Maximum Likelihood Estimation of Modal Parameters in Structures Using the Expectation Maximization Algorithm

This paper presents a time-domain stochastic system identification method based on maximum likelihood estimation (MLE) with the expectation maximization (EM) algorithm. The effectiveness of this structural identification method is evaluated through numerical simulation in the context of the ASCE benchmark problem on structural health monitoring. The benchmark structure is a four-story, two-bay by two-bay steel-frame scale model structure built in the Earthquake Engineering Research Laboratory at the University of British Columbia, Canada. This paper focuses on Phase I of the analytical benchmark studies. A MATLAB-based finite element analysis code obtained from the IASC-ASCE SHM Task Group web site is used to calculate the dynamic response of the prototype structure. A number of 100 simulations have been made using this MATLAB-based finite element analysis code in order to evaluate the proposed identification method. There are several techniques to realize system identification. In this work, stochastic subspace identification (SSI)method has been used for comparison. SSI identification method is a well known method and computes accurate estimates of the modal parameters. The principles of the SSI identification method has been introduced in the paper and next the proposed MLE with EM algorithm has been explained in detail. The advantages of the proposed structural identification method can be summarized as follows: (i) the method is based on maximum likelihood, that implies minimum variance estimates; (ii) EM is a computational simpler estimation procedure than other optimization algorithms; (iii) estimate more parameters than SSI, and these estimates are accurate. On the contrary, the main disadvantages of the method are: (i) EM algorithm is an iterative procedure and it consumes time until convergence is reached; and (ii) this method needs starting values for the parameters. Modal parameters (eigenfrequencies, damping ratios and mode shapes) of the benchmark structure have been estimated using both the SSI method and the proposed MLE + EM method. The numerical results show that the proposed method identifies eigenfrequencies, damping ratios and mode shapes reasonably well even in the presence of 10% measurement noises. These modal parameters are more accurate than the SSI estimated modal parameters.

Natural ventilation in underground wine cellars

The main objective of this research is to promote passive thermal design techniques in the construction of wineries. Natural ventilation in underground cellars is analyzed, focusing on the entrance tunnel, the ventilation chimney and the cave. A monitoring system was designed in order to detect changes in the indoor conditions and outdoor air infiltration. Monitoring process was carried out during one year. Results show the influence of outside temperature, ventilation chimney and access tunnel on the conditions inside the underground cellar. During hot periods, natural ventilation has a negligible influence on the indoor ambience, despite the permanently open vents in the door and chimney. The tunnel and ventilation chimney work as a temperature regulator, dampening outside fluctuations. Forced ventilation is necessary when a high air exchange ratio is needed. During cold periods, there is greater instability as a result of increased natural ventilation. The temperature differences along the tunnel are reduced, reflecting a homogenization and mixing of the air. The ventilation flow is sufficient to modify the temperature and relative humidity of the cave. Forced ventilation is not necessary in this period. During the intermediate periods --autumn and spring-- occurs different behaviors based on time of day.

Teaching in Landscape

“We, the landscape architects, concerned with the future development of our landscapes in a fast changing world, believe that everything, influencing the way in which the outdoor environment is created, used, and maintained is fundamental to sustainable development and human well-being. We, being responsible for the improvement of the education of future landscape architects to enable them to work for a sustainable environment within the context of our natural and cultural heritage”. (IFLA/UNESCO 2005) CHARTER FOR LANDSCAPE ARCHITECTURAL EDUCATION) Nowadays, respect and appreciation of the landscape sustainably grow. Landscape Architecture professionals are makers encourage and guide this growth. However, there is still no legal recognition of the landscape profession in Spain. For this reason we present this paper that shows the current framework in which appear a professional educated with masters and specialized courses, but it lacks a defined work area and unique in its business learned. We present the demands of both IFLA and the EFLA and teaching lines proposed by these organizations at European level are distinguished. Furthermore, we analyze the national scene and major training centres are distinguished.

Approximation to earth material from international normative

For centuries, earth has been used as a construction material. Nevertheless, the normative in this matter is very scattered, and the most developed countries, to carry out a construction with this material implies a variety of technical and legal problems. In this paper we review, in an international level, the normative panorama about earth constructions. It analyzes ninety one standards and regulations of countries all around the five continents. These standards represent the state of art that normalizes the earth as a construction material. In this research we analyze the international standards to earth construction, focusing on durability test (spray and drip erosion tests). It analyzes the differences between methods of test. Also we show all results about these tests in two types of compressed earth block.

Computational Methods for Identification of Vibrating Structures

System identification deals with the problem of building mathematical models of dynamical systems based on observed data from the system" [1]. In the context of civil engineering, the system refers to a large scale structure such as a building, bridge, or an offshore structure, and identification mostly involves the determination of modal parameters (the natural frequencies, damping ratios, and mode shapes). This paper presents some modal identification results obtained using a state-of-the-art time domain system identification method (data-driven stochastic subspace algorithms [2]) applied to the output-only data measured in a steel arch bridge. First, a three dimensional finite element model was developed for the numerical analysis of the structure using ANSYS. Modal analysis was carried out and modal parameters were extracted in the frequency range of interest, 0-10 Hz. The results obtained from the finite element modal analysis were used to determine the location of the sensors. After that, ambient vibration tests were conducted during April 23-24, 2009. The response of the structure was measured using eight accelerometers. Two stations of three sensors were formed (triaxial stations). These sensors were held stationary for reference during the test. The two remaining sensors were placed at the different measurement points along the bridge deck, in which only vertical and transversal measurements were conducted (biaxial stations). Point estimate and interval estimate have been carried out in the state space model using these ambient vibration measurements. In the case of parametric models (like state space), the dynamic behaviour of a system is described using mathematical models. Then, mathematical relationships can be established between modal parameters and estimated point parameters (thus, it is common to use experimental modal analysis as a synonym for system identification). Stable modal parameters are found using a stabilization diagram. Furthermore, this paper proposes a method for assessing the precision of estimates of the parameters of state-space models (confidence interval). This approach employs the nonparametric bootstrap procedure [3] and is applied to subspace parameter estimation algorithm. Using bootstrap results, a plot similar to a stabilization diagram is developed. These graphics differentiate system modes from spurious noise modes for a given order system. Additionally, using the modal assurance criterion, the experimental modes obtained have been compared with those evaluated from a finite element analysis. A quite good agreement between numerical and experimental results is observed.

Estudio de la fiabilidad software, según defectos, en controladores de velocidad para robots autónomos móviles

Tanto los robots autónomos móviles como los robots móviles remotamente operados se utilizan con éxito actualmente en un gran número de ámbitos, algunos de los cuales son tan dispares como la limpieza en el hogar, movimiento de productos en almacenes o la exploración espacial. Sin embargo, es difícil garantizar la ausencia de defectos en los programas que controlan dichos dispositivos, al igual que ocurre en otros sectores informáticos. Existen diferentes alternativas para medir la calidad de un sistema en el desempeño de las funciones para las que fue diseñado, siendo una de ellas la fiabilidad. En el caso de la mayoría de los sistemas físicos se detecta una degradación en la fiabilidad a medida que el sistema envejece. Esto es debido generalmente a efectos de desgaste. En el caso de los sistemas software esto no suele ocurrir, ya que los defectos que existen en ellos generalmente no han sido adquiridos con el paso del tiempo, sino que han sido insertados en el proceso de desarrollo de los mismos. Si dentro del proceso de generación de un sistema software se focaliza la atención en la etapa de codificación, podría plantearse un estudio que tratara de determinar la fiabilidad de distintos algoritmos, válidos para desempeñar el mismo cometido, según los posibles defectos que pudieran introducir los programadores. Este estudio básico podría tener diferentes aplicaciones, como por ejemplo elegir el algoritmo menos sensible a los defectos, para el desarrollo de un sistema crítico o establecer procedimientos de verificación y validación, más exigentes, si existe la necesidad de utilizar un algoritmo que tenga una alta sensibilidad a los defectos. En el presente trabajo de investigación se ha estudiado la influencia que tienen determinados tipos de defectos software en la fiabilidad de tres controladores de velocidad multivariable (PID, Fuzzy y LQR) al actuar en un robot móvil específico. La hipótesis planteada es que los controladores estudiados ofrecen distinta fiabilidad al verse afectados por similares patrones de defectos, lo cual ha sido confirmado por los resultados obtenidos. Desde el punto de vista de la planificación experimental, en primer lugar se realizaron los ensayos necesarios para determinar si los controladores de una misma familia (PID, Fuzzy o LQR) ofrecían una fiabilidad similar, bajo las mismas condiciones experimentales. Una vez confirmado este extremo, se eligió de forma aleatoria un representante de clase de cada familia de controladores, para efectuar una batería de pruebas más exhaustiva, con el objeto de obtener datos que permitieran comparar de una forma más completa la fiabilidad de los controladores bajo estudio. Ante la imposibilidad de realizar un elevado número de pruebas con un robot real, así como para evitar daños en un dispositivo que generalmente tiene un coste significativo, ha sido necesario construir un simulador multicomputador del robot. Dicho simulador ha sido utilizado tanto en las actividades de obtención de controladores bien ajustados, como en la realización de los diferentes ensayos necesarios para el experimento de fiabilidad. ABSTRACT Autonomous mobile robots and remotely operated robots are used successfully in very diverse scenarios, such as home cleaning, movement of goods in warehouses or space exploration. However, it is difficult to ensure the absence of defects in programs controlling these devices, as it happens in most computer sectors. There exist different quality measures of a system when performing the functions for which it was designed, among them, reliability. For most physical systems, a degradation occurs as the system ages. This is generally due to the wear effect. In software systems, this does not usually happen, and defects often come from system development and not from use. Let us assume that we focus on the coding stage in the software development pro¬cess. We could consider a study to find out the reliability of different and equally valid algorithms, taking into account any flaws that programmers may introduce. This basic study may have several applications, such as choosing the algorithm less sensitive to pro¬gramming defects for the development of a critical system. We could also establish more demanding procedures for verification and validation if we need an algorithm with high sensitivity to programming defects. In this thesis, we studied the influence of certain types of software defects in the reliability of three multivariable speed controllers (PID, Fuzzy and LQR) designed to work in a specific mobile robot. The hypothesis is that similar defect patterns affect differently the reliability of controllers, and it has been confirmed by the results. From the viewpoint of experimental planning, we followed these steps. First, we conducted the necessary test to determine if controllers of the same family (PID, Fuzzy or LQR) offered a similar reliability under the same experimental conditions. Then, a class representative was chosen at ramdom within each controller family to perform a more comprehensive test set, with the purpose of getting data to compare more extensively the reliability of the controllers under study. The impossibility of performing a large number of tests with a real robot and the need to prevent the damage of a device with a significant cost, lead us to construct a multicomputer robot simulator. This simulator has been used to obtain well adjusted controllers and to carry out the required reliability experiments.

Modelling of Topographic Irregularities for Seismic Site Response

Seismic evaluation methodology is applied to an existing viaduct in the south of Spain, near Granada, which is a medium seismicity region. The influence of both geology and topography in the spatial variability of ground motion are studied as well as seismic hazard analysis and ground motion characterization. Artificial hazard-consistent ground motion records are synthesised applying seismic hazard analysis and site effects are estimated through a diffraction study. Direct BEM is used to calculate the valley displacement response to vertically propagating SV waves and transfer functions are generated allowing the transformation of free field motion to motion at each support. A closed formulae is used to estimate these transfer function. Finally, the results obtained are compared.

Maximum likelihood estimation of new state space models for operational modal analysis

The modal analysis of a structural system consists on computing its vibrational modes. The experimental way to estimate these modes requires to excite the system with a measured or known input and then to measure the system output at different points using sensors. Finally, system inputs and outputs are used to compute the modes of vibration. When the system refers to large structures like buildings or bridges, the tests have to be performed in situ, so it is not possible to measure system inputs such as wind, traffic, . . .Even if a known input is applied, the procedure is usually difficult and expensive, and there are still uncontrolled disturbances acting at the time of the test. These facts led to the idea of computing the modes of vibration using only the measured vibrations and regardless of the inputs that originated them, whether they are ambient vibrations (wind, earthquakes, . . . ) or operational loads (traffic, human loading, . . . ). This procedure is usually called Operational Modal Analysis (OMA), and in general consists on to fit a mathematical model to the measured data assuming the unobserved excitations are realizations of a stationary stochastic process (usually white noise processes). Then, the modes of vibration are computed from the estimated model. The first issue investigated in this thesis is the performance of the Expectation- Maximization (EM) algorithm for the maximum likelihood estimation of the state space model in the field of OMA. The algorithm is described in detail and it is analysed how to apply it to vibration data. After that, it is compared to another well known method, the Stochastic Subspace Identification algorithm. The maximum likelihood estimate enjoys some optimal properties from a statistical point of view what makes it very attractive in practice, but the most remarkable property of the EM algorithm is that it can be used to address a wide range of situations in OMA. In this work, three additional state space models are proposed and estimated using the EM algorithm: • The first model is proposed to estimate the modes of vibration when several tests are performed in the same structural system. Instead of analyse record by record and then compute averages, the EM algorithm is extended for the joint estimation of the proposed state space model using all the available data. • The second state space model is used to estimate the modes of vibration when the number of available sensors is lower than the number of points to be tested. In these cases it is usual to perform several tests changing the position of the sensors from one test to the following (multiple setups of sensors). Here, the proposed state space model and the EM algorithm are used to estimate the modal parameters taking into account the data of all setups. • And last, a state space model is proposed to estimate the modes of vibration in the presence of unmeasured inputs that cannot be modelled as white noise processes. In these cases, the frequency components of the inputs cannot be separated from the eigenfrequencies of the system, and spurious modes are obtained in the identification process. The idea is to measure the response of the structure corresponding to different inputs; then, it is assumed that the parameters common to all the data correspond to the structure (modes of vibration), and the parameters found in a specific test correspond to the input in that test. The problem is solved using the proposed state space model and the EM algorithm. Resumen El análisis modal de un sistema estructural consiste en calcular sus modos de vibración. Para estimar estos modos experimentalmente es preciso excitar el sistema con entradas conocidas y registrar las salidas del sistema en diferentes puntos por medio de sensores. Finalmente, los modos de vibración se calculan utilizando las entradas y salidas registradas. Cuando el sistema es una gran estructura como un puente o un edificio, los experimentos tienen que realizarse in situ, por lo que no es posible registrar entradas al sistema tales como viento, tráfico, . . . Incluso si se aplica una entrada conocida, el procedimiento suele ser complicado y caro, y todavía están presentes perturbaciones no controladas que excitan el sistema durante el test. Estos hechos han llevado a la idea de calcular los modos de vibración utilizando sólo las vibraciones registradas en la estructura y sin tener en cuenta las cargas que las originan, ya sean cargas ambientales (viento, terremotos, . . . ) o cargas de explotación (tráfico, cargas humanas, . . . ). Este procedimiento se conoce en la literatura especializada como Análisis Modal Operacional, y en general consiste en ajustar un modelo matemático a los datos registrados adoptando la hipótesis de que las excitaciones no conocidas son realizaciones de un proceso estocástico estacionario (generalmente ruido blanco). Posteriormente, los modos de vibración se calculan a partir del modelo estimado. El primer problema que se ha investigado en esta tesis es la utilización de máxima verosimilitud y el algoritmo EM (Expectation-Maximization) para la estimación del modelo espacio de los estados en el ámbito del Análisis Modal Operacional. El algoritmo se describe en detalle y también se analiza como aplicarlo cuando se dispone de datos de vibraciones de una estructura. A continuación se compara con otro método muy conocido, el método de los Subespacios. Los estimadores máximo verosímiles presentan una serie de propiedades que los hacen óptimos desde un punto de vista estadístico, pero la propiedad más destacable del algoritmo EM es que puede utilizarse para resolver un amplio abanico de situaciones que se presentan en el Análisis Modal Operacional. En este trabajo se proponen y estiman tres modelos en el espacio de los estados: • El primer modelo se utiliza para estimar los modos de vibración cuando se dispone de datos correspondientes a varios experimentos realizados en la misma estructura. En lugar de analizar registro a registro y calcular promedios, se utiliza algoritmo EM para la estimación conjunta del modelo propuesto utilizando todos los datos disponibles. • El segundo modelo en el espacio de los estados propuesto se utiliza para estimar los modos de vibración cuando el número de sensores disponibles es menor que vi Resumen el número de puntos que se quieren analizar en la estructura. En estos casos es usual realizar varios ensayos cambiando la posición de los sensores de un ensayo a otro (múltiples configuraciones de sensores). En este trabajo se utiliza el algoritmo EM para estimar los parámetros modales teniendo en cuenta los datos de todas las configuraciones. • Por último, se propone otro modelo en el espacio de los estados para estimar los modos de vibración en la presencia de entradas al sistema que no pueden modelarse como procesos estocásticos de ruido blanco. En estos casos, las frecuencias de las entradas no se pueden separar de las frecuencias del sistema y se obtienen modos espurios en la fase de identificación. La idea es registrar la respuesta de la estructura correspondiente a diferentes entradas; entonces se adopta la hipótesis de que los parámetros comunes a todos los registros corresponden a la estructura (modos de vibración), y los parámetros encontrados en un registro específico corresponden a la entrada en dicho ensayo. El problema se resuelve utilizando el modelo propuesto y el algoritmo EM.

Medida de la conductividad térmica con el método de la aguja térmica, basado en la fuente lineal de calor transitorio, para su aplicación en los cerramientos de adobes y bloques de tierra comprimida

Existe un creciente interés internacional por el ahorro energético y la sostenibilidad en la edificación con importantes repercusiones en la Arquitectura. La inercia térmica es un parámetro fundamental para poder valorar energéticamente un edificio en condiciones reales. Para ello es necesario cambiar el enfoque tradicional de transmisión de calor en régimen estacionario por otro en régimen dinámico en el que se analizan las ondas térmicas y el flujo de calor oscilante que atraviesan los cerramientos. Los parámetros que definen la inercia térmica son: el espesor, la difusividad y el ciclo térmico. A su vez la difusividad está determinada por la conductividad térmica, la densidad y el calor específico del material. De estos parámetros la conductividad es el más complejo, variable y difícil de medir, especialmente en los cerramientos de tierra debido a su heterogeneidad y complejidad higrotérmica. En general, los métodos de medida de la conductividad o transmitancias en los paramentos presentan inconvenientes a la hora de medir un edificio construido con tierra: dificultades de implementación, el elevado coste o la fiabilidad de los resultados, principalmente. El Método de la Aguja Térmica (MAT) se basa en el principio de la evolución en el tiempo del calor emitido por una fuente lineal al insertarse en el seno de un material. Se ha escogido este método porque resulta práctico, de bajo coste y de fácil aplicación a gran escala pero tiene serios problemas de fiabilidad y exactitud. En esta tesis se desarrolla un método de medida de la conductividad térmica para Piezas de Albañilería de Tierra Cruda en laboratorio basado en el MAT, se mejora su fiabilidad, se analiza su incertidumbre, se compara con otros métodos de referencia y se aplica en adobes, Bloques de Tierra Comprimida y probetas de tierra estabilizada con distintas proporciones de paja. Este método servirá de base a una posterior aplicación in situ. Finalmente se proponen modelos matemáticos para mejorar la exactitud del dispositivo utilizado y para la estimación de la conductividad de cerramientos de tierra en función de su densidad. Con los resultados obtenidos se analizan las posibilidades de amortiguación y retardo de las ondas térmicas y capacidad de almacenaje de energía de los cerramientos en función de su densidad y humedad. There is growing international interest in energy saving and sustainability in buildings with significant impact on Architecture. Thermal inertia is a key parameter to assess energy in buildings in real conditions. This requires changing the traditional approach to heat transfer in steady state by another in dynamic regime which analyzes the thermal waves and oscillating heat flux passing through the external walls. The parameters defining the thermal inertia are: the thickness, the diffusivity and the thermal cycle. In turn, the diffusivity is determined by the thermal conductivity, density and specific heat of the material. Of these parameters, thermal conductivity is the most complex, variable and difficult to measure, especially in earth walls due to their heterogeneity and hygrothermal complexity. In general, the methods of measurement of conductivity and transmittance in walls have drawbacks when measuring a building with earth: implementation difficulties, high cost, or reliability of the results, mainly. The Thermal Needle Procedure (TNP) is based on the principle of evolution in time of heat from a line source when inserted within a material. This method was chosen because it is a practical, low cost and easy to implement on a large scale but has serious problems of reliability and accuracy. This thesis develops a laboratory method for measuring the thermal conductivity of Masonry Units Unfire Earth-based based on TNP, its uncertainty is analyzed, compared to other reference methods and applies in adobes, Compressed Earth Blocks and stabilized soil specimens with different proportions of straw. This method will form the basis of a subsequent application in situ. Finally, mathematical models are proposed to improve the accuracy of the device used, and to estimate the conductivity of earth enclosures depending on its density. With the results obtained earth enclosures are analyzed to estimate their possibilities of delay and buffer of termal waves and energy storage capacity according to their density and moisture.

Cálculo dinámico simplificado de muros de contención sometidos a empuje activo sísmico

En este trabajo se propone un método de cálculo dinámico para muros de gravedad en estado límite de equilibrio activo. El empuje sísmico se calcula mediante un método pseudo-dinámico que tiene en cuenta la naturaleza de la acción sísmica. El muro se introduce en las ecuaciones de equilibrio con masa, geometría e inercia a rotación. El terreno del cimiento se modela como un semiespacio de suelo homogéneo y se incluye en la formulación la rigidez de éste y la disipación de energía del sistema por radiación de ondas y por amortiguamiento material.

Estimating the modal parameters from multiple measurement setups using a joint state space model

Computing the modal parameters of structural systems often requires processing data from multiple non-simultaneously recorded setups of sensors. These setups share some sensors in common, the so-called reference sensors, which are fixed for all measurements, while the other sensors change their position from one setup to the next. One possibility is to process the setups separately resulting in different modal parameter estimates for each setup. Then, the reference sensors are used to merge or glue the different parts of the mode shapes to obtain global mode shapes, while the natural frequencies and damping ratios are usually averaged. In this paper we present a new state space model that processes all setups at once. The result is that the global mode shapes are obtained automatically, and only a value for the natural frequency and damping ratio of each mode is estimated. We also investigate the estimation of this model using maximum likelihood and the Expectation Maximization algorithm, and apply this technique to simulated and measured data corresponding to different structures.

Estimation of modal parameters in structures using multiple time history records

In Operational Modal Analysis of structures we often have multiple time history records of vibrations measured at different time instants. This work presents a procedure for estimating the modal parameters of the structure processing all the records, that is, using all available information to obtain a single estimate of the modal parameters. The method uses Maximum Likelihood Estimation and the Expectation Maximization algorithm. Finally, it has been applied to various problems for both simulated and real structures and the results show the advantage of the joint analysis proposed.

Operational Modal Analysis using Expectation Maximization Algorithm

This paper presents a time-domain stochastic system identification method based on Maximum Likelihood Estimation and the Expectation Maximization algorithm. The effectiveness of this structural identification method is evaluated through numerical simulation in the context of the ASCE benchmark problem on structural health monitoring. Modal parameters (eigenfrequencies, damping ratios and mode shapes) of the benchmark structure have been estimated applying the proposed identification method to a set of 100 simulated cases. The numerical results show that the proposed method estimates all the modal parameters reasonably well in the presence of 30% measurement noise even. Finally, advantages and disadvantages of the method have been discussed.

Operational modal analisys using joint statistical analysis of multiple records

In Operational Modal Analysis (OMA) of a structure, the data acquisition process may be repeated many times. In these cases, the analyst has several similar records for the modal analysis of the structure that have been obtained at di�erent time instants (multiple records). The solution obtained varies from one record to another, sometimes considerably. The differences are due to several reasons: statistical errors of estimation, changes in the external forces (unmeasured forces) that modify the output spectra, appearance of spurious modes, etc. Combining the results of the di�erent individual analysis is not straightforward. To solve the problem, we propose to make the joint estimation of the parameters using all the records. This can be done in a very simple way using state space models and computing the estimates by maximum-likelihood. The method provides a single result for the modal parameters that combines optimally all the records.

Joint statistical analysis of multiple measurement setups for the estimation of vibrational modes

Computing the modal parameters of large structures in Operational Modal Analysis often requires to process data from multiple non simultaneously recorded setups of sensors. These setups share some sensors in common, the so-called reference sensors that are fixed for all the measurements, while the other sensors are moved from one setup to the next. One possibility is to process the setups separately what result in different modal parameter estimates for each setup. Then the reference sensors are used to merge or glue the different parts of the mode shapes to obtain global modes, while the natural frequencies and damping ratios are usually averaged. In this paper we present a state space model that can be used to process all setups at once so the global mode shapes are obtained automatically and subsequently only a value for the natural frequency and damping ratio of each mode is computed. We also present how this model can be estimated using maximum likelihood and the Expectation Maximization algorithm. We apply this technique to real data measured at a footbridge.