. The reduction of active phase rainfall is linked with an increased stability of the atmosphere over central India. Lastly, we propose a possible mechanism for the reduction of rainfall in active phase. We found that there is a significant reduction in the strength of the vertical easterly shear over the northern Indian region during break–active transition phase. This basically weakens the conditions for the growth of Rossby wave instability, thereby elongating break phase and reducing the rainfall intensity in the following active phase. This study highlights the redistribution of rainfall intensity among periodic (low-frequency) and non-periodic (extreme) modes in a changing climate scenario, which is further tested in a modeling study. The results presented in this thesis will provide a pathway to understand, using observations and numerical model simulations, the ISV and its relative contribution to the Indian summer monsoon. It can also be used for model evaluation.

The Mara River Basin (MRB) is endowed with pristine biodiversity, socio-cultural heritage and natural resources. The purpose of my study is to develop and apply an integrated water resource allocation framework for the MRB based on the hydrological processes, water demand and economic factors. The basin was partitioned into twelve sub-basins and the rainfall runoff processes was modeled using the Soil and Water Assessment Tool (SWAT) after satisfactory Nash-Sutcliff efficiency of 0.68 for calibration and 0.43 for validation at Mara Mines station. The impact and uncertainty of climate change on the hydrology of the MRB was assessed using SWAT and three scenarios of statistically downscaled outputs from twenty Global Circulation Models. Results predicted the wet season getting more wet and the dry season getting drier, with a general increasing trend of annual rainfall through 2050. Three blocks of water demand (environmental, normal and flood) were estimated from consumptive water use by human, wildlife, livestock, tourism, irrigation and industry. Water demand projections suggest human consumption is expected to surpass irrigation as the highest water demand sector by 2030. Monthly volume of water was estimated in three blocks of current minimum reliability, reserve (>95%), normal (80–95%) and flood (40%) for more than 5 months in a year. The assessment of water price and marginal productivity showed that current water use hardly responds to a change in price or productivity of water. Finally, a water allocation model was developed and applied to investigate the optimum monthly allocation among sectors and sub-basins by maximizing the use value and hydrological reliability of water. Model results demonstrated that the status on reserve and normal volumes can be improved to ‘low’ or ‘moderate’ by updating the existing reliability to meet prevailing demand. Flow volumes and rates for four scenarios of reliability were presented. Results showed that the water allocation framework can be used as comprehensive tool in the management of MRB, and possibly be extended similar watersheds.

A Growth Curve Model (GCM) is a multivariate linear model used for analyzing longitudinal data with short to moderate time series. It is a special case of Generalized Multivariate Analysis of Variance (GMANOVA) models. Analysis using the GCM involves comparison of mean growths among different groups. The classical GCM, however, possesses some limitations including distributional assumptions, assumption of identical degree of polynomials for all groups and it requires larger sample size than the number of time points. In this thesis, we relax some of the assumptions of the traditional GCM and develop appropriate inferential tools for its analysis, with the aim of reducing bias, improving precision and to gain increased power as well as overcome limitations of high-dimensionality. Existing methods for estimating the parameters of the GCM assume that the underlying distribution for the error terms is multivariate normal. In practical problems, however, we often come across skewed data and hence estimation techniques developed under the normality assumption may not be optimal. Simulation studies conducted in this thesis, in fact, show that existing methods are sensitive to the presence of skewness in the data, where estimators are associated with increased bias and mean square error (MSE), when the normality assumption is violated. Methods appropriate for skewed distributions are, therefore, required. In this thesis, we relax the distributional assumption of the GCM and provide estimators for the mean and covariance matrices of the GCM under multivariate skew normal (MSN) distribution. An estimator for the additional skewness parameter of the MSN distribution is also provided. The estimators are derived using the expectation maximization (EM) algorithm and extensive simulations are performed to examine the performance of the estimators. Comparisons with existing estimators show that our estimators perform better than existing estimators, when the underlying distribution is multivariate skew normal. Illustration using real data set is also provided, wherein Triglyceride levels from the Framingham Heart Study is modelled over time. The GCM assumes equal degree of polynomial for each group. Therefore, when groups means follow different shapes of polynomials, the GCM fails to accommodate this difference in one model. We consider an extension of the GCM, wherein mean responses from different groups can have different shapes, represented by polynomials of different degree. Such a model is referred to as Extended Growth Curve Model (EGCM). We extend our work on GCM to EGCM, and develop estimators for the mean and covariance matrices under MSN errors. We adopted the Restricted Expectation Maximization (REM) algorithm, which is based on the multivariate Newton-Raphson (NR) method and Lagrangian optimization. However, the multivariate NR method and hence, the existing REM algorithm are applicable to vector parameters and the parameters of interest in this study are matrices. We, therefore, extended the NR approach to matrix parameters, which consequently allowed us to extend the REM algorithm to matrix parameters. The performance of the proposed estimators were examined using extensive simulations and a motivating real data example was provided to illustrate the application of the proposed estimators. Finally, this thesis deals with high-dimensional application of GCM. Existing methods for a GCM are developed under the assumption of ‘small p large n’ (n >> p) and are not appropriate for analyzing high-dimensional longitudinal data, due to singularity of the sample covariance matrix. In a previous work, we used Moore-Penrose generalized inverse to overcome this challenge. However, the method has some limitations around near singularity, when p~n. In this thesis, a Bayesian framework was used to derive a test for testing the linear hypothesis on the mean parameter of the GCM, which is applicable in high-dimensional situations. Extensive simulations are performed to investigate the performance of the test statistic and establish optimality characteristics. Results show that this test performs well, under different conditions, including the near singularity zone. Sensitivity of the test to mis-specification of the parameters of the prior distribution are also examined empirically. A numerical example is provided to illustrate the usefulness of the proposed method in practical situations. / Thesis / Doctor of Philosophy (PhD)

Wind power presently plays a minor role in Western Canada as compared to hydroelectric power in British Columbia and coal and natural gas thermal power generation in Alberta. However, ongoing reductions in the cost of wind power generation facilities and the increasing costs of conventional power generation, particularly if the cost to the environment is included, suggest that assessment of the present and future wind field in Western Canada is of some importance. To assess present wind power, raw hourly wind speeds and homogenized monthly mean wind speeds from 30 stations in Western Canada were analyzed over the period 1971-2000 (past). The hourly data were adjusted using the homogenized monthly means to attempt to compensate for differences in anemometer height from the standard height of 10m and changes in observing equipment at stations. A regional reanalysis product, the North American Regional Reanalysis (NARR), and simulations conducted with the Canadian Regional Climate Model (CRCM) driven with global reanalysis boundary forcing, were compared to the adjusted station wind-speed time-series and probability distributions. The NARR had a better temporal correlation with the observations, than the CRCM. We posit this is due to the NARR assimilating regional observations, whereas the CRCM did not. The NARR was generally worse than the CRCM in reproducing the observed speed distribution, possibly due to the crude representation of the regional topography in NARR. While the CRCM was run at both standard (45 km) and fine (15 km) resolution, the fine grid spacing does not always provide better results: the character of the surrounding topography appears to be an important factor for determining the level of agreement. Multiple simulations of the CRCM at the 45 km resolution were also driven by two global climate models (GCMs) over the periods 1971-2000 (using only historic emissions) and 2031-2060 (using the A2 emissions scenario). In light of the CRCM biases relative to the observations, these simulations were calibrated using quantile-quantile matching to the adjusted station observations to obtain ensembles of 9 and 25 projected wind speed distributions for the 2031-2060 period (future) at the station locations. Both bias correction and change factor techniques were used for calibration. At most station locations modest increases in mean wind speed were found for most of the projected distributions, but with a large variance. Estimates of wind power density for the projected speed distributions were made using a relationship between wind speed and power from a CRCM simulation for both time periods using the 15km grid. As would be expected from the wind speed results and the proportionality of wind power to the cube of wind speed, wind power at the station locations is more likely than not to increase in the 2031-2060 period from the 1971-2000 period. Relative changes in mean wind speeds at station locations were found to be insensitive to the station observations and choice of calibration technique, suggesting that we estimate relative change at all 45km grid points using all pairs of past/future mean wind speeds from the CRCM simulations. Overall, our results suggest that wind energy resources in Western Canada are reasonably likely to increase at least modestly in the future. / Graduate / 0725 / 0608 / jtdaines@uvic.ca