1) Atmospheric transport is driven by three principal forces: gravity, pressure gradients, and the Coriolis force. Horizontal air movement is caused by pressure gradients and balanced by the Coriolis force, resulting in geostrophic flow.
2) Vertical transport in the atmosphere is determined by buoyancy, which depends on temperature lapse rates. Air parcels experiencing adiabatic and isothermal changes cycle through temperatures.
3) Turbulence plays a key role in atmospheric transport. Flows are generally turbulent due to large characteristic lengths and speeds. Turbulent fluxes are parameterized using diffusion coefficients to model spreading.
This document outlines various laws and equations relating to gas mixtures, including:
1) Dalton's Law which states that the total pressure of a gas mixture is equal to the sum of the partial pressures of the individual components.
2) Amagat's Law which describes how the total volume of a gas mixture is equal to the sum of the volumes the components would occupy individually at the same temperature and pressure.
3) Equations for determining the molecular weight, gas constant, and specific heat of a gas mixture based on the properties of its individual components.
Motion is defined as a change in an object's position over time relative to a reference point. It can be described using terms like displacement, velocity, acceleration, and time. Work is defined as the transfer of energy when a force causes an object to move in the direction of the applied force. The SI unit for work is the joule, which is calculated as the product of the applied force and displacement. For example, lifting a 1kg object 2m against gravity performs 20J of work.
1. The document discusses concepts from engineering chemistry including the laws of thermodynamics, kinetics, and related topics.
2. It explains key thermodynamic concepts such as state functions, path functions, the four laws of thermodynamics, entropy, enthalpy, and Gibbs free energy.
3. The document also discusses kinetic concepts such as activation energy, the Arrhenius equation, and enzyme catalysis using the Michaelis-Menten mechanism.
Thermodynamics describes the relationship between heat and work for a system and its surroundings. A system is separated from its surroundings by walls that may allow heat transfer (diathermal) or not (adiabatic). A system's state is specified by properties like pressure, volume, temperature, and mass. The zeroth law states that if two systems are in thermal equilibrium with a third, they are in equilibrium with each other. The first law relates changes in a system's internal energy to heat and work. Thermal processes like isobaric, isochoric, isothermal, and adiabatic involve constant pressure, volume, temperature, or no heat transfer respectively.
1. The document defines key concepts related to ideal gases including the ideal gas law, gas constant, Boyle's law, Charles' law, Avogadro's law, specific heat, ratio of specific heats, entropy change, gas mixtures, and processes involving gases.
2. It provides equations of state for ideal gases, gas mixtures, and non-ideal gases. Equations are given for properties of gas mixtures including volume, pressure, molecular weight, and specific heat.
3. Key thermodynamic processes involving gases are summarized including isobaric and isometric processes for both closed and open systems, with equations provided for heat, work, internal energy, and entropy changes.
The document provides information about physics concepts related to pressure and temperature measurement. It discusses:
- Definitions of pressure, standard pressure units like Pascal and atmospheric pressure in different units.
- Relationship between pressure, density, height of liquid and gravity.
- Conversion between different pressure units.
- Operation of barometers, manometers and gas thermometers to measure pressure and temperature. Barometers use mercury columns, while manometers use liquid columns differentially to measure gas pressures. Gas thermometers use the direct relationship between gas pressure and temperature at constant volume.
The first law of thermodynamics states that energy can be transformed from one form to another, but cannot be created or destroyed. It provides a necessary but not sufficient condition for a process to occur. The first law was established through experiments by Joule showing that work input is proportional to heat output. The first law applies to closed systems and describes the various forms energy can take, such as work, heat, internal energy, and how changes in these forms are related through the principle of conservation of energy.
1) Atmospheric transport is driven by three principal forces: gravity, pressure gradients, and the Coriolis force. Horizontal air movement is caused by pressure gradients and balanced by the Coriolis force, resulting in geostrophic flow.
2) Vertical transport in the atmosphere is determined by buoyancy, which depends on temperature lapse rates. Air parcels experiencing adiabatic and isothermal changes cycle through temperatures.
3) Turbulence plays a key role in atmospheric transport. Flows are generally turbulent due to large characteristic lengths and speeds. Turbulent fluxes are parameterized using diffusion coefficients to model spreading.
This document outlines various laws and equations relating to gas mixtures, including:
1) Dalton's Law which states that the total pressure of a gas mixture is equal to the sum of the partial pressures of the individual components.
2) Amagat's Law which describes how the total volume of a gas mixture is equal to the sum of the volumes the components would occupy individually at the same temperature and pressure.
3) Equations for determining the molecular weight, gas constant, and specific heat of a gas mixture based on the properties of its individual components.
Motion is defined as a change in an object's position over time relative to a reference point. It can be described using terms like displacement, velocity, acceleration, and time. Work is defined as the transfer of energy when a force causes an object to move in the direction of the applied force. The SI unit for work is the joule, which is calculated as the product of the applied force and displacement. For example, lifting a 1kg object 2m against gravity performs 20J of work.
1. The document discusses concepts from engineering chemistry including the laws of thermodynamics, kinetics, and related topics.
2. It explains key thermodynamic concepts such as state functions, path functions, the four laws of thermodynamics, entropy, enthalpy, and Gibbs free energy.
3. The document also discusses kinetic concepts such as activation energy, the Arrhenius equation, and enzyme catalysis using the Michaelis-Menten mechanism.
Thermodynamics describes the relationship between heat and work for a system and its surroundings. A system is separated from its surroundings by walls that may allow heat transfer (diathermal) or not (adiabatic). A system's state is specified by properties like pressure, volume, temperature, and mass. The zeroth law states that if two systems are in thermal equilibrium with a third, they are in equilibrium with each other. The first law relates changes in a system's internal energy to heat and work. Thermal processes like isobaric, isochoric, isothermal, and adiabatic involve constant pressure, volume, temperature, or no heat transfer respectively.
1. The document defines key concepts related to ideal gases including the ideal gas law, gas constant, Boyle's law, Charles' law, Avogadro's law, specific heat, ratio of specific heats, entropy change, gas mixtures, and processes involving gases.
2. It provides equations of state for ideal gases, gas mixtures, and non-ideal gases. Equations are given for properties of gas mixtures including volume, pressure, molecular weight, and specific heat.
3. Key thermodynamic processes involving gases are summarized including isobaric and isometric processes for both closed and open systems, with equations provided for heat, work, internal energy, and entropy changes.
The document provides information about physics concepts related to pressure and temperature measurement. It discusses:
- Definitions of pressure, standard pressure units like Pascal and atmospheric pressure in different units.
- Relationship between pressure, density, height of liquid and gravity.
- Conversion between different pressure units.
- Operation of barometers, manometers and gas thermometers to measure pressure and temperature. Barometers use mercury columns, while manometers use liquid columns differentially to measure gas pressures. Gas thermometers use the direct relationship between gas pressure and temperature at constant volume.
The first law of thermodynamics states that energy can be transformed from one form to another, but cannot be created or destroyed. It provides a necessary but not sufficient condition for a process to occur. The first law was established through experiments by Joule showing that work input is proportional to heat output. The first law applies to closed systems and describes the various forms energy can take, such as work, heat, internal energy, and how changes in these forms are related through the principle of conservation of energy.
The document discusses water and energy budgets. It explains that a budget represents the variation of a given quantity within a control volume over a time interval, and is the algebraic sum of inputs and outputs. It provides examples of water budgets for soil volumes and atmospheric layers, accounting for precipitation, evapotranspiration, runoff and other fluxes. It also discusses the components of an energy budget, including net radiation, heat conduction, heat of vaporization and more.
This document provides an introduction to concepts related to delineating a hydrographic catchment from a digital elevation model. It discusses how a DEM is discretized into a grid with elevation values for each cell. Primary topographic attributes that can be derived from the DEM like altitude, slope, gradient, and curvature are described. It explains how drainage directions and hydrographic networks can be determined from the DEM and how this allows delineation of contributing areas and hydrographic catchments. The objectives are to introduce these concepts and lay the groundwork for subsequent lectures on using software like JGrass to perform catchment delineation.
The document discusses The Real Book, which refers to collections of lead sheets that contain standard jazz songs. It provides background on the original Real Book from the 1970s, which was compiled illegally by students at Berklee College of Music. The document then explains that the title "Real Book" is being used for this collection of hydrology lecture slides, which provide systematic knowledge about the topic beyond textbooks. It aims to direct students towards further resources while communicating information.
The document provides an introduction to hydrology, including:
- Defining hydrology as the science studying the water cycle and flows between the atmosphere, land, and oceans.
- Describing the key elements of the water cycle, including precipitation, infiltration, evaporation, and the spatial and temporal scales involved.
- Noting that the water cycle sustains life on Earth, shapes its surface, and regulates the climate.
Tropical climates have warm temperatures without frost year-round and seasonal variations in rainfall. Vegetation includes lower, spiny plants that reduce their leaves during long dry seasons. Fauna ranges from insects and rodents to large herbivores and carnivores. People live in tribes and engage in subsistence farming and livestock grazing, especially cattle, adapted to the climate. Economic activities focus on small-scale agriculture when rainfall allows and cattle ranching in drier areas.
The document discusses the subtropical climate and its key characteristics. It is located between the tropics and has warm temperatures year-round, with annual averages above 18°C. Precipitation varies between dry and damp subtropical climates, but is generally moderate. Vegetation includes forests, steppe lands, and plants adapted to heat and drought. Fauna is diverse and nocturnal. Habitants include some tribes and farmers, though population density is generally low. The economy traditionally involved hunting and agriculture but now also includes intensive farming and tourism.
This document describes a graphical language for representing reservoir systems using time-continuous Petri nets (TCPN).
Places in the TCPN represent water storages such as volumes of groundwater or energy/momentum contents. Transitions represent fluxes between storages. The TCPN uses colors to distinguish different types of quantities (mass, energy, etc.) and storages. Connections between places and transitions represent differential equations governing the system.
An example TCPN represents a system of three differential equations with three storages, inputs, and both linear and nonlinear fluxes. Additional information like parameter values can be provided in tables. Adjacency matrixes describe the connections between places and transitions. TCPNs provide an algebraic framework for conceptual
This document defines key vocabulary terms related to weather and climate, including weather, atmosphere, climate, precipitation, wind, anemometer, meteorologist, and wind vane. It explains that weather is the current atmospheric conditions, atmosphere is the air surrounding Earth, and climate is the combination of weather over time in a region. Precipitation refers to rain, snow or hail falling to the surface. An anemometer measures wind speed and a wind vane measures wind direction. Meteorologists are scientists who study weather and the atmosphere.
Higher concentrations of heavy metal ions, phosphates, and nitrates can be removed from water through precipitation. Precipitation occurs when soluble compounds are added to water causing the dissolved ions to form insoluble solids that settle out. Common precipitating agents include sulfides, which form insoluble sulfide salts, and hydroxides, which form insoluble hydroxide salts. The solubility of precipitates depends on the solubility product constant (Ksp), with precipitation occurring when the product of the ion concentrations exceeds the Ksp value.
Presented by Mark Giordano
Integrated Water Resources Management provides a set of reasoned principles that, if followed, would lead us to an improved water future. This promise plus the backing of important international organizations has allowed IWRM ideals to acquire a near monopoly on water management discourse. This is unfortunate because, while the potential benefits of IWRM are large, its implementation comes with its own set of economic, political and time costs, costs which are not always considered in IWRM policy advocacy. Failure to recognize these costs can sometimes result in outcomes counter to the goals of water sector reform. The ubiquity of IWRM in policy discussions means that lower cost and potentially more effective options are sometimes not considered. This presentation highlights these points by describing the sometimes neglected costs of IWRM implementation, particularly in developing country contexts and provides a set of case studies (in India, Central Asia and China) examining solutions to water problems whose methods run counter to IWRM.
Daily evapotranspiration by combining remote sensing with ground observations...CIMMYT
This document discusses combining remote sensing data with ground observations to estimate daily evapotranspiration (ET) for agricultural water management. It summarizes using remote sensing to model spatial land surface temperature and vegetation cover hourly, integrating them to compute daily ET. It also describes using wireless sensors at an experimental cotton field in Maricopa, Arizona to monitor crops and irrigation as part of an integrated monitoring system for irrigation scheduling. The goal is to provide reasonably accurate and cost-effective daily ET estimates at resolutions useful to growers.
A sensitivity Analysis of Eddy Covariance Data Processing Methods for Evapotr...Troy Bernier
The document discusses sensor errors found in a microclimate station in the Florida Everglades wetlands that could lead to inaccurate water budget calculations. Errors up to 22.34% were found in rainfall and evapotranspiration sensors, which could result in water budgets being off by over 7 inches in a year. Such large errors from sensors in small watersheds can create seriously inaccurate water budgets that could cause problems like perceived drought conditions or poor infrastructure planning.
The document discusses various hydrological measurement quantities and instruments. It describes 8 main hydrological quantities of interest: temperature, humidity, precipitation, radiation, wind, pressure, wetting, and evapotranspiration. It then explains principles and instruments for measuring temperature, humidity, and soil moisture, including thermometers, hygrometers, psychrometers, lysimeters, tensiometers, and instruments measuring electrical conductivity, thermal conductivity, and dielectric constants.
The document discusses the measurement and representation of hydrological quantities. It notes that hydrological data has complex trends that are nonlinear and influenced by many factors. Statistical tools must be used to describe hydrological quantities given their spatiotemporal variability. Examples of typical problems in measuring quantities like precipitation, river flows, and soil moisture are provided.
This document discusses peak river flows and flow hydrology. It introduces the concept of a peak flow, shows a graph of discharge over time as an example, and discusses precipitation patterns and the calculation of effective precipitation. It also discusses the instantaneous unit hydrograph method for summing surface runoff over a basin to determine discharge at the basin outlet.
The document summarizes the activities of the Platform Water Management in the Alps over the past two years and outlines its planned activities for the next period. It discusses workshops held on sediment management, hydropeaking, and hydropower that brought together administrators, practitioners, and stakeholders. It also describes dissemination of guidelines on small hydropower and platform meetings. Going forward, the document outlines three planned workshops on local adaptation to climate change, flood risk prevention, and river management, as well as a conference on water in the Alps. The goals are to address EU directives in an alpine context and local adaptation to climate change.
An evapotranspiration (ET) bed uses evaporation and plant transpiration to treat wastewater. It consists of storage trenches filled with crushed stone or other media, surrounded by loamy soil and planted with grass. Wastewater flows from the septic tank into the distribution pipes in the trenches. The water then evaporates or is absorbed by plant roots and transpired out of their leaves. Proper maintenance of the grass cover and diversion of rainfall runoff are needed for the system to function effectively.
Crop Et And Implications For Irrigationcarterjfranz
Crop coefficient studies were conducted at the Tal Amara Research Station in Lebanon's Bekka Valley to determine optimal irrigation volumes for sunflowers, soybeans, wheat, and corn. Deficit irrigation experiments on sunflowers found that yield was reduced by 25% during early flowering but only 14% during mid-flowering. Seed yield actually increased with deficit irrigation during seed formation. The studies provide crop water use data and coefficients to inform sustainable irrigation planning for farmers in the water-stressed Bekka Valley region.
1. The document outlines the class schedule for an advanced soil and water engineering course, covering topics like physical characteristics of soil, micrometeorology, soil water, and applications of soil and water engineering.
2. It discusses the concepts of surface radiation balance and heat transfer, including equations for net radiation, solar radiation, reflected radiation, and outgoing longwave radiation.
3. It explains the components of surface heat balance - net radiation, ground heat flux, sensible heat flux, and latent heat flux - and the equations governing heat and vapor transfer through atmospheric resistances.
The document discusses water and energy budgets. It explains that a budget represents the variation of a given quantity within a control volume over a time interval, and is the algebraic sum of inputs and outputs. It provides examples of water budgets for soil volumes and atmospheric layers, accounting for precipitation, evapotranspiration, runoff and other fluxes. It also discusses the components of an energy budget, including net radiation, heat conduction, heat of vaporization and more.
This document provides an introduction to concepts related to delineating a hydrographic catchment from a digital elevation model. It discusses how a DEM is discretized into a grid with elevation values for each cell. Primary topographic attributes that can be derived from the DEM like altitude, slope, gradient, and curvature are described. It explains how drainage directions and hydrographic networks can be determined from the DEM and how this allows delineation of contributing areas and hydrographic catchments. The objectives are to introduce these concepts and lay the groundwork for subsequent lectures on using software like JGrass to perform catchment delineation.
The document discusses The Real Book, which refers to collections of lead sheets that contain standard jazz songs. It provides background on the original Real Book from the 1970s, which was compiled illegally by students at Berklee College of Music. The document then explains that the title "Real Book" is being used for this collection of hydrology lecture slides, which provide systematic knowledge about the topic beyond textbooks. It aims to direct students towards further resources while communicating information.
The document provides an introduction to hydrology, including:
- Defining hydrology as the science studying the water cycle and flows between the atmosphere, land, and oceans.
- Describing the key elements of the water cycle, including precipitation, infiltration, evaporation, and the spatial and temporal scales involved.
- Noting that the water cycle sustains life on Earth, shapes its surface, and regulates the climate.
Tropical climates have warm temperatures without frost year-round and seasonal variations in rainfall. Vegetation includes lower, spiny plants that reduce their leaves during long dry seasons. Fauna ranges from insects and rodents to large herbivores and carnivores. People live in tribes and engage in subsistence farming and livestock grazing, especially cattle, adapted to the climate. Economic activities focus on small-scale agriculture when rainfall allows and cattle ranching in drier areas.
The document discusses the subtropical climate and its key characteristics. It is located between the tropics and has warm temperatures year-round, with annual averages above 18°C. Precipitation varies between dry and damp subtropical climates, but is generally moderate. Vegetation includes forests, steppe lands, and plants adapted to heat and drought. Fauna is diverse and nocturnal. Habitants include some tribes and farmers, though population density is generally low. The economy traditionally involved hunting and agriculture but now also includes intensive farming and tourism.
This document describes a graphical language for representing reservoir systems using time-continuous Petri nets (TCPN).
Places in the TCPN represent water storages such as volumes of groundwater or energy/momentum contents. Transitions represent fluxes between storages. The TCPN uses colors to distinguish different types of quantities (mass, energy, etc.) and storages. Connections between places and transitions represent differential equations governing the system.
An example TCPN represents a system of three differential equations with three storages, inputs, and both linear and nonlinear fluxes. Additional information like parameter values can be provided in tables. Adjacency matrixes describe the connections between places and transitions. TCPNs provide an algebraic framework for conceptual
This document defines key vocabulary terms related to weather and climate, including weather, atmosphere, climate, precipitation, wind, anemometer, meteorologist, and wind vane. It explains that weather is the current atmospheric conditions, atmosphere is the air surrounding Earth, and climate is the combination of weather over time in a region. Precipitation refers to rain, snow or hail falling to the surface. An anemometer measures wind speed and a wind vane measures wind direction. Meteorologists are scientists who study weather and the atmosphere.
Higher concentrations of heavy metal ions, phosphates, and nitrates can be removed from water through precipitation. Precipitation occurs when soluble compounds are added to water causing the dissolved ions to form insoluble solids that settle out. Common precipitating agents include sulfides, which form insoluble sulfide salts, and hydroxides, which form insoluble hydroxide salts. The solubility of precipitates depends on the solubility product constant (Ksp), with precipitation occurring when the product of the ion concentrations exceeds the Ksp value.
Presented by Mark Giordano
Integrated Water Resources Management provides a set of reasoned principles that, if followed, would lead us to an improved water future. This promise plus the backing of important international organizations has allowed IWRM ideals to acquire a near monopoly on water management discourse. This is unfortunate because, while the potential benefits of IWRM are large, its implementation comes with its own set of economic, political and time costs, costs which are not always considered in IWRM policy advocacy. Failure to recognize these costs can sometimes result in outcomes counter to the goals of water sector reform. The ubiquity of IWRM in policy discussions means that lower cost and potentially more effective options are sometimes not considered. This presentation highlights these points by describing the sometimes neglected costs of IWRM implementation, particularly in developing country contexts and provides a set of case studies (in India, Central Asia and China) examining solutions to water problems whose methods run counter to IWRM.
Daily evapotranspiration by combining remote sensing with ground observations...CIMMYT
This document discusses combining remote sensing data with ground observations to estimate daily evapotranspiration (ET) for agricultural water management. It summarizes using remote sensing to model spatial land surface temperature and vegetation cover hourly, integrating them to compute daily ET. It also describes using wireless sensors at an experimental cotton field in Maricopa, Arizona to monitor crops and irrigation as part of an integrated monitoring system for irrigation scheduling. The goal is to provide reasonably accurate and cost-effective daily ET estimates at resolutions useful to growers.
A sensitivity Analysis of Eddy Covariance Data Processing Methods for Evapotr...Troy Bernier
The document discusses sensor errors found in a microclimate station in the Florida Everglades wetlands that could lead to inaccurate water budget calculations. Errors up to 22.34% were found in rainfall and evapotranspiration sensors, which could result in water budgets being off by over 7 inches in a year. Such large errors from sensors in small watersheds can create seriously inaccurate water budgets that could cause problems like perceived drought conditions or poor infrastructure planning.
The document discusses various hydrological measurement quantities and instruments. It describes 8 main hydrological quantities of interest: temperature, humidity, precipitation, radiation, wind, pressure, wetting, and evapotranspiration. It then explains principles and instruments for measuring temperature, humidity, and soil moisture, including thermometers, hygrometers, psychrometers, lysimeters, tensiometers, and instruments measuring electrical conductivity, thermal conductivity, and dielectric constants.
The document discusses the measurement and representation of hydrological quantities. It notes that hydrological data has complex trends that are nonlinear and influenced by many factors. Statistical tools must be used to describe hydrological quantities given their spatiotemporal variability. Examples of typical problems in measuring quantities like precipitation, river flows, and soil moisture are provided.
This document discusses peak river flows and flow hydrology. It introduces the concept of a peak flow, shows a graph of discharge over time as an example, and discusses precipitation patterns and the calculation of effective precipitation. It also discusses the instantaneous unit hydrograph method for summing surface runoff over a basin to determine discharge at the basin outlet.
The document summarizes the activities of the Platform Water Management in the Alps over the past two years and outlines its planned activities for the next period. It discusses workshops held on sediment management, hydropeaking, and hydropower that brought together administrators, practitioners, and stakeholders. It also describes dissemination of guidelines on small hydropower and platform meetings. Going forward, the document outlines three planned workshops on local adaptation to climate change, flood risk prevention, and river management, as well as a conference on water in the Alps. The goals are to address EU directives in an alpine context and local adaptation to climate change.
An evapotranspiration (ET) bed uses evaporation and plant transpiration to treat wastewater. It consists of storage trenches filled with crushed stone or other media, surrounded by loamy soil and planted with grass. Wastewater flows from the septic tank into the distribution pipes in the trenches. The water then evaporates or is absorbed by plant roots and transpired out of their leaves. Proper maintenance of the grass cover and diversion of rainfall runoff are needed for the system to function effectively.
Crop Et And Implications For Irrigationcarterjfranz
Crop coefficient studies were conducted at the Tal Amara Research Station in Lebanon's Bekka Valley to determine optimal irrigation volumes for sunflowers, soybeans, wheat, and corn. Deficit irrigation experiments on sunflowers found that yield was reduced by 25% during early flowering but only 14% during mid-flowering. Seed yield actually increased with deficit irrigation during seed formation. The studies provide crop water use data and coefficients to inform sustainable irrigation planning for farmers in the water-stressed Bekka Valley region.
1. The document outlines the class schedule for an advanced soil and water engineering course, covering topics like physical characteristics of soil, micrometeorology, soil water, and applications of soil and water engineering.
2. It discusses the concepts of surface radiation balance and heat transfer, including equations for net radiation, solar radiation, reflected radiation, and outgoing longwave radiation.
3. It explains the components of surface heat balance - net radiation, ground heat flux, sensible heat flux, and latent heat flux - and the equations governing heat and vapor transfer through atmospheric resistances.
1. The document discusses long wave radiation emitted by the Earth's surface and atmosphere. It describes the Earth as a gray body that emits radiation in the infrared band given its average surface temperature of 288K.
2. It explains that the atmosphere absorbs and re-emits long wave radiation from the Earth's surface, and without this greenhouse effect the average surface temperature would be around -17C instead of 15C.
3. It provides equations to calculate long wave radiation from a surface based on the surface temperature and the atmospheric emissivity and temperature, noting that multiple parameterizations exist to estimate the atmospheric emissivity.
Climate Sensitivity, Forcings, And Feedbacks_2.pptxObulReddy61
This document discusses climate forcings and feedbacks. It provides examples of climate forcings such as changes in greenhouse gases, solar irradiance, volcanic eruptions and aerosols. It also discusses examples of feedbacks like water vapor, ice-albedo, and clouds. The document shows how feedbacks amplify the initial response to forcing and increase climate sensitivity. It presents estimates of climate sensitivity from 1D models and comparisons of feedbacks simulated in climate models.
The document provides instructions for 8 tasks to complete related to Isaac Newton and his laws of motion. The tasks direct the student to various websites to find information about Newton's life, contributions to science, his 3 laws of motion, and examples of each law. The final task involves calculating the time it would take a car to stop if the engine suddenly stopped while traveling at a given speed, taking into account various friction forces.
This document discusses the key equations and concepts related to tropical climatology. It includes:
1) The seven equations that describe the seven unknowns in the complete tropical climate model, including momentum, continuity, thermodynamic energy, equation of state, and moisture equations.
2) Descriptions of key forces like pressure gradient force, Coriolis force, and geostrophic balance between these two forces.
3) How differential heating between land and oceans drives atmospheric circulations like sea breezes and land breezes through differences in specific heat.
4) Definitions of the tropics based on factors like the sun's overhead position, net radiation balances, and predominant easterly wind patterns between 30
The document provides an overview of hillslope hydrology and summarizes key concepts. It discusses Richards' equation, which describes water movement in unsaturated soils. Through a series of assumptions and simplifications, the equation can be used to model hydrological processes on hillslopes. It also presents the derivation of simplified forms of Richards' equation that describe saturated vertical flow and lateral flow on hillslopes.
Environmental Studies. Environmental Issues.Jobin Abraham
This document discusses several environmental issues including climate change, global warming, the greenhouse effect, acid rain, and depletion of the ozone layer. It provides definitions and explanations of these topics, noting that they are interdependent and can have similar causes from both human activities and natural events. Specific impacts of climate change and global warming are outlined, such as rising temperatures, sea levels, and extreme weather events. The greenhouse effect is explained as a process that occurs naturally but has been intensified by human activities like burning fossil fuels.
The Sun is a star located at the center of our solar system. It is about 4.6 billion years old and has a mass of 2x1030 kg. Its surface temperature is around 5,500°C and is composed primarily of hydrogen and helium. The Sun has an interior core and atmosphere made up of the photosphere, chromosphere, and corona. It undergoes a solar cycle of sunspot activity every 11 years and can produce powerful solar flares. The Sun's magnetic field and solar wind influence the entire solar system and are responsible for phenomena like the Northern Lights. The Sun provides Earth with heat and light that are essential to sustaining life.
Global warming is caused by greenhouse gases like carbon dioxide trapping heat in the atmosphere that would otherwise escape to space. This occurs when fossil fuels are burned, releasing CO2. Effects of global warming include melting glaciers and ice sheets, rising sea levels, more extreme weather, and disruption of habitats. The Kyoto Protocol committed developed countries to reducing greenhouse gas emissions, though impacts are already being felt globally through changes in water availability, food security, and energy access. Estimates indicate the Earth's average surface temperature has risen about 0.9°C from pre-industrial levels due to increased CO2 in the atmosphere from human activities like burning fossil fuels.
The document describes characteristics of Earth's atmosphere including its composition of nitrogen, oxygen, argon, and other gases. It discusses air pressure and how it is created by the weight of the atmosphere above. The atmosphere provides services like moderating temperatures, protecting from radiation, and powering weather systems. It describes how energy is transferred through the atmosphere using radiation, conduction, and convection. Specific layers of the atmosphere and processes like the greenhouse effect and water cycle are also outlined.
This document provides an overview of piezoelectricity including its history, internal working, materials, effects, and applications. It describes how certain crystals produce an electric charge when mechanically stressed (piezoelectric effect) or deform when an electric field is applied (reverse piezoelectric effect). Common piezoelectric materials include quartz, barium titanate, lead zirconate titanate. The document also discusses piezoelectric sensors, actuators, transducers and their uses in applications such as energy harvesting floor tiles, medical devices, and ignition lighters.
This document provides an overview of piezoelectricity including its history, internal working, materials, effects, and applications. It describes how certain crystals produce an electric charge when mechanically stressed (direct piezoelectric effect) or change shape when exposed to an electric field (reverse effect). Common piezoelectric materials include quartz, ceramics, and polymers. The document outlines key piezoelectric applications such as sensors, actuators, generators, and transducers used in devices like lighters, microphones, and medical equipment.
Irreversibility of mechanical and hydrodynamic instabilitiesAlejandro Jenkins
The literature on dynamical systems has, for the most part, considered self-oscillators (i.e., systems capable of generating and maintaining a periodic motion at the expense of an external energy source with no corresponding periodicity) either as applications of the concepts of limit cycle and Hopf bifurcation in the theory of differential equations, or else as instability problems in feedback control systems. Here we outline a complementary approach, based on physical considerations of work extraction and thermodynamic irreversibility. We illustrate the power of this method with two concrete examples: the mechanical instability of elastic discs that spin at super-critical speeds, and the hydrodynamic Kelvin-Helmholtz instability of the interface between fluid layers with different tangential velocities. Our treatment clarifies the necessary role of frictional or viscous dissipation (and therefore of irreversibility), while revealing an underlying unity to the physics of many irreversible processes that generate mechanical work and an autonomous temporal structure (periodic, quasi-periodic, or chaotic) in the presence of an out-of equilibrium background.
The document provides an introduction to climate change, covering the science of climate change including the greenhouse effect and greenhouse gases, climate change impacts, and climate change policies and response measures. It discusses the natural greenhouse effect, the key greenhouse gases, and how increased greenhouse gases are leading to global warming and climate change impacts. It also briefly outlines climate change modeling and projections for future temperature and precipitation changes, as well as some potential impacts of climate change.
Carbon dioxide may not be the main cause of northern hemisphere warming. Heat generated by human activities may be a significant contributor. Polar ice variations are compared.
Hollow earth, contrails & global warming calculations lectureMarcus 2012
http://marcusvannini2012.blogspot.com/
http://www.marcusmoon2022.org/designcontest.htm
Shoot for the moon and if you miss you'll land among the stars...
This document discusses the use of satellite soil moisture data for hydrological applications. It summarizes research validating satellite soil moisture products against in situ observations across different scales. It also describes a method called SM2RAIN that estimates rainfall from satellite soil moisture observations by inverting the soil water balance equation. Initial tests of SM2RAIN show good agreement between estimated and observed rainfall.
This contains the lecture about how to read data from the console. And obviously it contains also other information: about UML, about TextIO class and other stuff. See also http://abouthydrology.blogspot.it/2013/07/java-for-hydrologists-101.html for more information and for the other slides
The document discusses a Java program that solves linear equations. It begins by outlining objectives and analyzing the problem of solving for one variable in an equation of the form "ax + b = 0". It then shows the initial coding of a simple program to solve a specific case. The document goes on to discuss improving the program by making it more general and introducing object-oriented programming concepts like classes, methods and information hiding. It provides annotated code and explanations for a class called LinearEquationSolver that takes parameters to solve any linear equation, unless the coefficient of x is 0.
This is the implementation with explanations of a Hello World simple program. It is useful to document keyword and Java modifiers, as well as how to execute a program.
The document provides an introduction to using the Eclipse Java IDE for beginners learning Java. It recommends first understanding basic Java concepts by reading introductory books before using an IDE. It then directs the reader to an external website that provides instructions on installing and using Eclipse's basic features. The document stresses the importance of self-practice and mentions several other tools like Git, Ant, and Maven that programmers should learn but doesn't provide details as the author is also still learning.
This document provides an introduction to solar radiation and its role in powering the water cycle. It discusses the composition and structure of the Sun, and how it produces radiation through nuclear fusion. While solar radiation is generally constant, it exhibits variations in the form of solar spots and an 11-year activity cycle. The amount of radiation emitted by any body is determined by the Stefan-Boltzmann law, which relates radiation to the body's temperature and emissivity.
This introduces the Open Source GIS JGrass. Other useful tools are the udig Walkthrough -1 and 2 from the udig site, and obviously the main resources are on www.jgrass.org. Other presentations about JGrass are available from slideshare. Serach them!
The document provides an introduction to geographic information systems (GIS) and land information systems. It defines GIS as a set of tools for collecting, modeling, manipulating, analyzing and presenting spatially referenced data. GIS allows for the overlay of different data layers to gain a better understanding of the factors that characterize an area. The document discusses the history of GIS, its components and functions, as well as how it represents spatial data through raster files, vector files, and other methods.
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it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
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Temple of Asclepius in Thrace. Excavation resultsKrassimira Luka
The temple and the sanctuary around were dedicated to Asklepios Zmidrenus. This name has been known since 1875 when an inscription dedicated to him was discovered in Rome. The inscription is dated in 227 AD and was left by soldiers originating from the city of Philippopolis (modern Plovdiv).
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(𝐓𝐋𝐄 𝟏𝟎𝟎) (𝐋𝐞𝐬𝐬𝐨𝐧 𝟏)-𝐏𝐫𝐞𝐥𝐢𝐦𝐬
𝐃𝐢𝐬𝐜𝐮𝐬𝐬 𝐭𝐡𝐞 𝐄𝐏𝐏 𝐂𝐮𝐫𝐫𝐢𝐜𝐮𝐥𝐮𝐦 𝐢𝐧 𝐭𝐡𝐞 𝐏𝐡𝐢𝐥𝐢𝐩𝐩𝐢𝐧𝐞𝐬:
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𝐄𝐱𝐩𝐥𝐚𝐢𝐧 𝐭𝐡𝐞 𝐍𝐚𝐭𝐮𝐫𝐞 𝐚𝐧𝐝 𝐒𝐜𝐨𝐩𝐞 𝐨𝐟 𝐚𝐧 𝐄𝐧𝐭𝐫𝐞𝐩𝐫𝐞𝐧𝐞𝐮𝐫:
-Define entrepreneurship, distinguishing it from general business activities by emphasizing its focus on innovation, risk-taking, and value creation. Students will describe the characteristics and traits of successful entrepreneurs, including their roles and responsibilities, and discuss the broader economic and social impacts of entrepreneurial activities on both local and global scales.
This document provides an overview of wound healing, its functions, stages, mechanisms, factors affecting it, and complications.
A wound is a break in the integrity of the skin or tissues, which may be associated with disruption of the structure and function.
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2. “The rain patters, the leaf quivers”
Rabindranath Tagore
Saturday, September 11, 2010
3. Precipitations
Riccardo Rigon
Objectives:
3
•To Give an introduction to general circulation phenomena and a description
of the atmospheric phenomena that are correlated to precipitation
•To introduce a minimum of atmospheric thermodynamics and some clues
regarding cloud formation
•To speak of precipitations, their formation in the atmosphere, and their
characterisations on the ground
Saturday, September 11, 2010
5. Some Atmospheric Physics
Riccardo Rigon
!"#$#"%"#& '%($()"*#$(+%,-./'/#(./#./$(#
('$",-%'(#0'%+#(./#/12$(%'#(%#(./#-%3/,#
4%235#6/#7$'')/5#%2(#68#$#5)'/7(#(./'+$3#7/33
Foufula-Georgiou,2008
5
Saturday, September 11, 2010
6. Some Atmospheric Physics
Riccardo Rigon
D = 2 ω V sin φ
6
But the Earth rotates on its own axis
And this means that all bodies are subject to the Coriolis force
In the northern hemisphere, a body moving at non-null velocity is deviated
to the right. In the southern hemisphere, to the left.
Saturday, September 11, 2010
7. Some Atmospheric Physics
Riccardo Rigon
D = 2 ω V sin φ
7
But the Earth rotates on its own axis
And this means that all bodies are subject to the Coriolis force
Saturday, September 11, 2010
8. Some Atmospheric Physics
Riccardo Rigon
D = 2 ω V sin φ
7
Coriolis Force
But the Earth rotates on its own axis
And this means that all bodies are subject to the Coriolis force
Saturday, September 11, 2010
9. Some Atmospheric Physics
Riccardo Rigon
D = 2 ω V sin φ
7
Coriolis Force
Rotational velocity of
the Earth
But the Earth rotates on its own axis
And this means that all bodies are subject to the Coriolis force
Saturday, September 11, 2010
10. Some Atmospheric Physics
Riccardo Rigon
D = 2 ω V sin φ
7
Coriolis Force
Rotational velocity of
the Earth
Relative velocity of
the object considered
But the Earth rotates on its own axis
And this means that all bodies are subject to the Coriolis force
Saturday, September 11, 2010
11. Some Atmospheric Physics
Riccardo Rigon
D = 2 ω V sin φ
7
Coriolis Force
Rotational velocity of
the Earth
Relative velocity of
the object considered
Latitude of the object
considered
But the Earth rotates on its own axis
And this means that all bodies are subject to the Coriolis force
Saturday, September 11, 2010
12. Some Atmospheric Physics
Riccardo Rigon
8
Thus, the air masses rotate around the
centres of low and high pressure
High pressure
polar, cold
Easterlies
cold
Westerlies, warm
High pressure
subtropical
warm
Polar
front
Low pressure
zone
Saturday, September 11, 2010
14. Some Atmospheric Physics
Riccardo Rigon
Foufula-Georgiou,2008
10
!"#$%#&#'()$*+'*,)(-+.&
+&$($'.-(-+&%$(-/.01"#'#
Forming a complex global
circulation system
Saturday, September 11, 2010
16. Some Atmospheric Physics
Riccardo Rigon
12
The forces of the pressure gradient...
Pressure, mb
Isobaric surfaces
surface of the ground
surface of the ground
Pressure, mb
pressure gradienthigher
pressure
lower
pressure
map at 1,000m altitude
isobar
Saturday, September 11, 2010
17. Some Atmospheric Physics
Riccardo Rigon
13
...generate winds
The sea breeze
Sea Land
Day
Night
Sea Land
Plane
Valley
Plane
Valley
WarmWarm
ColdCold
Pressure
gradient
Pressure
gradient
Saturday, September 11, 2010
19. Some Atmospheric Physics
Riccardo Rigon
15
The hydrostatic equilibrium of the atmosphere
Column with
section of unit area
Ground
Pressure = p + dp
Pressure = p
Saturday, September 11, 2010
20. Some Atmospheric Physics
Riccardo Rigon
16
dp = −g(z) ρ(z)dz
The hydrostatic equilibrium of the atmosphere
Saturday, September 11, 2010
21. Some Atmospheric Physics
Riccardo Rigon
16
dp = −g(z) ρ(z)dz
V a r i a t i o n i n
pressure
The hydrostatic equilibrium of the atmosphere
Saturday, September 11, 2010
22. Some Atmospheric Physics
Riccardo Rigon
16
dp = −g(z) ρ(z)dz
V a r i a t i o n i n
pressure
Acceleration
due to gravity
The hydrostatic equilibrium of the atmosphere
Saturday, September 11, 2010
23. Some Atmospheric Physics
Riccardo Rigon
16
dp = −g(z) ρ(z)dz
V a r i a t i o n i n
pressure
Acceleration
due to gravity
Air density
The hydrostatic equilibrium of the atmosphere
Saturday, September 11, 2010
24. Some Atmospheric Physics
Riccardo Rigon
16
dp = −g(z) ρ(z)dz
V a r i a t i o n i n
pressure
Acceleration
due to gravity
Air density
Thickness of the
air layer
The hydrostatic equilibrium of the atmosphere
Saturday, September 11, 2010
25. Some Atmospheric Physics
Riccardo Rigon
17
dp = −g(z) ρ(z)dz
Ideal Gas Law
ρ(z) =
p(z)
R T(z)
The hydrostatic equilibrium of the atmosphere
Saturday, September 11, 2010
26. Some Atmospheric Physics
Riccardo Rigon
18
dp = −g(z) ρ(z)dz
Temperature
Pressure
ρ(z) =
p(z)
R T(z)
The hydrostatic equilibrium of the atmosphere
Saturday, September 11, 2010
27. Some Atmospheric Physics
Riccardo Rigon
18
dp = −g(z) ρ(z)dz
Air constant
Temperature
Pressure
ρ(z) =
p(z)
R T(z)
The hydrostatic equilibrium of the atmosphere
Saturday, September 11, 2010
28. Some Atmospheric Physics
Riccardo Rigon
18
dp = −g(z) ρ(z)dz
Air constant
Temperature
Air density
Pressure
ρ(z) =
p(z)
R T(z)
The hydrostatic equilibrium of the atmosphere
Saturday, September 11, 2010
29. Some Atmospheric Physics
Riccardo Rigon
19
dp(z) = −g(z)
p(z)
R T(z)
dz
dp
p
= −g(z)
p(z)
R T(z)
dz
p(z)
p(0)
dp
p
= −
z
0
g(z)
p(z)
R T(z)
dz
The hydrostatic equilibrium of the atmosphere
Saturday, September 11, 2010
30. Some Atmospheric Physics
Riccardo Rigon
20
log
p(z)
p(0)
= −
z
0
g(z)
R T(z)
dz
log
p(z)
p(0)
≈
g
R
z
0
1
T(z)
dz
The hydrostatic equilibrium of the atmosphere
Saturday, September 11, 2010
31. Some Atmospheric Physics
Riccardo Rigon
The first law of thermodynamics
with the help of the second
U = U(S, V )
Equilibrium thermodynamics states that the internal energy of a system is a
function of Entropy and Volume:
As a consequence, every variation in internal energy is given by:
∂U()
∂S
:= T(S, V )
dU() = T()dS − pU ()dV
∂U()
∂V
:= −pU (S, V )
21
Saturday, September 11, 2010
32. Some Atmospheric Physics
Riccardo Rigon
The first law of thermodynamics
with the help of the second
U = U(S, V )
Equilibrium thermodynamics states that the internal energy of a system is a
function of Entropy and Volume:
As a consequence, every variation in internal energy is given by:
∂U()
∂S
:= T(S, V )
Temperature
dU() = T()dS − pU ()dV
∂U()
∂V
:= −pU (S, V )
21
Saturday, September 11, 2010
33. Some Atmospheric Physics
Riccardo Rigon
The first law of thermodynamics
with the help of the second
U = U(S, V )
Equilibrium thermodynamics states that the internal energy of a system is a
function of Entropy and Volume:
As a consequence, every variation in internal energy is given by:
∂U()
∂S
:= T(S, V )
Temperature pressure
dU() = T()dS − pU ()dV
∂U()
∂V
:= −pU (S, V )
21
Saturday, September 11, 2010
34. Some Atmospheric Physics
Riccardo Rigon
U = U(S, V )
Variation of
internal
energy
heat
exchanged by
the system
work done by the
system
dU() = T()dS − pU ()dV
The first law of thermodynamics
with the help of the second
As a consequence, every variation in internal energy is given by:
22
Equilibrium thermodynamics states that the internal energy of a system is a
function of Entropy and Volume:
Saturday, September 11, 2010
35. Some Atmospheric Physics
Riccardo Rigon
UT := U(S(T, V ), V )
However, while temperature is directly measurable, entropy is not - a
consequence of the second law of thermodynamics. For this reason it is
preferred to express entropy as a function of temperature, by means of a
change of variables. In this case, it should be observed that entropy is not
solely a function of temperature, but also of volume:
pS() :=
∂U()
∂S
∂S()
∂V
dUT = CV ()dT + (pS() − pU ())dV
The first law of thermodynamics
with the help of the second
23
Saturday, September 11, 2010
36. Some Atmospheric Physics
Riccardo Rigon
UT := U(S(T, V ), V )
However, while temperature is directly measurable, entropy is not - a
consequence of the second law of thermodynamics. For this reason it is
preferred to express entropy as a function of temperature, by means of a
change of variables. In this case, it should be observed that entropy is not
solely a function of temperature, but also of volume:
Entropic
PressurepS() :=
∂U()
∂S
∂S()
∂V
dUT = CV ()dT + (pS() − pU ())dV
The first law of thermodynamics
with the help of the second
23
Saturday, September 11, 2010
37. Some Atmospheric Physics
Riccardo Rigon
The sum of the two pressures, entropic ed energetic, if so they can be defined,
is the normal pressure:
p() := pS() − pU ()
The first law of thermodynamics
with the help of the second
24
Saturday, September 11, 2010
38. Some Atmospheric Physics
Riccardo Rigon
By definition (!) the internal energy of an ideal gas does NOT explicitly
depend on the volume. Therefore:
Variation of
internal
energy
heat
exchanged by
the system
U = U(S)
dU() = T()dS !!!!!!! =⇒ dQ() = dU()
The first law of thermodynamics
with the help of the second
As a consequence, every variation in internal energy is given by:
25
Saturday, September 11, 2010
39. Some Atmospheric Physics
Riccardo Rigon
Therefore, for an ideal gas:
CV () :=
∂UT
∂T
or:
dividing the expression by the mass of air present in the volume:
dUT = dQ() = CV ()dT + ps()dV
dUT = CV ()dT + d(ps() V ) − V dps()
The first law of thermodynamics
with the help of the second
26
Saturday, September 11, 2010
40. Some Atmospheric Physics
Riccardo Rigon
Therefore, for an ideal gas:
CV () :=
∂UT
∂T
or:
dividing the expression by the mass of air present in the volume:
dUT = dQ() = CV ()dT + ps()dV
dUT = CV ()dT + d(ps() V ) − V dps()
The first law of thermodynamics
with the help of the second
26
specific heat at
constant volume
Saturday, September 11, 2010
41. Some Atmospheric Physics
Riccardo Rigon
v :=
1
ρ
duT = cV ()dT + d(ps() v) − v dps()
dividing the expression by the mass of air present in the volume:
The first law of thermodynamics
with the help of the second
27
Saturday, September 11, 2010
42. Some Atmospheric Physics
Riccardo Rigon
v :=
1
ρ
specific
volume
duT = cV ()dT + d(ps() v) − v dps()
dividing the expression by the mass of air present in the volume:
The first law of thermodynamics
with the help of the second
27
Saturday, September 11, 2010
43. Some Atmospheric Physics
Riccardo Rigon
And using the ideal gas law per unit of mass:
ps() v = R T
The following results:
duT = cV ()dT + d(R T) − v dps()
duT = cV ()dT − d(ps() v) + v dps()
The first law of thermodynamics
with the help of the second
28
Saturday, September 11, 2010
44. Some Atmospheric Physics
Riccardo Rigon
Which can be rewritten as (in this case being du = dq):
During isobaric transformations, by definition, dp() = 0, and
dq|p = (cV () + R) dT = cpdT
cp() := cv() + R
cp is known as specific heat at constant pressure
dq = (cV () + R) dT − v dp()
The first law of thermodynamics
with the help of the second
29
Saturday, September 11, 2010
45. Some Atmospheric Physics
Riccardo Rigon
Adiabatic lapse rate
The information given in the first law of thermodynamics can be
combined with that obtained from the law of hydrostatics. In fact,
assuming that a rising parcel of air is subject to an adiabatic
process, then:
v dps() = −g dz
dq() = cp() dT + v dps()
dq() = 0
30
Saturday, September 11, 2010
46. Some Atmospheric Physics
Riccardo Rigon
Resolving the previous system results in:
dT
dz
= −Γd
Γd :=
g
cp
≈ 9.8◦
K Km−1
Adiabatic lapse rate
31
Saturday, September 11, 2010
48. Some Atmospheric Physics
Riccardo Rigon
33
The conditions of atmospheric stability
Temperature
STABLE AIR
Altitude Temperature
GROUND LEVEL
1. The wind pushes
the parcels of air at
21°C up the hill
2. The moving air
cools to 18.3°C
3. The air is cooler
than the surrounding
air and therefore it
drops
Altitude
Saturday, September 11, 2010
49. Some Atmospheric Physics
Riccardo Rigon
34
The conditions of atmospheric stability
Temperature
STABLE AIR
Altitude Temperature
GROUND LEVEL
1. The wind pushes
the parcels of air at
21°C up the hill
2. The moving air
cools to 18.3°C
3. The air is cooler
than the surrounding
air and therefore it
drops
Altitude
Saturday, September 11, 2010
50. Some Atmospheric Physics
Riccardo Rigon
35
The conditions of atmospheric stability
Temperature
STABLE AIR
Altitude Temperature
GROUND LEVEL
1. The wind pushes
the parcels of air at
21°C up the hill
2. The moving air
cools to 18.3°C
3. The air is cooler
than the surrounding
air and therefore it
drops
Altitude
Saturday, September 11, 2010
51. Some Atmospheric Physics
Riccardo Rigon
36
The conditions of atmospheric instability
Temperature
UNSTABLE AIR
Altitude Temperature
GROUND LEVEL
1. The wind pushes
the parcels of air at
21°C up the hill
2. The
moving air
cools to
18.1°C
3. The air is warmer
than the surrounding
air and therefore
continues to rise
4. The air at 15.1°C
continues to rise
5. The air at
12.1°C continues
to rise
6. The air at
9.1°C continues
to rise
Altitude
At altitude the air is relatively cool
Saturday, September 11, 2010
52. Some Atmospheric Physics
Riccardo Rigon
37
The conditions of atmospheric instability
Temperature
UNSTABLE AIR
Altitude Temperature
GROUND LEVEL
1. The wind pushes
the parcels of air at
21°C up the hill
2. The
moving air
cools to
18.1°C
3. The air is warmer
than the surrounding
air and therefore
continues to rise
4. The air at 15.1°C
continues to rise
5. The air at
12.1°C continues
to rise
6. The air at
9.1°C continues
to rise
Altitude
At altitude the air is relatively cool
Saturday, September 11, 2010
53. Some Atmospheric Physics
Riccardo Rigon
38
What happens when water vapour is added?
The water content of the atmosphere is specified by the mixing
ratio w :
w =
Mv
Md
=
ρv
ρd
where Mv is the mass of vapour and Md is the mass of dry air.
Alternatively, one can refer to the specific humidity, q:
q =
Mv
Md + Mv
=
ρv
ρd + ρv
≈ w
where the last equality is valid for MvMd, which is generally true.
Given that humid air can be considered, in good approximation, an
ideal gas its degrees of freedom are restricted once more by the
ideal gas law:
p = ρRT
where the value of the constant depends on the humidity. At the
extremes the values are Rd=287J.K-1kg-1 for dry air and
Rv=461J.K-1kg-1 for vapour.
Saturday, September 11, 2010
54. Some Atmospheric Physics
Riccardo Rigon
39
What happens when water vapour is added?
Let us now introduce a thermodynamic parameter, the potential
temperature θ, that takes account of this phenomenon. It is
defined as the temperature of a parcel of air that has moved
adiabatically from a starting point with temperature T and
pressure p to a reference altitude (and therefore reference
pressure), conventionally set at p0=1,000hPa (sea level). In other
words it describes an adiabatic transformation from (p,T) to
(p0, θ). Qualitatively, the potential temperature represents a
temperature correction based on the altitude.
θv = Tv
p0
p
Rd/co
p
Saturday, September 11, 2010
69. Some Atmospheric Physics
Riccardo Rigon
54
High pressure
polar, cold
Easterlies
cold
Westerlies,
warm
High pressure
subtropical
warm
Polar
front
Low pressure
zone
DejaVu
Saturday, September 11, 2010
79. Precipitations
Riccardo Rigon
Why it rains
•Large-scale atmospheric movements are caused by the variability of solar
radiation at the Earth’s surface, due to the spherical shape of the Earth.
Saturday, September 11, 2010
80. Precipitations
Riccardo Rigon
Why it rains
•Large-scale atmospheric movements are caused by the variability of solar
radiation at the Earth’s surface, due to the spherical shape of the Earth.
•Also, given the rotation of the Earth about its own axis, every air mass in
movement is deflected because of the (apparent) Coriolis force.
Saturday, September 11, 2010
81. Precipitations
Riccardo Rigon
Why it rains
•Large-scale atmospheric movements are caused by the variability of solar
radiation at the Earth’s surface, due to the spherical shape of the Earth.
•This situation:
•generates movements between “quasi-stable” positions of high and low
pressures
•causes large-scale discontinuities in the air’s flow field and discontinuities
of the thermodynamic properties of the air masses in contact with one
another
•generates, therefore, the situation where the lighter masses of air “slide”
over heavier ones, being lifted upwards in the process.
•Also, given the rotation of the Earth about its own axis, every air mass in
movement is deflected because of the (apparent) Coriolis force.
Saturday, September 11, 2010
83. Precipitations
Riccardo Rigon
•The surface of the Earth is composed of various material masses (air, water,
soil) that are oriented differently. They each respond to solar radiation in
different ways causing further movements of the air masses (at the scale of the
variability that presents itself) in order to redistribute the incoming radiant
energy.
Why it rains
Saturday, September 11, 2010
84. Precipitations
Riccardo Rigon
•The surface of the Earth is composed of various material masses (air, water,
soil) that are oriented differently. They each respond to solar radiation in
different ways causing further movements of the air masses (at the scale of the
variability that presents itself) in order to redistribute the incoming radiant
energy.
•Because of these movements, localised lifting of air masses can occur.
Why it rains
Saturday, September 11, 2010
85. Precipitations
Riccardo Rigon
•The surface of the Earth is composed of various material masses (air, water,
soil) that are oriented differently. They each respond to solar radiation in
different ways causing further movements of the air masses (at the scale of the
variability that presents itself) in order to redistribute the incoming radiant
energy.
•Because of these movements, localised lifting of air masses can occur.
•Moving masses of air are lifted by the presence of orography.
Why it rains
Saturday, September 11, 2010
86. Precipitations
Riccardo Rigon
•The surface of the Earth is composed of various material masses (air, water,
soil) that are oriented differently. They each respond to solar radiation in
different ways causing further movements of the air masses (at the scale of the
variability that presents itself) in order to redistribute the incoming radiant
energy.
•Because of these movements, localised lifting of air masses can occur.
•Moving masses of air are lifted by the presence of orography.
• Heating of the Earth’s surface also causes air to be lifted, causing conditions
of atmospheric instability.
Why it rains
Saturday, September 11, 2010
88. Precipitations
Riccardo Rigon
•As air rises it cools, due to adiabatic (isentropic) expansion, and the
equilibrium vapour pressure is reduced. Hence, the condensation of water
vapour becomes possible (though not always probable).
Why it rains
Saturday, September 11, 2010
89. Precipitations
Riccardo Rigon
•As air rises it cools, due to adiabatic (isentropic) expansion, and the
equilibrium vapour pressure is reduced. Hence, the condensation of water
vapour becomes possible (though not always probable).
•In this way, at a suitable altitude above the ground, clouds are formed: particles
of liquid or solid water suspended in the air.
Why it rains
Saturday, September 11, 2010
90. Precipitations
Riccardo Rigon
•As air rises it cools, due to adiabatic (isentropic) expansion, and the
equilibrium vapour pressure is reduced. Hence, the condensation of water
vapour becomes possible (though not always probable).
•In this way, at a suitable altitude above the ground, clouds are formed: particles
of liquid or solid water suspended in the air.
Why it rains
Saturday, September 11, 2010
100. Precipitations
Riccardo Rigon
Factors that influence the nature and quantity of
precipitation at the ground
•Latitude: precipitations are distributed over the surface of the Earth in
function of the general circulation systems.
•Altitude: precipitation (mean annual) tends to grow with altitude - up to a
limit (the highest altitudes are arid, on average).
•Position with respect to the oceanic masses, the prevalent winds, and the
general orographic position.
Saturday, September 11, 2010
103. Precipitations
Riccardo Rigon
Precipitation exhibits spatial variability at
a large range of scales
(mm/hr)
512km
pixel = 4 km
0 4 9 13 17 21 26 30
R (mm/hr)
2
km
4
km
pixel = 125 m
Foufula-Georgiou,2008
77
Spatialdistribution
Saturday, September 11, 2010
106. Precipitations
Riccardo Rigon
Characteristics of precipitation at the ground
•The physical state (rain, snow, hail, dew)
•Depth: the quantity of precipitation per unit area (projection),
often expressed in mm or cm.
•Duration: the time interval during which continuous precipitation is
registered, or, depending on the context, the duration to register a
certain amount of precipitation (independently of its continuity)
•Cumulative depth, the depth of precipitation measured in a pre-fixed
time interval, even if due to more than one event.
Saturday, September 11, 2010
107. Precipitations
Riccardo Rigon
•Storm inter-arrival time
•The spatial distribution of the rain volumes
•The frequency or return period of a certain precipitation event with
assigned depth and duration
•The quality, that is to say the chemical composition of the
precipitation
Characteristics of precipitation at the ground
Saturday, September 11, 2010
117. Extreme precipitations
Riccardo Rigon
Let is consider the maximum annual precipitations
These can be found in hydrological records, registered by characteristic durations:
1h, 3h, 6h,12h 24 h and they represent the maximum cumulative rainfall over the
pre-fixed time.
91
year 1h 3h 6h 12h 24h
1 1925 50.0 NA NA NA NA
2 1928 35.0 47.0 50.0 50.4 67.6
......................................
......................................
46 1979 38.6 52.8 54.8 70.2 84.2
47 1980 28.2 42.4 71.4 97.4 107.4
51 1987 32.6 40.6 64.6 77.2 81.2
52 1988 89.2 102.0 102.0 102.0 104.2
Saturday, September 11, 2010
118. Extreme precipitations
Riccardo Rigon
92
Let is consider the maximum annual precipitations
for each duration there is a precipitation distribution
Precipitazioni Massime a Paperopoli
durata
Precipitazione(mm)
1 3 6 12 24
5010015050100150
Precipitation(mm)
Duration
Maximum Precipitations at Toontown
Saturday, September 11, 2010
119. Extreme precipitations
Riccardo Rigon
1 3 6 12 24
50100150
Precipitazioni Massime a Paperopoli
durata
Precipitazione(mm)
Median
boxplot(hh ~ h,xlab=duration,ylab=Precipitation
(mm),main=Maximum Precipitations at Toontown) 93
Let is consider the maximum annual precipitations
Precipitation(mm)
Duration
Maximum Precipitations at Toontown
Saturday, September 11, 2010
120. Extreme precipitations
Riccardo Rigon
1 3 6 12 24
50100150
Precipitazioni Massime a Paperopoli
durata
Precipitazione(mm)
upper quantile
94
Let is consider the maximum annual precipitations
Precipitation(mm)
Duration
Maximum Precipitations at Toontown
Saturday, September 11, 2010
121. Extreme precipitations
Riccardo Rigon
1 3 6 12 24
50100150
Precipitazioni Massime a Paperopoli
durata
Precipitazione(mm)
lower quantile
95
Let is consider the maximum annual precipitations
Precipitation(mm)
Duration
Maximum Precipitations at Toontown
Saturday, September 11, 2010
122. Extreme precipitations
Riccardo Rigon
1 ora
Precipitazion in mm
Frequenza
20 40 60 80
0510152025
3 ore
Precipitazion in mm
Frequenza
20 40 60 80 100
051015
6 ore
Precipitazion in mm
Frequenza
40 60 80 100
051015
96
Frequency
Precipitation (mm)
Frequency
Frequency
Precipitation (mm) Precipitation (mm)
6 hours3 hour1 hour
Saturday, September 11, 2010
123. Extreme precipitations
Riccardo Rigon
12 ore
Precipitazion in mm
Frequenza
40 60 80 100 120
02468
24 ore
Precipitazion in mm
Frequenza
40 80 120 160
024681012
97
Frequency
Precipitation (mm)
12 hours
Frequency
Precipitation (mm)
24 hours
Saturday, September 11, 2010
124. Extreme precipitations
Riccardo Rigon
Return period
It is the average time interval in which a certain precipitation intensity is
repeated (or exceeded).
Let:
T
be the time interval for which a certain measure is available.
Let:
n
be the measurements made in T.
And let:
m=T/n
be the sampling interval of a single measurement (the duration of the event in
consideration).
98
Saturday, September 11, 2010
125. Extreme precipitations
Riccardo Rigon
Then, the return period for the depth h* is:
99
where Fr= l/n is the success frequency (depths greater or equal to h*).
If the sampling interval is unitary (m=1), then the return period is the
inverse of the exceedance frequency for the value h*.
Tr :=
T
l
= n
m
l
=
m
ECDF(h∗)
=
m
1 − Fr(H h∗)
N.B. On the basis of the above, there is a bijective relation between
quantiles and return period
Return period
Saturday, September 11, 2010
126. Extreme precipitations
Riccardo Rigon
1 3 6 12 24
50100150
Precipitazioni Massime a Paperopoli
durata
Precipitazione(mm)
Median - q(0.5) - Tr = 2 years
q(0.25) - Tr = 1.33 years
100
Precipitation(mm)
Duration
Maximum Precipitations at Toontown
q(0.75) - Tr = 4 years
Saturday, September 11, 2010
128. Extreme precipitations
Riccardo Rigon
h(tp, Tr) = a(Tr) tn
p
102
depth of
precipitation
power law
Rainfall Depth-Duration-Frequency (DDF) curves
Saturday, September 11, 2010
129. Extreme precipitations
Riccardo Rigon
h(tp, Tr) = a(Tr) tn
p
103
coefficient
dependent on
the return
period
depth of
precipitation
Rainfall Depth-Duration-Frequency (DDF) curves
Saturday, September 11, 2010
130. Extreme precipitations
Riccardo Rigon
h(tp, Tr) = a(Tr) tn
p
104
duration
considered
depth of
precipitation
Rainfall Depth-Duration-Frequency (DDF) curves
Saturday, September 11, 2010
131. Extreme precipitations
Riccardo Rigon
h(tp, Tr) = a(Tr) tn
p
105
exponent (not
dependent on
t h e r e t u r n
period)
depth of
precipitation
Rainfall Depth-Duration-Frequency (DDF) curves
Saturday, September 11, 2010
132. Extreme precipitations
Riccardo Rigon
h(tp, Tr) = a(Tr) tn
p
Given that the depth of cumulated precipitation is a non-decreasing function
of duration, it therefore stands that n 0
Also, it is known that average intensity of precipitation:
J(tp, Tr) :=
h(tp, Tr)
tp
= a(Tr) tn−1
p
decreases as the duration increases. Therefore, we also have n 1
Rainfall Depth-Duration-Frequency (DDF) curves
Saturday, September 11, 2010
133. Extreme precipitations
Riccardo Rigon
Tr = 50 years a = 36.46 n = 0.472
Tr = 100 years a = 40.31
Tr = 200 years a = 44.14
curve di possibilità pluviometrica
1.4
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
1 10 100tp[h]
log(prec) [mm]
tr=50 anni
tr=100 anni
tr=200 anni
a 50
a 100
a 200
107
Rainfall Depth-Duration-Frequency (DDF) curves
Tr=50 years
Tr=100 years
Tr=200 years
DDF Curve
Saturday, September 11, 2010
134. Extreme precipitations
Riccardo Rigon
curve di possibilità pluviometrica
1.4
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
1 10 100tp[h]
log(prec) [mm]
tr=50 anni
tr=100 anni
tr=200 anni
a 50
a 100
a 200
DDF curves are parallel to each other in the
bilogarithmic plane
108
Tr=50 years
Tr=100 years
Tr=200 years
DDF Curve
Saturday, September 11, 2010
135. Extreme precipitations
Riccardo Rigon
curve di possibilità pluviometrica
1.4
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
1 10 100tp[h]
log(prec) [mm]
tr=50 anni
tr=100 anni
tr=200 anni
a 50
a 100
a 200
tr = 500 years
tr = 200 years
h(,500) h(200)
109
DDF curves are parallel to each other in the
bilogarithmic plane
Tr=50 years
Tr=100 years
Tr=200 years
DDF Curve
Saturday, September 11, 2010
136. Extreme precipitations
Riccardo Rigon
curve di possibilità pluviometrica
1.4
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
1 10 100tp[h]
log(prec) [mm]
tr=50 anni
tr=100 anni
tr=200 anni
a 50
a 100
a 200
tr = 500 years
tr = 200 years
Invece h(,500) h(200) !!!!
110
DDF curves are parallel to each other in the
bilogarithmic plane
Tr=50 years
Tr=100 years
Tr=200 years
DDF Curve
Saturday, September 11, 2010
137. Extreme precipitations
Riccardo Rigon
The problem to solve using
probability theory and statistical analysis...
...is, therefore, to determine, for each duration, the correspondence between
quantiles (assigned return periods) and the depth of precipitation
For each duration, the data will need to be interpolated to a probability
distribution. The family of distribution curves suitable to this scope is the Type I
Extreme Value Distribution, or the Gumbel Distribution
b is a form parameter, a is a position parameter (it is, in effect, the mode)
P[H h; a, b] = e−e− h−a
b
− ∞ h ∞
Saturday, September 11, 2010
140. Extreme precipitations
Riccardo Rigon
The distribution mean is given by:
E[X] = bγ + a
where:
is the Euler-Mascheroni constant
γ ≈ 0.57721566490153228606
Gumbel Distribution
Saturday, September 11, 2010
142. Extreme precipitations
Riccardo Rigon
The standard form of the distribution (with respect to which there are tables
of the significant values) is
P[Y y] = ee−y
Gumbel Distribution
Saturday, September 11, 2010
144. Extreme precipitations
Riccardo Rigon
In order to adapt the family of Gumbel distributions to the data of interest
methods of adjusting the parameters are used.
We shall use three:
- The method of the least squares
- The method of moments
- The method of maximum likelihood
Let us consider, therefore, a series of n measures, h = {h1, ....., hn}
118
Methods of adjusting parameters
with respect to the Gumbel distribution but having general validity
Saturday, September 11, 2010
145. Extreme precipitations
Riccardo Rigon
The method of moments consists in equalising the moments of the sample
with the moments of the population. For example, let us consider
The mean and the variance and
the t-th moment of the SAMPLE
119
µH
σ2
H
M
(t)
H
Methods of adjusting parameters
with respect to the Gumbel distribution but having general validity
Saturday, September 11, 2010
146. Extreme precipitations
Riccardo Rigon
If the probabilistic model has t parameters, then the method of
moments consists in equalising the t sample moments with the t
population moments, which are defined by:
In order to obtain a sufficient number of equations one must consider as
many moments as there are parameters. Even though, in principle, the
resulting parameter function can be solved numerically by points, the
method becomes effective when the integral in the second member
admits an analytical solution.
120
MH[t; θ] =
∞
−∞
(h − EH[h])t
pdfH(h; θ) dh t 1
MH[1; θ] = EH[h] =
∞
−∞
h pdfH(h; θ) dh
Methods of adjusting parameters
with respect to the Gumbel distribution but having general validity
Saturday, September 11, 2010
147. Extreme precipitations
Riccardo Rigon
The application of the method of moments to the Gumbel distribution
consists, therefore, in imposing:
or:
bγ + a = µH
b2 π2
6 = σ2
H
MH[1; a, b] = µH
MH[2; a, b] = σ2
H
Methods of adjusting parameters
with respect to the Gumbel distribution but having general validity
Saturday, September 11, 2010
148. Extreme precipitations
Riccardo Rigon
The method is based on the evaluation of the (compound) probability of
obtaining the recorded temporal series:
P[{h1, · · ·, hN }; a, b]
In the hypothesis of independence of observations, the probability is:
P[{h1, · · ·, hN }; a, b] =
N
i=1
P[hi; a, b]
The method of maximum likelihood
with respect to the Gumbel distribution but having general validity
Saturday, September 11, 2010
149. Extreme precipitations
Riccardo Rigon
This probability is also called the likelihood function - it is evidently a
function of the parameters. In order to simplify calculation the log-
likelihood is also defined:
123
P[{h1, · · ·, hN }; a, b] =
N
i=1
P[hi; a, b]
log(P[{h1, · · ·, hN }; a, b]) =
N
i=1
log(P[hi; a, b])
The method of maximum likelihood
with respect to the Gumbel distribution but having general validity
Saturday, September 11, 2010
150. Extreme precipitations
Riccardo Rigon
124
If the observed series is sufficiently long, it is assumed that it must be such that
the probability of observing it is maximum. Then, the parameters of the curve
that describe the population can be obtained from:
∂ log(P [{h1,···,hN };a,b])
∂a = 0
∂ log(P [{h1,···,hN };a,b])
∂b = 0
Which gives a system of two non-linear equations with two unknowns.
The method of maximum likelihood
with respect to the Gumbel distribution but having general validity
Saturday, September 11, 2010
151. Extreme precipitations
Riccardo Rigon
125
e.g. Adjusting the Gumbel Distribution
The logarithm of the likelihood function, in this case, assumes the form:
Deriving with respect to u and α the following relations are obtained:
That is:
Saturday, September 11, 2010
152. Extreme precipitations
Riccardo Rigon
The method of least squares
It consists of defining the the standard deviation of the measures, the ECDF,
and the probability of non-exceedance:
δ2
(θ) =
n
i=1
(Fi − P[H hi; θ])
2
and then minimising it
126
Saturday, September 11, 2010
153. Extreme precipitations
Riccardo Rigon
Standard
deviation
The method of least squares
It consists of defining the the standard deviation of the measures, the ECDF,
and the probability of non-exceedance:
δ2
(θ) =
n
i=1
(Fi − P[H hi; θ])
2
and then minimising it
126
Saturday, September 11, 2010
154. Extreme precipitations
Riccardo Rigon
ECDF
Standard
deviation
The method of least squares
It consists of defining the the standard deviation of the measures, the ECDF,
and the probability of non-exceedance:
δ2
(θ) =
n
i=1
(Fi − P[H hi; θ])
2
and then minimising it
126
Saturday, September 11, 2010
155. Extreme precipitations
Riccardo Rigon
ProbabilityECDF
Standard
deviation
The method of least squares
It consists of defining the the standard deviation of the measures, the ECDF,
and the probability of non-exceedance:
δ2
(θ) =
n
i=1
(Fi − P[H hi; θ])
2
and then minimising it
126
Saturday, September 11, 2010
156. Extreme precipitations
Riccardo Rigon
∂δ2
(θj)
∂θj
= 0 j = 1 · · · m
The minimisation is obtained by deriving the standard deviation expression
with respect to the m parameters
so obtaining the m equations, with m unknowns, that are necessary.
127
The method of least squares
Saturday, September 11, 2010
157. Extreme precipitations
Riccardo Rigon
we have, as a result, three pairs of parameters which are all, to a certain extent,
optimal. In order to distinguish which of these sets of parameters is the best
we must use a confrontation criterion (a non-parametric test). We will use
Pearson’s Test.
128
After the application of the various adjusting methods...
Saturday, September 11, 2010
158. Extreme precipitations
Riccardo Rigon
Pearson’s test is NON-parametric and consists in:
1 - Sub-dividing the probability field into k parts. These can be, for example, of equal
size.
129
Pearson’s Test
Saturday, September 11, 2010
160. Extreme precipitations
Riccardo Rigon
131
Pearson’s Test
Pearson’s test is NON-parametric and consists in:
3 - Counting the number of data in each interval (of the five in the figure).
Saturday, September 11, 2010
161. Extreme precipitations
Riccardo Rigon
Pearson’s test is NON-parametric and consists in:
4 - Evaluating the function:
P[H h0] = P[H 0]
P[H hn+1] = P[H ∞]
where:
in the case of the figure of the previous slides we have:
(P[H hj+1] − P[H hj]) = 0.2
X2
=
1
n + 1
n+1
j=0
(Nj − n (P[H hj+1] − P[H hj])2
n (P[H hj+1] − P[H hj])
132
Pearson’s Test
Saturday, September 11, 2010
162. Extreme precipitations
Riccardo Rigon
0 50 100 150
0.00.20.40.60.81.0
Precipitazione [mm]
P[h]
1h
3h
6h
12h
24h
133
After having applied Pearson’s test...
Precipitation (mm)
Saturday, September 11, 2010
163. Extreme precipitations
Riccardo Rigon
0 50 100 150
0.00.20.40.60.81.0
Precipitazione [mm]
P[h]
1h
3h
6h
12h
24h
Tr = 10 anni
h1 h3 h6 h12 h24
134
After having applied Pearson’s test...
Precipitation (mm)
Tr = 10 years
Saturday, September 11, 2010
164. Extreme precipitations
Riccardo Rigon
0 5 10 15 20 25 30 35
406080100120140160180
Linee Segnalitrici di Possibilita' Pluviometrica
h [mm]
t[ore]
135
By interpolation one obtains...
DDF Curves
t(hours)
Saturday, September 11, 2010
165. Extreme precipitations
Riccardo Rigon
0.5 1.0 2.0 5.0 10.0 20.0
6080100120140160
Linee Segnalitrici di Possibilita' Pluviometrica
t [ore]
h[mm]
136
By interpolation one obtains...
DDF Curves
t (hours)
Saturday, September 11, 2010
166. Extreme precipitations - addendum
Riccardo Rigon
χ2
If a variable, X, is distributed normally with null mean and unit variance,
then the variable
is distributed according to the “Chi squared” distribution (as proved by Ernst
Abbe, 1840-1905) and it is indicated
which is a monoparametric distribution of the Gamma family of
distributions. The only parameter is called “degrees of freedom”.
137
Saturday, September 11, 2010
167. Extreme precipitations - addendum
Riccardo Rigon
In fact, the distribution is:
And its cumulated probability is:
where is the incomplete “gamma” functionγ()
χ2
from Wikipedia
138
Saturday, September 11, 2010
168. Extreme precipitations - addendum
Riccardo Rigon
γ(s, z) :=
x
0
ts−1
e−t
dt
The incomplete gamma function
Saturday, September 11, 2010
169. Extreme precipitations - addendum
Riccardo Rigon
χ2
from Wikipedia
140
Saturday, September 11, 2010
170. Extreme precipitations - addendum
Riccardo Rigon
The expected value of the distribution is equal to the number of degrees of
freedom
χ2
The variance is equal to twice the number of degrees of freedom
E(χk) = k
V ar(χk) = 2k
from Wikipedia
141
Saturday, September 11, 2010
171. Extreme precipitations - addendum
Riccardo Rigon
Generally, the distribution is used in statistics to estimate the goodness of an
inference. Its general form is:
χ2
Assuming that the root of the variables represented in the summation has a
gaussian distribution, then it is expected that the sum of squares variable is
distributed according to with a number of degrees of freedom equal to
the number of addenda reduced by 1.
χ2
χ2
from Wikipedia
142
χ2
=
(Observed − Expected)2
Expected
Saturday, September 11, 2010
172. Extreme precipitations - addendum
Riccardo Rigon
The distribution is important because we can make two mutually
exclusive hypotheses. The null hypothesis:
χ2
It is conventionally assumed that the alternative hypothesis can be excluded
from being valid if X^2 is inferior to the 0.05 quantile of the
distribution with the appropriate number of degrees of freedom.
χ2
from Wikipedia
And its opposite, the alternative hypothesis:
that the sample and the population have the same distribution
that the sample and the population do NOT have the same
distribution
χ2
143
Saturday, September 11, 2010
174. Extreme Events - GEV
Riccardo Rigon
A little more formally
The choice of the Gumbel distribution is not a whim, it is due to a
Theorem which states that, under quite general hypotheses, the
distribution of maxima chosen from samples that are sufficiently
numerous can only belong to one of the following families of
distributions:
I) The Gumbel Distribution
G(z) = e−e− z−b
a
− ∞ z ∞
a 0
145
Saturday, September 11, 2010
175. Extreme Events - GEV
Riccardo Rigon
II) The Frechèt Distribution
G(z) =
0 z ≤ b
e−(z−b
a )
−α
z b
α 0a 0
146
A little more formally
The choice of the Gumbel distribution is not a whim, it is due to a
Theorem, which states that, under quite general hypotheses, the
distribution of maxima chosen from samples that are sufficiently
numerous can only belong to one of the following families of
distributions:
Saturday, September 11, 2010
176. Extreme Events - GEV
Riccardo Rigon
Mean
Mode
Median
Variance
P[X x] = e−x−α
II) The Frechèt Distribution
from Wikipedia
147
A little more formally
Saturday, September 11, 2010
178. Extreme Events - GEV
Riccardo Rigon
α 0
a 0
G(z) =
e−[−(z−b
a )]
−α
z b
1 z ≥ b
III) The Weibull Distribution
149
A little more formally
The choice of the Gumbel distribution is not a whim, it is due to a
Theorem, which states that, under quite general hypotheses, the
distribution of maxima chosen from samples that are sufficiently
numerous can only belong to one of the following families of
distributions:
Saturday, September 11, 2010
179. Extreme Events - GEV
Riccardo Rigon
from Wikipedia
III) The Weibull Distribution
(P. Rosin and E. Rammler, 1933)
150
A little more formally
Saturday, September 11, 2010
180. Extreme Events - GEV
Riccardo Rigon
When k = 1, the Weibull distribution
reduces to the exponential distribution.
When k = 3.4, the Weibull distribution
becomes very similar to the normal
distribution.
Mean
Mode
Median
Variance
from Wikipedia
151
A little more formally
III) The Weibull Distribution
(P. Rosin and E. Rammler, 1933)
Saturday, September 11, 2010
182. Extreme Events - GEV
Riccardo Rigon
For the distribution reduces to the Gumbel distribution
For the distribution becomes a Frechèt distribution
For the distribution becomes a Weibull distribution
ξ = 0
ξ 0
ξ 0
The aforementioned theorem can be reformulated in terms of a three-parameter
distribution called the Generalised Extreme Values (GEV) Distribution.
G(z) = e−[1+ξ(z−µ
σ )]−1/ξ
z : 1 + ξ(z − µ)/σ 0
−∞ µ ∞ σ 0
−∞ ξ ∞
153
A little more formally
Saturday, September 11, 2010
183. Extreme Events - GEV
Riccardo Rigon
G(z) = e−[1+ξ(z−µ
σ )]−1/ξ
z : 1 + ξ(z − µ)/σ 0
−∞ µ ∞ σ 0
−∞ ξ ∞
154
A little more formally
The aforementioned theorem can be reformulated in terms of a three-parameter
distribution called the Generalised Extreme Values (GEV) Distribution.
Saturday, September 11, 2010
184. Extreme Events - GEV
Riccardo Rigon
gk = Γ(1 − kξ)
155
A little more formally
The aforementioned theorem can be reformulated in terms of a three-parameter
distribution called the Generalised Extreme Values (GEV) Distribution.
Saturday, September 11, 2010
185. Extreme Events - GEV
Riccardo Rigon
dgev(x, loc=0, scale=1, shape=0, log = FALSE)
pgev(q, loc=0, scale=1, shape=0, lower.tail = TRUE)
qgev(p, loc=0, scale=1, shape=0, lower.tail = TRUE)
rgev(n, loc=0, scale=1, shape=0)
R
156
A little more formally
Saturday, September 11, 2010
186. Bibliography and Further Reading
Riccardo Rigon
•Albertson, J., and M. Parlange, Surface Length Scales and Shear Stress: Implications
for Land-Atmosphere Interaction Over Complex Terrain, Water Resour. Res., vol. 35,
n. 7, p. 2121-2132, 1999
•Burlando, P. and R. Rosso, (1992) Extreme storm rainfall and climatic change,
Atmospheric Res., 27 (1-3), 169-189.
•Burlando, P. and R. Rosso, (1993) Stochastic Models of Temporal Rainfall:
Reproducibility, Estimation and Prediction of Extreme Events, in: Salas, J.D., R.
Harboe, e J. Marco-Segura (eds.), Stochastic Hydrology in its Use in Water Resources
Systems Simulation and Optimization, Proc. of NATO-ASI Workshop, Peniscola,
Spain, September 18-29, 1989, Kluwer, pp. 137-173.
Bibliography and Further Reading
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187. Bibliography and Further Reading
Riccardo Rigon
•Burlando, P. e R. Rosso, (1996) Scaling and multiscaling Depth-Duration-Frequency
curves of storm precipitation, J. Hydrol., vol. 187/1-2, pp. 45-64.
•Burlando, P. and R. Rosso, (2002) Effects of transient climate change on basin
hydrology. 1. Precipitation scenarios for the Arno River, central Italy, Hydrol.
Process., 16, 1151-1175.
•Burlando, P. and R. Rosso, (2002) Effects of transient climate change on basin
hydrology. 2. Impacts on runoff variability of the Arno River, central Italy, Hydrol.
Process., 16, 1177-1199.
• Coles S.,ʻʻAn Introduction to Statistical Modeling of Extreme Values, Springer,
2001
• Coles, S., and Davinson E., Statistical Modelling of Extreme Values, 2008
Saturday, September 11, 2010
188. Bibliography and Further Reading
Riccardo Rigon
•Foufula-Georgiou, Lectures at 2008 Summer School on Environmental Dynamics,
2008
•Fréchet M., Sur la loi de probabilité de l'écart maximum, Annales de la Société
Polonaise de Mathematique, Crocovie, vol. 6, p. 93-116, 1927
•Gumbel, On the criterion that a given system of deviations from the probable in
the case of a correlated system of variables is such that it can be reasonably
supposed to have arisen from random sampling, Phil. Mag. vol. 6, p. 157-175, 1900
• Houze, Clouds Dynamics, Academic Press, 1994
Saturday, September 11, 2010
189. Bibliography and Further Reading
Riccardo Rigon
•Kleissl J., V. Kumar, C. Meneveau, M. B. Parlange, Numerical study of dynamic
Smagorinsky models in large-eddy simulation of the atmospheric boundary layer:
Validation in stable and unstable conditions, Water Resour. Res., 42, W06D10, doi:
10.1029/2005WR004685, 2006
•Kottegoda and R. Rosso, Applied statistics for civil and environmental engineers,
Blackwell, 2008
•Kumar V., J. Kleissl, C. Meneveau, M. B. Parlange, Large-eddy simulation of a diurnal
cycle of the atmospheric boundary layer: Atmospheric stability and scaling issues,
Water Resour. Res., 42, W06D09, doi:10.1029/2005WR004651, 2006
•Lettenmaier D., Stochastic modeling of precipitation with applications to climate
model downscaling, in von Storch and, Navarra A., Analysis of Climate Variability:
Applications and Statistical Techniques,1995
Saturday, September 11, 2010
190. Bibliography and Further Reading
Riccardo Rigon
•Salzman, William R. (2001-08-21). Clapeyron and Clausius–Clapeyron
Equations (in English). Chemical Thermodynamics. University of Arizona. Archived
from the original on 2007-07-07. http://web.archive.org/web/20070607143600/
http://www.chem.arizona.edu/~salzmanr/480a/480ants/clapeyro/clapeyro.html.
Retrieved 2007-10-11.
•von Storch H, and Zwiers F. W, Statistical Analysis in climate Research, Cambridge
University Press, 2001
•Whiteman, Mountain Meteorology, Oxford University Press, p. 355, 2000
Saturday, September 11, 2010