Variational ensemble Kalman filtering applied to data assimilation problems in computational fluid dynamics

One challenge on data assimilation (DA) methods is how the error covariance for the model state is computed. Ensemble methods have been proposed for producing error covariance estimates, as error is propagated in time using the non-linear model. Variational methods, on the other hand, use the concepts of control theory, whereby the state estimate is optimized from both the background and the measurements. Numerical optimization schemes are applied which solve the problem of memory storage and huge matrix inversion needed by classical Kalman filter methods. Variational Ensemble Kalman filter (VEnKF), as a method inspired the Variational Kalman Filter (VKF), enjoys the benefits from both ensemble methods and variational methods. It avoids filter inbreeding problems which emerge when the ensemble spread underestimates the true error covariance. In VEnKF this is tackled by resampling the ensemble every time measurements are available. One advantage of VEnKF over VKF is that it needs neither tangent linear code nor adjoint code. In this thesis, VEnKF has been applied to a two-dimensional shallow water model simulating a dam-break experiment. The model is a public code with water height measurements recorded in seven stations along the 21:2 m long 1:4 m wide flume’s mid-line. Because the data were too sparse to assimilate the 30 171 model state vector, we chose to interpolate the data both in time and in space. The results of the assimilation were compared with that of a pure simulation. We have found that the results revealed by the VEnKF were more realistic, without numerical artifacts present in the pure simulation. Creating a wrapper code for a model and DA scheme might be challenging, especially when the two were designed independently or are poorly documented. In this thesis we have presented a non-intrusive approach of coupling the model and a DA scheme. An external program is used to send and receive information between the model and DA procedure using files. The advantage of this method is that the model code changes needed are minimal, only a few lines which facilitate input and output. Apart from being simple to coupling, the approach can be employed even if the two were written in different programming languages, because the communication is not through code. The non-intrusive approach is made to accommodate parallel computing by just telling the control program to wait until all the processes have ended before the DA procedure is invoked. It is worth mentioning the overhead increase caused by the approach, as at every assimilation cycle both the model and the DA procedure have to be initialized. Nonetheless, the method can be an ideal approach for a benchmark platform in testing DA methods. The non-intrusive VEnKF has been applied to a multi-purpose hydrodynamic model COHERENS to assimilate Total Suspended Matter (TSM) in lake Säkylän Pyhäjärvi. The lake has an area of 154 km2 with an average depth of 5:4 m. Turbidity and chlorophyll-a concentrations from MERIS satellite images for 7 days between May 16 and July 6 2009 were available. The effect of the organic matter has been computationally eliminated to obtain TSM data. Because of computational demands from both COHERENS and VEnKF, we have chosen to use 1 km grid resolution. The results of the VEnKF have been compared with the measurements recorded at an automatic station located at the North-Western part of the lake. However, due to TSM data sparsity in both time and space, it could not be well matched. The use of multiple automatic stations with real time data is important to elude the time sparsity problem. With DA, this will help in better understanding the environmental hazard variables for instance. We have found that using a very high ensemble size does not necessarily improve the results, because there is a limit whereby additional ensemble members add very little to the performance. Successful implementation of the non-intrusive VEnKF and the ensemble size limit for performance leads to an emerging area of Reduced Order Modeling (ROM). To save computational resources, running full-blown model in ROM is avoided. When the ROM is applied with the non-intrusive DA approach, it might result in a cheaper algorithm that will relax computation challenges existing in the field of modelling and DA.

Computational modeling of laser-induced delamination testing of thin films

Thin film adhesion often determines microelectronic device reliability and it is therefore essential to have experimental techniques that accurately and efficiently characterize it. Laser-induced delamination is a novel technique that uses laser-generated stress waves to load thin films at high strain rates and extract the fracture toughness of the film/substrate interface. The effectiveness of the technique in measuring the interface properties of metallic films has been documented in previous studies. The objective of the current effort is to model the effect of residual stresses on the dynamic delamination of thin films. Residual stresses can be high enough to affect the crack advance and the mode mixity of the delimitation event, and must therefore be adequately modeled to make accurate and repeatable predictions of fracture toughness. The equivalent axial force and bending moment generated by the residual stresses are included in a dynamic, nonlinear finite element model of the delaminating film, and the impact of residual stresses on the final extent of the interfacial crack, the relative contribution of shear failure, and the deformed shape of the delaminated film is studied in detail. Another objective of the study is to develop techniques to address issues related to the testing of polymeric films. These type of films adhere well to silicon and the resulting crack advance is often much smaller than for metallic films, making the extraction of the interface fracture toughness more difficult. The use of an inertial layer which enhances the amount of kinetic energy trapped in the film and thus the crack advance is examined. It is determined that the inertial layer does improve the crack advance, although in a relatively limited fashion. The high interface toughness of polymer films often causes the film to fail cohesively when the crack front leaves the weakly bonded region and enters the strong interface. The use of a tapered pre-crack region that provides a more gradual transition to the strong interface is examined. The tapered triangular pre-crack geometry is found to be effective in reducing the stresses induced thereby making it an attractive option. We conclude by studying the impact of modifying the pre-crack geometry to enable the testing of multiple polymer films.

Computational Analysis of Intelligent Agents: Social and Strategic Settings

The central motif of this work is prediction and optimization in presence of multiple interacting intelligent agents. We use the phrase `intelligent agents' to imply in some sense, a `bounded rationality', the exact meaning of which varies depending on the setting. Our agents may not be `rational' in the classical game theoretic sense, in that they don't always optimize a global objective. Rather, they rely on heuristics, as is natural for human agents or even software agents operating in the real-world. Within this broad framework we study the problem of influence maximization in social networks where behavior of agents is myopic, but complication stems from the structure of interaction networks. In this setting, we generalize two well-known models and give new algorithms and hardness results for our models. Then we move on to models where the agents reason strategically but are faced with considerable uncertainty. For such games, we give a new solution concept and analyze a real-world game using out techniques. Finally, the richest model we consider is that of Network Cournot Competition which deals with strategic resource allocation in hypergraphs, where agents reason strategically and their interaction is specified indirectly via player's utility functions. For this model, we give the first equilibrium computability results. In all of the above problems, we assume that payoffs for the agents are known. However, for real-world games, getting the payoffs can be quite challenging. To this end, we also study the inverse problem of inferring payoffs, given game history. We propose and evaluate a data analytic framework and we show that it is fast and performant.

Mécanismes psychophysiques et neuronaux de la compensation dynamique de multiples champs de force : facilitation et anticipation liée à des indices de couleur

Dans cette thèse, nous abordons le contrôle moteur du mouvement du coude à travers deux approches expérimentales : une première étude psychophysique a été effectuée chez les sujets humains, et une seconde implique des enregistrements neurophysiologiques chez le singe. Nous avons recensé plusieurs aspects non résolus jusqu’à présent dans l’apprentissage moteur, particulièrement concernant l’interférence survenant lors de l’adaptation à deux ou plusieurs champs de force anti-corrélés. Nous avons conçu un paradigme où des stimuli de couleur aident les sujets à prédire la nature du champ de force externe actuel avant qu’ils ne l’expérimentent physiquement durant des mouvements d’atteinte. Ces connaissances contextuelles faciliteraient l’adaptation à des champs de forces en diminuant l’interférence. Selon le modèle computationnel de l’apprentissage moteur MOSAIC (MOdular Selection And Identification model for Control), les stimuli de couleur aident les sujets à former « un modèle interne » de chaque champ de forces, à s’en rappeler et à faire la transition entre deux champs de force différents, sans interférence. Dans l’expérience psychophysique, quatre groupes de sujets humains ont exécuté des mouvements de flexion/extension du coude contre deux champs de forces. Chaque force visqueuse était associée à une couleur de l’écran de l’ordinateur et les deux forces étaient anti-corrélées : une force résistante (Vr) a été associée à la couleur rouge de l’écran et l’autre, assistante (Va), à la couleur verte de l’écran. Les deux premiers groupes de sujets étaient des groupes témoins : la couleur de l’écran changeait à chaque bloc de 4 essais, tandis que le champ de force ne changeait pas. Les sujets du groupe témoin Va ne rencontraient que la force assistante Va et les sujets du groupe témoin Vr performaient leurs mouvements uniquement contre une force résistante Vr. Ainsi, dans ces deux groupes témoins, les stimuli de couleur n’étaient pas pertinents pour adapter le mouvement et les sujets ne s’adaptaient qu’à une seule force (Va ou Vr). Dans les deux groupes expérimentaux, cependant, les sujets expérimentaient deux champs de forces différents dans les différents blocs d’essais (4 par bloc), associés à ces couleurs. Dans le premier groupe expérimental (groupe « indice certain », IC), la relation entre le champ de force et le stimulus (couleur de l’écran) était constante. La couleur rouge signalait toujours la force Vr tandis que la force Va était signalée par la couleur verte. L’adaptation aux deux forces anti-corrélées pour le groupe IC s’est avérée significative au cours des 10 jours d’entraînement et leurs mouvements étaient presque aussi bien ajustés que ceux des deux groupes témoins qui n’avaient expérimenté qu’une seule des deux forces. De plus, les sujets du groupe IC ont rapidement démontré des changements adaptatifs prédictifs dans leurs sorties motrices à chaque changement de couleur de l’écran, et ceci même durant leur première journée d’entraînement. Ceci démontre qu’ils pouvaient utiliser les stimuli de couleur afin de se rappeler de la commande motrice adéquate. Dans le deuxième groupe expérimental, la couleur de l’écran changeait régulièrement de vert à rouge à chaque transition de blocs d’essais, mais le changement des champs de forces était randomisé par rapport aux changements de couleur (groupe « indice-incertain », II). Ces sujets ont pris plus de temps à s’adapter aux champs de forces que les 3 autres groupes et ne pouvaient pas utiliser les stimuli de couleurs, qui n’étaient pas fiables puisque non systématiquement reliés aux champs de forces, pour faire des changements prédictifs dans leurs sorties motrices. Toutefois, tous les sujets de ce groupe ont développé une stratégie ingénieuse leur permettant d’émettre une réponse motrice « par défaut » afin de palper ou de sentir le type de la force qu’ils allaient rencontrer dans le premier essai de chaque bloc, à chaque changement de couleur. En effet, ils utilisaient la rétroaction proprioceptive liée à la nature du champ de force afin de prédire la sortie motrice appropriée pour les essais qui suivent, jusqu’au prochain changement de couleur d’écran qui signifiait la possibilité de changement de force. Cette stratégie était efficace puisque la force demeurait la même dans chaque bloc, pendant lequel la couleur de l’écran restait inchangée. Cette étude a démontré que les sujets du groupe II étaient capables d’utiliser les stimuli de couleur pour extraire des informations implicites et explicites nécessaires à la réalisation des mouvements, et qu’ils pouvaient utiliser ces informations pour diminuer l’interférence lors de l’adaptation aux forces anti-corrélées. Les résultats de cette première étude nous ont encouragés à étudier les mécanismes permettant aux sujets de se rappeler d’habiletés motrices multiples jumelées à des stimuli contextuels de couleur. Dans le cadre de notre deuxième étude, nos expériences ont été effectuées au niveau neuronal chez le singe. Notre but était alors d’élucider à quel point les neurones du cortex moteur primaire (M1) peuvent contribuer à la compensation d’un large éventail de différentes forces externes durant un mouvement de flexion/extension du coude. Par cette étude, nous avons testé l’hypothèse liée au modèle MOSAIC, selon laquelle il existe plusieurs modules contrôleurs dans le cervelet qui peuvent prédire chaque contexte et produire un signal de sortie motrice approprié pour un nombre restreint de conditions. Selon ce modèle, les neurones de M1 recevraient des entrées de la part de plusieurs contrôleurs cérébelleux spécialisés et montreraient ensuite une modulation appropriée de la réponse pour une large variété de conditions. Nous avons entraîné deux singes à adapter leurs mouvements de flexion/extension du coude dans le cadre de 5 champs de force différents : un champ nul ne présentant aucune perturbation, deux forces visqueuses anti-corrélées (assistante et résistante) qui dépendaient de la vitesse du mouvement et qui ressemblaient à celles utilisées dans notre étude psychophysique chez l’homme, une force élastique résistante qui dépendait de la position de l’articulation du coude et, finalement, un champ viscoélastique comportant une sommation linéaire de la force élastique et de la force visqueuse. Chaque champ de force était couplé à une couleur d’écran de l’ordinateur, donc nous avions un total de 5 couleurs différentes associées chacune à un champ de force (relation fixe). Les singes étaient bien adaptés aux 5 conditions de champs de forces et utilisaient les stimuli contextuels de couleur pour se rappeler de la sortie motrice appropriée au contexte de forces associé à chaque couleur, prédisant ainsi leur sortie motrice avant de sentir les effets du champ de force. Les enregistrements d’EMG ont permis d’éliminer la possibilité de co-contractions sous-tendant ces adaptations, étant donné que le patron des EMG était approprié pour compenser chaque condition de champ de force. En parallèle, les neurones de M1 ont montré des changements systématiques dans leurs activités, sur le plan unitaire et populationnel, dans chaque condition de champ de force, signalant les changements requis dans la direction, l’amplitude et le décours temporel de la sortie de force musculaire nécessaire pour compenser les 5 conditions de champs de force. Les changements dans le patron de réponse pour chaque champ de force étaient assez cohérents entre les divers neurones de M1, ce qui suggère que la plupart des neurones de M1 contribuent à la compensation de toutes les conditions de champs de force, conformément aux prédictions du modèle MOSAIC. Aussi, cette modulation de l’activité neuronale ne supporte pas l’hypothèse d’une organisation fortement modulaire de M1.

Computational approaches to emotional decision making

Findings on the role that emotion plays in human behavior have transformed Artificial Intelligence computations. Modern research explores how to simulate more intelligent and flexible systems. Several studies focus on the role that emotion has in order to establish values for alternative decision and decision outcomes. For instance, Busemeyer et al. (2007) argued that emotional state affects the subjectivity value of alternative choice. However, emotional concepts in these theories are generally not defined formally and it is difficult to describe in systematic detail how processes work. In this sense, structures and processes cannot be explicitly implemented. Some attempts have been incorporated into larger computational systems that try to model how emotion affects human mental processes and behavior (Becker-Asano & Wachsmuth, 2008; Marinier, Laird & Lewis, 2009; Marsella & Gratch, 2009; Parkinson, 2009; Sander, Grandjean & Scherer, 2005). As we will see, some tutoring systems have explored this potential to inform user models. Likewise, dialogue systems, mixed-initiative planning systems, or systems that learn from observation could also benefit from such an approach (Dickinson, Brew & Meurers, 2013; Jurafsky & Martin, 2009). That is, considering emotion as interaction can be relevant in order to explain the dynamic role it plays in action and cognition (see Boehner et al., 2007).

Mode locking in a spatially extended neuron model: active soma and compartmental tree

Understanding the mode-locked response of excitable systems to periodic forcing has important applications in neuroscience. For example it is known that spatially extended place cells in the hippocampus are driven by the theta rhythm to generate a code conveying information about spatial location. Thus it is important to explore the role of neuronal dendrites in generating the response to periodic current injection. In this paper we pursue this using a compartmental model, with linear dynamics for each compartment, coupled to an active soma model that generates action potentials. By working with the piece-wise linear McKean model for the soma we show how the response of the whole neuron model (soma and dendrites) can be written in closed form. We exploit this to construct a stroboscopic map describing the response of the spatially extended model to periodic forcing. A linear stability analysis of this map, together with a careful treatment of the non-differentiability of the soma model, allows us to construct the Arnol'd tongue structure for 1:q states (one action potential for q cycles of forcing). Importantly we show how the presence of quasi-active membrane in the dendrites can influence the shape of tongues. Direct numerical simulations confirm our theory and further indicate that resonant dendritic membrane can enlarge the windows in parameter space for chaotic behavior. These simulations also show that the spatially extended neuron model responds differently to global as opposed to point forcing. In the former case spatio-temporal patterns of activity within an Arnol'd tongue are standing waves, whilst in the latter they are traveling waves.

Development of a global IT charging model for BASF Group

With ‘GS Strategy 2025’ BASF Business Services GmbH was formed to centrally steer all IT related topics of BASF group. Thus, a global charging system has to be designed, which complies to international transfer price regulations and the strategy of BASF SE. This work project develops a charging system with a following evaluation. The direct charging system benefits from its cost transparency upsides but comes with a higher administrative effort due to volume-based charging. In contrast, the indirect charging system convinces because of easy handling, which is the result of the application of suitable allocation keys. Regarding the complex group structure of BASF SE with more than 300 legal entities in 80 countries, the lower administrative effort of the indirect charging system outweighs the benefits of the direct charging model and should be used by BASF group.

Computational and experimental study of the behavior of cyano-based ionic liquids in aqueous solution

The solvation of cyano- (CN-) based ionic liquids (ILs) and their capacity to establish hydrogen bonds (H-bonds) with water was studied by means of experimental and computational approaches. Experimentally, water activity data were measured for aqueous solutions of ILs based on 1-butyl-3-methylimidazolium ([BMIM](+)) cation combined with one of the following anions: thiocyanate ([SCN](-)), dicyanamide ([DCA](-)), or tricyanomethanide ([TCM](-)), and of 1-ethyl-3-methylimidazolium tetracyanoborate ([EMIM][TCB]). From the latter data, water activity coefficients were estimated showing that [BMIM][SCN] and [BMIM][DCA], unlike [BMIM][TCM] and [EMIM][TCB], are able to establish favorable interactions with water. Computationally, the conductor like screening model for real solvents (COSMO-RS) was used to estimate the water activity coefficients which compare well with the experimental ones. From the COSMO-RS results, it is suggested that the polarity of each ion composing the ILs has a strong effect on the solvation phenomena. Furthermore, classical molecular dynamics (MD) simulations were performed for obtaining an atomic level picture of the local molecular neighborhood of the different species. From the experimental and computational data it is showed that increasing the number of CN groups in the ILs' anions does not enhance their ability to establish H-bonds with water but decreases their polarities, being [BMIM][DCA] and [BMIM][SCN] the ones presenting higher propensity to interact.

Designing a S2-Enterprise (Smart x Sensing) Reference Model

Part 11: Reference and Conceptual Models

A NATURALISTIC COMPUTATIONAL MODEL OF HUMAN BEHAVIOR IN NAVIGATION AND SEARCH TASKS

Planning, navigation, and search are fundamental human cognitive abilities central to spatial problem solving in search and rescue, law enforcement, and military operations. Despite a wealth of literature concerning naturalistic spatial problem solving in animals, literature on naturalistic spatial problem solving in humans is comparatively lacking and generally conducted by separate camps among which there is little crosstalk. Addressing this deficiency will allow us to predict spatial decision making in operational environments, and understand the factors leading to those decisions. The present dissertation is comprised of two related efforts, (1) a set of empirical research studies intended to identify characteristics of planning, execution, and memory in naturalistic spatial problem solving tasks, and (2) a computational modeling effort to develop a model of naturalistic spatial problem solving. The results of the behavioral studies indicate that problem space hierarchical representations are linear in shape, and that human solutions are produced according to multiple optimization criteria. The Mixed Criteria Model presented in this dissertation accounts for global and local human performance in a traditional and naturalistic Traveling Salesman Problem. The results of the empirical and modeling efforts hold implications for basic and applied science in domains such as problem solving, operations research, human-computer interaction, and artificial intelligence.

Computational Evaluation of Wind Loads on Low- and High- Rise Buildings

Buildings and other infrastructures located in the coastal regions of the US have a higher level of wind vulnerability. Reducing the increasing property losses and causalities associated with severe windstorms has been the central research focus of the wind engineering community. The present wind engineering toolbox consists of building codes and standards, laboratory experiments, and field measurements. The American Society of Civil Engineers (ASCE) 7 standard provides wind loads only for buildings with common shapes. For complex cases it refers to physical modeling. Although this option can be economically viable for large projects, it is not cost-effective for low-rise residential houses. To circumvent these limitations, a numerical approach based on the techniques of Computational Fluid Dynamics (CFD) has been developed. The recent advance in computing technology and significant developments in turbulence modeling is making numerical evaluation of wind effects a more affordable approach. The present study targeted those cases that are not addressed by the standards. These include wind loads on complex roofs for low-rise buildings, aerodynamics of tall buildings, and effects of complex surrounding buildings. Among all the turbulence models investigated, the large eddy simulation (LES) model performed the best in predicting wind loads. The application of a spatially evolving time-dependent wind velocity field with the relevant turbulence structures at the inlet boundaries was found to be essential. All the results were compared and validated with experimental data. The study also revealed CFD’s unique flow visualization and aerodynamic data generation capabilities along with a better understanding of the complex three-dimensional aerodynamics of wind-structure interactions. With the proper modeling that realistically represents the actual turbulent atmospheric boundary layer flow, CFD can offer an economical alternative to the existing wind engineering tools. CFD’s easy accessibility is expected to transform the practice of structural design for wind, resulting in more wind-resilient and sustainable systems by encouraging optimal aerodynamic and sustainable structural/building design. Thus, this method will help ensure public safety and reduce economic losses due to wind perils.

Development of a Coupling Model for Fluid-Structure Interaction using the Mesh-free Finite Element Method and the Lattice Boltzmann Method

In the presented thesis work, the meshfree method with distance fields was coupled with the lattice Boltzmann method to obtain solutions of fluid-structure interaction problems. The thesis work involved development and implementation of numerical algorithms, data structure, and software. Numerical and computational properties of the coupling algorithm combining the meshfree method with distance fields and the lattice Boltzmann method were investigated. Convergence and accuracy of the methodology was validated by analytical solutions. The research was focused on fluid-structure interaction solutions in complex, mesh-resistant domains as both the lattice Boltzmann method and the meshfree method with distance fields are particularly adept in these situations. Furthermore, the fluid solution provided by the lattice Boltzmann method is massively scalable, allowing extensive use of cutting edge parallel computing resources to accelerate this phase of the solution process. The meshfree method with distance fields allows for exact satisfaction of boundary conditions making it possible to exactly capture the effects of the fluid field on the solid structure.

Good is generally more alike than bad in the information ecology: Investigating the case of (the ABC model of) group stereotypes

On most if not all evaluatively relevant dimensions such as the temperature level, taste intensity, and nutritional value of a meal, one range of adequate, positive states is framed by two ranges of inadequate, negative states, namely too much and too little. This distribution of positive and negative states in the information ecology results in a higher similarity of positive objects, people, and events to other positive stimuli as compared to the similarity of negative stimuli to other negative stimuli. In other words, there are fewer ways in which an object, a person, or an event can be positive as compared to negative. Oftentimes, there is only one way in which a stimulus can be positive (e.g., a good meal has to have an adequate temperature level, taste intensity, and nutritional value). In contrast, there are many different ways in which a stimulus can be negative (e.g., a bad meal can be too hot or too cold, too spicy or too bland, or too fat or too lean). This higher similarity of positive as compared to negative stimuli is important, as similarity greatly impacts speed and accuracy on virtually all levels of information processing, including attention, classification, categorization, judgment and decision making, and recognition and recall memory. Thus, if the difference in similarity between positive and negative stimuli is a general phenomenon, it predicts and may explain a variety of valence asymmetries in cognitive processing (e.g., positive as compared to negative stimuli are processed faster but less accurately). In my dissertation, I show that the similarity asymmetry is indeed a general phenomenon that is observed in thousands of words and pictures. Further, I show that the similarity asymmetry applies to social groups. Groups stereotyped as average on the two dimensions agency / socio-economic success (A) and conservative-progressive beliefs (B) are stereotyped as positive or high on communion (C), while groups stereotyped as extreme on A and B (e.g., managers, homeless people, punks, and religious people) are stereotyped as negative or low on C. As average groups are more similar to one another than extreme groups, according to this ABC model of group stereotypes, positive groups are mentally represented as more similar to one another than negative groups. Finally, I discuss implications of the ABC model of group stereotypes, pointing to avenues for future research on how stereotype content shapes social perception, cognition, and behavior.

Numeric model to estimate pesticide deposition and losses from aerial spraying.

Pesticides applications have been described by many researches as a very inefficient process. In some cases, there are reports that only 0.02% of the applied products are used for the effective control of the problem. The main factor that influences pesticides applications is the droplet size formed on spraying nozzles. Many parameters affects the dynamic of the droplets, like wind, temperature, relative humidity, and others. Small droplets are biologically more active, but they are affected by evaporation and drift. On the other hand, the great droplets do not promote a good distribution of the product on the target. In this sense, associated with the risk of non target areas contamination and with the high costs involved in applications, the knowledge of the droplet size is of fundamental importance in the application technology. When sophisticated technology for droplets analysis is unavailable, is common the use of artificial targets like water-sensitive paper to sample droplets. On field sampling, water-sensitive papers are placed on the trials where product will be applied. When droplets impinging on it, the yellow surface of this paper will be stained dark blue, making easy their recognition. Collected droplets on this papers have different kinds of sizes. In this sense, the determination of the droplet size distribution gives a mass distribution of the material and so, the efficience of the application of the product. The stains produced by droplets shows a spread factor proportional to their respectives initial sizes. One of methodologies to analyse the droplets is a counting and measure of the droplets made in microscope. The Porton N-G12 graticule, that shows equaly spaces class intervals on geometric progression of square 2, are coulpled to the lens of the microscope. The droplet size parameters frequently used are the Volumetric Median Diameter (VMD) and the Numeric Median Diameter. On VMD value, a representative droplets sample is divided in two equal parts of volume, in such away one part contains droplets of sizes smaller than VMD and the other part contains droplets of sizes greater that VMD. The same process is done to obtaining the NMD, which divide the sample in two equal parts in relation to the droplets size. The ratio between VMD and NMD allows the droplets uniformity evaluation. After that, the graphics of accumulated probability of the volume and size droplets are plotted on log scale paper (accumulated probability versus median diameter of each size class). The graphics provides the NMD on the x-axes point corresponding to the value of 50% founded on the y-axes. All this process is very slow and subjected to operator error. So, in order to decrease the difficulty envolved with droplets measuring it was developed a numeric model, implemented on easy and accessfull computational language, which allows approximate VMD and NMD values, with good precision. The inputs to this model are the frequences of the droplets sizes colected on the water-sensitive paper, observed on the Porton N-G12 graticule fitted on microscope. With these data, the accumulated distribution of the droplet medium volumes and sizes are evaluated. The graphics obtained by plotting this distributions allow to obtain the VMD and NMD using linear interpolation, seen that on the middle of the distributions the shape of the curves are linear. These values are essential to evaluate the uniformity of droplets and to estimate the volume deposited on the observed paper by the density (droplets/cm2). This methodology to estimate the droplets volume was developed by 11.0.94.224 Project of the CNPMA/EMBRAPA. Observed data of herbicides aerial spraying samples, realized by Project on Pelotas/RS county, were used to compare values obtained manual graphic method and with those obtained by model has shown, with great precision, the values of VMD and NMD on each sampled collector, allowing to estimate a quantities of deposited product and, by consequence, the quantities losses by drifty. The graphics of variability of VMD and NMD showed that the quantity of droplets that reachs the collectors had a short dispersion, while the deposited volume shows a great interval of variation, probably because the strong action of air turbulence on the droplets distribution, enfasizing the necessity of a deeper study to verify this influences on drift.