Publicações

This paper presents a comparative analysis of Multi-Temporal Interferometric Synthetic Aperture Radar (MT-InSAR) algorithms in a corner reflector located in Lisbon, Portugal. The reflector was monitored using daily observations from the Global Navigation Satellite System (GNSS) technique with submillimeter precision. The study focuses on comparing the performance and accuracy of different MT-InSAR approaches i) the Persistent Scatterer Interferometry (PSI) method using the open-source software StaMPS, ii) the PSI method using the commercial software SARPROZ, iii) the Quasi-PS method implemented with the SARPROZ software, as well as iv) a hybrid method Persistent scatterer

Interferometric Synthetic Aperture Radar (InSAR) has proved its efficiency for displacementmonitoring in urban areas. However, the large volume of data generated by this technology turnsthe retrieval of information useful for structure monitoring into a big data problem. In this study, anew tool (SARClust) to analyze InSAR displacement time series is proposed. The tool performs theclustering of persistent scatterers (PSs) based on dissimilarities between their displacement time seriesevaluated through dynamic time warping. This strategy leads to the formation of clusters containingPSs with similar displacements, which can be analyzed together, reducing data dimensionality, andfacilitating the identification of displacement patterns potentially related to structural damage. Aproof of concept was performed for downtown Lisbon, Portugal, where ten distinct displacementpatterns were identified. A relationship between clusters presenting centimeter-level displacementsand buildings located on steep slopes was observed. The results were validated through visualinspections and comparison with another tool for time series analysis. Agreement was found inboth cases. The innovation in this study is the attention brought to SARClust

Interferometric synthetic aperture radar (InSAR) is a cost-effective method for displacement measurement at dams and reservoir banks, and is especially relevant for structures without embedded equipment or rarely monitored. However, the lack of redundant observations prevents the determination of a posteriori variances for InSAR measurements, which limits the uptake of the InSAR technology for structure monitoring. This paper presents a methodology for the integration of persistent scatterer interferometry (PSI) and Global NavigationSatellite System (GNSS) displacement measurements. Double differences between PSI and GNSS data are used as observations in a redundant network connecting the points of interest, with GNSS data used to constrain PSI measurements. This procedure allows the determination of adjusted displacements and a posteriori variances at all network points. Maximum uncertainty of 3 mm was achieved for displacementsalong the SAR sensor line-of-sight at both the dam and its surrounding slopes. The proposed method is of interest for monitoring purposes, because it allows displacement measurement at both the structure and its surroundings, together with the displacement uncertainty, which aids decision makers in the planning of maintenance operations for the whole area influenced by the dam.

The hydrostatic-season-time (HST) model is a widely used method in the safety assessment of dams and for the estimation of reversible and irreversible deformations due to different load scenarios. After implementing an optimal HST-model to observational data, the residual time series can still expose some underlying periodicities. In our contribution, the underlying periodicities and their contribution to residual minimisation are further analysed by the Lomb

The three authors of this paper work at the Applied Geodetic Division of the National Laboratory for Civil Engineering (LNEC), in Lisbon, Portugal. LNEC is a state owned research and development institution founded in 1946. It works in the various domains of civil engineering (structures, hydraulic, geotechnics, environment, materials, among others), giving it a unique multidisciplinary perspective in this field. The main goals of the LNEC are to carry out innovative research and development and to contribute to the best practices in civil engineering.The Applied Geodetic Division nowadays develops works in two domains: the geodetic surveying of large dams and other engineering structures for monitoring purposes, and the processing of digital images with applications in several domains, which includes the study of the evolution of pathologies in engineering works. Originally the processing of digital images made use mostly of the chromatic information included in the images (from satellite images to close range photographs). But recently it has evolved to extract information of the geometry of the objects by the generation of point clouds. This use of close range photographs (from distances of decimetres to a few meters), which started in the summer of 2014, looks very promising and we, the authors, are identifying possible areas where the use of point clouds and orthomosaics that can be of interest to our colleagues of LNEC, civil engineers mostly. This paper presents the first examples where point clouds and orthomosaics, generated from close range photographs, can help civil engineers on their studies. The photogrammetic products were all generated using the free open-source software Micmac (Multi-Image Correspondances, Méthodes Automatiques de Corrélation) from IGN (Institut National de l

Les ports et autres infrastructures construits sur des côtes exposées à de fortes houles ont besoin d

This work presents several approaches in the validation of the hydrodynamic model Tuflow on the simulation of flood extents and water levels, based on satellite SAR imagery. A methodology that uses a simplified bathymetry in the river main course is employed, which proves to be reliable and accurate for high-flow events. This was made possible as the digital terrain model was acquired in a dry period, accounting for large dry areas in the river bed, avoiding in this way the need of expensive river bathymetry surveys. Also, two methods are applied to the SAR imagery to extract the flood boundaries: visual interpretation followed by manual delimitation and an object-based algorithm approach. The hydraulic model is tested on a reach of the Tagus River, Portugal, where the largest flood inundation areas occur, using a historical flood event to verify its robustness and reliability. The accuracy of model prediction is done through comparisons of water levels at a hydrometric station and the determination of commission and omission errors of flood extent, between the reference SAR image and the predicted inundation. It was concluded that the methodology followed in this work is well suited for the hydraulic model validation.

This article describes the implementation of a monitoring system on the breakwater at the Funchal Port (Madeira Island, Portugal) for the purpose of understanding its dynamic behavior and the temporal evolution of structural health. The monitoring system focuses on the breakwater superstructure and aims to identify the evolution of damage on the breakwater and analyze the impact of cruise ships docking and undocking. The system includes the measurement of displacements through geodetic techniques using global navigation satellite systems and the measurement of vibrations using accelerometers. Ten observation campaigns were performed over almost 3 years. As the measured displacements during this period were less than 7.5 mm it can be concluded that, despite some visible cracks, the breakwater superstructure did not suffer significant structural changes. It was shown from the measured acceleration records that some of the main modal parameters could be identified, namely the main natural frequencies of the superstructure. Through an analysis of the natural frequencies progression over time, it will be possible to identify deterioration effects on the foundation and/or the superstructure. The comparison of natural frequencies identified could be useful to evaluate the impact of these events on structural health.

A fusão de dados provenientes de diferentes tipos de instrumentos pode dar uma informação mais completa para a monitorização do comportamento de estruturas de engenharia civil, em particular nas pontes de grande vão. O objectivo da fusão de deslocamentos medidos com o GNSS e de acelerações, é tirar partido da precisão conseguida com o GNSS, em baixas frequências, e da grande sensibilidade dos acelerómetros, em altas frequências, e obter uma única série temporal de deslocamentos, longa e de grande resolução. Através dum filtro de Kalman será possível integrar deslocamentos medidos, com uma frequência de amostragem de, por exemplo, 1 Hz, e acelerações medidas, com uma frequência de amostragem de, por exemplo, 100 Hz, de modo a obter uma série temporal de deslocamentos de longa duração e com uma resolução de 100 Hz. Nesta comunicação apresentam-se os resultados preliminares dum ensaio realizado num modelo estrutural simples correspondente a um edifício de três pisos, instrumentado com antenas GNSS da Topcon e acelerómetros da Kinemetrics, modelo Episensor ES-U2, e da Nanometrics, modelo Titan SMA, tendo em vista a futura aplicação desta metodologia em pontes de grande vão.

There is an increasing trend to apply GNSS continuous observation of short baselines to the monitoring of engineering works, such as bridges and dams, for their structural analysis and safety control. In the case of large dams, one important application of the GNSS continuous observation is the establishment of early warning systems that demand accurate, frequently updated information and where the analysis of the baseline time series, in order to separate signal from noise is mandatory. The paper presents a study on the performance of linear filters of the asymmetric moving average (AMA) type to smooth baseline time series. The transfer function of the AMA is adopted as a smoothing criterion to define an adequate order for the AMA, in face of the spectral density function of the baseline time series. One series of measurements of a short test baseline (325 m), materialized in the campus of the National Laboratory for Civil Engineering, is used as an example of the proposed strategy.

O trabalho aqui apresentado resultou da análise de um inquérito, que teve como objectivocaracterizar a atitude dos serviços/departamentos autárquicos face à informação geográfica (IG).Três questões basilares orientaram a sua elaboração: A IG é usada regularmente? Constitui umrecurso importante? Os seus atributos espaciais e temporais são apropriados à actividade municipal?Analisando as respostas a estas questões, pretendeu-se avaliar a utilidade e valor da IG, bem comoanalisar as prioridades e perspectivas do governo local em matéria de dados geográficos.O universo estatístico foi constituido por 308 Municípios. A taxa de resposta foi de 45%, com 128respostas em Portugal Continental, 3 na Região Autónoma da Madeira e 9 na Região Autónoma dosAçores, perfazendo um total de 140 inquéritos preenchidos e validados.Os resultados permitiram concluir: i) que os tipos de utilização da IG são fortemente dependentes dotipo de serviços; ii) que a actualização de IG é particularmente importante por efeito da Lei; iii) quea introdução de dados orbitais de alta resolução espacial nos processos de actualização de IG é aindapouco frequente.

A comunicação apresenta um critério numérico Bayesiano - o Factor de Bayes - para comparar o mérito de distribuições alternativas, com base nas correspondentes distribuições preditivas anteriores.

O trabalho que aqui se apresenta tem lugar no âmbito do projecto GeoSat, em curso, que visa explorar o potencial das imagens de satélite de alta resolução espacial para extrair informação geográfica e actualizar cartografia existente para apoio às actividades dos municípios. Neste contexto, testam-se metodologias para extracção de edifícios de tipologias diversas em áreas da cidade de Lisboa com diferentes características, caracterizadas com recurso a métricas espaciais, e investiga-se o impacto da heterogeneidade destes elementos e do contexto urbano no processo de extracção

Na presente comunicação descreve-se o apoio dado pelos métodos Geodésicos ao acompanhamento da abertura e monitorização do túnel rodoviário que está a ser construído no Marão, um túnel que terá 5.6 km de extensão. A abertura de túneis com uma extensão de vários quilómetros, exige um acompanhamento da Geodesia Aplicada, de modo a serem respeitados os níveis de precisão desejáveis. Os critérios geodésicos devem intervir no estabelecimento das redes de apoio exterior e interior, necessárias para definir a orientação da escavação e na monitorização de convergências e assentamentos do túnel. As redes de apoio exteriores, necessitam de metodologias geodésicas de observação e de processamento dos dados que assegurem incertezas reduzidas. Relativamente à monitorização, a obtenção de resultados com incerteza milimétrica ou mesmo submilimétrica, exige também cuidados especiais, tanto na concepção, como na materialização do sistema e, também, durante a observação.

This paper presents a comprehensive approach to effectively extract cartographic urban data from high resolution satellite imagery. Itconsists of a sequence of image processing techniques, for image segmentation, based on RGB band separation, analysis and preprocessing,followed by a morphological-based approach for data segmentation. The chosen image objects for this study are roof-tilebuildings and road network. The first step of this approach consists of a spectral response evaluation for roof-tile building objects ina dense urban environment, being those enhanced through proper sequence of standard arithmetic operators, applied to RGB bands,segmented and generalized. The second step aims at urban road network segmentation for cartographic representation purposes, bycombining watershed and dual reconstruction morphological transformations, which characterize the hierarchical waterfall concept.For the latter concept, a new approach is developed in order to improve hierarchical segmentation procedure, to better induce objectdiscrimination. Each one of the referred objects will be segmented in separate. The road network segmentation will have inconsideration the previous result of roof-tile buildings extraction. Finally, segmented objects will be compared with other extractionresults in order to do proper validation. The method is applied over high-resolution ortho-rectified images, taken from satellite, of thecity of Lisbon. Results will show the practicality of the method for purposes of cartographic data structures acquisition, from highresolutionsatellite imagery, aiming urban management and GIS applications.

No presente trabalho, a classificação temática das imagens realiza-se ao nível do elemento. A experimentação teve como objectivo a extracção automática de elementos de interesse municipal para detecção de alterações. Dos elementos com maior relevância para a actualização cartográfica de bases de dados municipais podem-se distinguir os edifícios, as vias de comunicação e áreas verdes. Numa área urbana com elevada dinâmica como a do concelho de Lisboa, a informação actualizada sobre o edificado é necessária para dar cumprimento aos pedidos de licenciamento, à emissão de licenças de construção ou para emissão de plantas. Actualmente, a cartografia municipal é actualizada durante a fase de revisão do Plano Director Municipal (PDM), o que acontece geralmente todos os 10 anos, por imposição legal. Porém, muitas vezes este período é superior (e.g., o actual PDM de Lisboa data de 1994), o que conduz a um grande desfasamento entre as bases de dados cartográficas e a realidade.

No âmbito do projecto GeoSat, a aplicação Feature Analyst é testada para extracção automática de elementos ao nível municipal a partir dados espectrais e altimétricos numa área de estudo localizada em Lisboa.A metodologia apresentada baseia-se na extracção semi-automática de features, tendo por base informação proveniente de uma imagem QuickBird e de dados LiDAR. A imagem QuickBird permite delimitar os diferentes elementos à superfície com base no seu comportamento espectral. Porém, esta tarefa é tanto melhor sucedida quanto mais rica for a informação auxiliar utilizada no processo de extracção. Assim, com a introdução de dados LiDAR, pretende-se estudar o efeito da contribuição desta informação no processo de classificação de dados orbitais.Os resultados obtidos mostram que a introdução da informação altimétrica, no conjunto de dados espectrais, permite uma melhoria na qualidade da informação extraída por meios semi-automátcios.

The methodology is based on semi-automatic extraction of features, using information from a QuickBird image and LIDAR data.The QuickBird image is suitable to delineate the different elements on the surface based on their spectral characteristics. However,this task is much more successful and richer when altimetric information is also used in the extraction process. Thus, with theintroduction of LIDAR data, we intend to study the effect of the contribution of information on the height of urban elements, whenclassifying orbital data. The thematic information under analysis is the land cover class Buildings.Two maps are produced and compared. One map is obtained by a classification based only on spectral data while the other map isobtained using spectral and altimetric data. The same methodology is applied in both scenarios, using the same training features. Toevaluate the quality of the two maps, a comparison with a cartographic layer is carried out. This reference layer is obtained by visualanalysis of the QuickBird image and is used to calculate the Overall Accuracy of the maps obtained with the different data sets.

Many municipal activities require updated large-scale maps that include both topographic and thematic information. For thispurpose, the efficient use of very high spatial resolution (VHR) satellite imagery suggests the development of approaches that enablea timely discrimination, counting and delineation of urban elements according to legal technical specifications and quality standards.Therefore, the nature of this data source and expanding range of applications calls for additional methods and metrics to assess thequality of the extracted information which go beyond traditional thematic accuracy alone. The present work concerns the feasibilityof VHR satellite imagery as an alternative source of geospatial information for large scale mapping to assist urban planning inPortugal. Feature extraction software was employed to map buildings present in a pansharpened QuickBird image of Lisbon. Qualityassessment was exhaustive and involved comparisons of extracted features against a reference dataset, introducing cartographicconstraints from scales 1:1000, 1:5000, and 1:10000. The spatial data quality elements subject to evaluation were: thematic(attribute) accuracy, completeness, and geometric quality based on planimetric deviation from the reference. Tests were developedand metrics analyzed considering thresholds and standards for the different mapping scales most used by municipalities. Resultsshow that values for completeness varied with mapping scales and results were only slightly better for scale 1:10000. Concerning thegeometric quality, a large percentage of extracted features met the strict topographic standards of planimetric deviation for scale1:10000, while no buildings were compliant with the specification for scale 1:1000.

Lisbon is both a historical and modern city having a dynamic landscape, where increasingly diverse urban forms and materials coexist. This complex reality is possibly causing the feature extraction process from imagery to become more challenging. This study tests the semi-automated extraction of buildings from a QuickBird image in urban areas with different characteristics, and explores the impact of the heterogeneity of these features in the extraction process. Spatial metrics are used to characterize types of buildings present in the study areas. Results show that the study areas display different levels of heterogeneity even for the same type of building and suggest that the extraction may be affected by the spatial configuration of target features.

O calendário Gregoriano não é mais do que uma reforma do calendárioJuliano, devido a Júlio César, que adaptou os formalismos romanos a uma antigaestrutura solar de origem egípcia. Este ensaio descreve a história do calendárioGregoriano e dos calendários da bacia do Mediterrâneo e do Oriente Próximo,com ele mais ou menos aparentados.

Interferometric synthetic aperture RADAR is an applied geodesy technique, which enables displacement measurement through satellite images. A literature review of the technique principles and of previous studies regarding its application to structural health monitoring allowed the identification of a few bottlenecks to the operational usage of this cost-effective technology for monitoring purposes.Three research lines were defined. First, a data mining strategy was developed to extract structural behaviour information from displacement time series from thousands of points on the region of interest, with tens of observation epochs each. Second, displacements from this technique were integrated into a previously existing monitoring system and were used to expand it to a larger number of points on the structure and to its surroundings. Third, an experimental infrastructure was deployed in order to perform several tests for displacement validation and uncertainty assessment. The infrastructure included a new model of artificial reflector to the satellite signals, purposefully developed in the scope of this study.The developed methods enabled the successful monitoring of different types of civil and geotechnical structures. Slope behaviour around water reservoirs was assessed during both the reservoir first impoundment and regular operation. Differential displacements were detected on building-blocks at a cultural heritage urban area and concrete arch dam responses to temperature loads were identified. The experimental work showed the achieved displacements agreed with those of other geodetic techniques and uncertainties around 2 mm were obtained for the measurements.In conclusion, interferometric synthetic aperture RADAR is a promising technique, able to provide useful information for structure monitoring, such as aiding in the planning of in situ monitoring activities and in identifying space-time patterns in structural behaviour. Thanks to its spatial coverage, observation period of a few days and achieved displacement uncertainty, it is advantageous to complement existing monitoring systems with data from this technique.

Este relatório apresenta uma metodologia para controlo dos erros instrumentais e avaliação da incerteza de medição dos distanciómetros eletromagnéticos (DEM) usados na observação geodésica de barragens de betão. A metodologia, que é baseada numa base de teste existente no campus do LNEC, é ilustrada com a análise de seis medições da base, entre 2010 e 2014, com os dois DEM utilizados pelo Núcleo de Geodesia Aplicada.

The monitoring systems have two main objectives: (i) to evaluate the safety conditions and durability of the infrastructures and to develop a better understanding about their day-to-day behaviour; (ii) to detect the occurrence of special events that might jeopardize the safety of the structure. To reach the first objective it is mandatory to create a database as most extensive as possible to support an analysis of the structural behaviour. Regarding the second objective near real-time automatic systems must be created in order to analyse the recorded data and establish limits of alert with alarm warnings when those limits are exceeded. This R&D project aims to develop a measurement system, based on GNSS (Global Navigation Satellite Systems), suitable to monitor large scale civil engineering structures and infrastructures, particularly bridges and dams. This monitoring system final objective is to measure the displacements of some selected points of the structures. These points should be the most important ones to characterize the general behaviour of the monitored structure. The advantages of using a GNSS system are undeniable: this equipment is robust, operates under all weather conditions, does not require intervisibility between the monitored points and the reference points, uses an external tridimensional reference systems, and provides an extremely accurate time-reference frame. The GNSS positioning is affected by a numerous set of errors due to natural and artificial phenomena. These errors should be reduced as much as possible, in order to obtain good estimated distances between the observed satellites and the antenna of the receivers. There are well known observation and observation combination techniques (phase differences measurements, relative positioning and Precise Point Positioning - PPP) that allow to strongly reduce, or sometimes even eliminate, most of the errors. Using carefully selected observation and processing techniques it is possible to obtain high-precision GNSS positioning. The RTK (Real Time Kinematics) is the more recent high-precision positioning GNSS technique. It allows us to obtain high quality coordinates in almost real-time. When used in conjunction with top grade receivers (able to do observations at high frequencies) the RTK seems to be able to monitor high frequencies displacements of structures. Unfortunately the positioning precision is only at the centimetre level. To reduce this uncertainty it will be used short baselines and processing techniques based on the use of various filters and methodologies for multi-sensor observations fusion. It is precisely in the development of these techniques that the actual research project is focused. After the monitoring system specification, several techniques will be developed and tested in order to be included in a reliable measurement system that will meet the specified requirements. To define the monitoring system demands the following items have been addressed: - Requirements of the monitoring system in view of the behaviour of the selected structures; - Acquisition of know-how on data acquisition systems and in-site monitoring equipment; - Identification, characterization and analysis of software available for data processing.

A inspeção visual assistida pode ser utilizada para a monitorização de ensaios de flexão atéà rotura de vigas de madeira, tendo sido usado um sistema de observação com fotografia evídeo. Foi desenvolvido um programa para identificação automática das imagens da viga,que registaram alterações significativas, tendo a fratura sido delineada através de técnicasde análise de imagem orientada por objetos.

A extração de informação a partir de imagens digitais tem-se tornado numa prática comum em váriasáreas do conhecimento. Foi desenvolvido um programa em MATLAB para permitir a um utilizador nãoespecializado a realização de tarefas simples de processamento digital sobre imagens adquiridas porvários géneros de sensores.

O relatório apresenta os princípios gerais da termografia e a sua aplicação prática no levantamentotérmico de paramentos de barragens de betão para apoio ao acompanhamento da evolução depatologias. Apresentam-se aplicações relativas ao levantamento térmico dos paramentos de jusantedas barragens do Alto Ceira II, Alto Ceira e Cabril. O levantamento realizado na primeira barragem,que se encontrava na fase final de construção e ainda vazia, permitiu a avaliação da qualidade dosresultados obtidos (temperaturas superficiais do paramento) por comparação com valores medidospor termómetros embebidos no betão. Nas outras duas barragens, ambas apresentando problemasde infiltrações, permitiu avaliar a aplicabilidade das imagens térmicas na deteção de ressurgimentosde água na superfície do betão, geralmente indicativas da existência de repasses de água.

O presente relatório apresenta uma experiência de aquisição de fotografias aéreasutilizando um veículo aéreo não tripulado (VANT) para recolha de informação geográfica doquebra-mar da Ericeira, com o objetivo de testar a capacidade destes dados para produçãode ortofotomapas e modelos numéricos de superfície para monitorização de infraestruturasao longo do tempo. A experiência foi realizada numa colaboração entre o LNEC e aempresa SINFIC, SA

Neste relatório faz-se a síntese dos métodos disponíveis para realizar a Inspecção Visual Assistida(IVISA) de paramentos de barragens de betão, no que diz respeito às várias fases envolvidas: aquisiçãode imagens, processamento, monitorização temporal e arquivo. Apresentam-se os recursos existentesno LNEC e a adquirir, assim como os custos associados.

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Este relatório apresenta um procedimento para a avaliação da incerteza de medição de equipamento de nivelamento geométrico de precisão e ainda regras para o teste estatístico de hipóteses sobre essa incerteza de medição, preconizadas na segunda parte da norma ISO 17123, exemplificados com um ensaio comparativo de um nível automático numérico Wild NA 3003 e de um nível automático Wild NA2, com um micrómetro óptico acoplado.

O trabalho aborda simultaneamente a distribuição do desvio padrão da amostra aleatória normal e a distribuição do desvio padrão do modelo de Gauss-Markov, designados genericamente por desvio padrão empírico. Após mostrar que o desvio padrão empírico é um estimador excêntrico que subestima o desvio padrão da amostra ou do modelo, conforme o caso, são apresentadas correções de excentricidade. É também tratada a estimação do desvio padrão da amostra e do modelo em cenário Bayesiano, com diferentes distribuições anteriores. O trabalho é rematado por casos de estudo baseados na simulação de amostras normais e por um exemplo concreto relacionado com o controlo da precisão de um distanciómetro eletromagnético.

O presente trabalho apresenta uma metodologia genérica para a regularizaçãode problemas inversos lineares mal colocados e a sua aplicação às redes altimétricas eplanimétricas usadas na monitorização de deslocamentos de pontos notáveis de grandesbarragens por métodos da Geodesia Aplicada. A metodologia é ilustrada por intermédiode três casos de estudo.

This paper deals with the beta-binomial model, which is an important stochasticmodel within the scope of the Bayesian statistical inference. After a short review of the binomialand beta distributions, the paper presents the closed formulae for the posterior probabilitydensity function of the binomial trial parameter and for the prior predictive and forthe posterior predictive probabilities of the successes in binomial experiments. A descriptionof the Bayesian methodology to test different hyper-parameters for the conjugate priorprobability density functions of the beta family is also presented. Finally, the beta-binomialmodel is illustrated with a study case related to the quality control of the geodetic observationof large dams.

O presente trabalho, além de uma revisão sintética da abordagem frequencista,trata da abordagem Bayesiana da análise de regressão linear multivariada. É apresentadauma compilação de procedimentos e fórmulas para a estimação dos coeficientes da regressão,sob três cenários de informação anterior. Finalmente, o trabalho ocupa-se dosproblemas da predição de respostas e da comparação de modelos de regressão.

The work presents the theoretical definitions of the gamma, multi-gamma, digammaand trigamma functions as well as operational approximations to compute their values.A more precise alternative, due to Robert Windschitl, to the traditional approximationof the gamma function by the Stirling

O trabalho descreve os princípios da análise Bayesiana, usando como caso deestudo a amostra aleatória normal identicamente distribuída. Após um curto preâmbulohistórico, são resumidos, na Secção 2, os conceitos mais importantes da análise frequencista.Nas secções seguintes, são abordados os temas Bayesianos mais importantes, taiscomo a estimação, pontual e regional (Secção 5), e o teste de hipóteses sobre osparâmetros da distribuição da amostra (Secção 6). As distribuições anteriores de Laplace,Jeffreys e conjugada normal-gama invertida são apresentadas na Secção 4. O problemada eliciação paramétrica das distribuições anteriores conjugadas é tratado na Secção 7.Palavras Chave: distribuições anteriores e posteriores, divergência de Kullback-Leibler,eliciação paramétrica, factor de Bayes, fórmula de Bayes, verosimilhança.

Este trabalho define uma estratégia, apoiada o mais possível nas disponibilidades do LNEC, para a calibragem dos distanciómetros electromagnéticos (DEM) usados pelo Núcleo de Geodesia Aplicada na observação geodésica de barragens.

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