in a direction to both of the fields as shown in Figure 1. In wave model, the electromagnetic radiation is commonly associated with wavelength and frequency, exressed mathematically as: c f...(1) f Wave s frequency (H) c Seed of light (3 1 8 ms -1 ) Wavelength (m) h c E...(3) The amount of energy contains in a hoton defines the tye of electromagnetic radiation. Based on its wavelength, frequency, and the amount of energy it carries, electromagnetic radiation is classified into the electromagnetic sectrum which consists of radio waves, microwaves, infrared radiation, visible light, ultraviolet radiation, x rays, and gamma rays. Figure 1 Electric (E) and magnetic (M) fields roagation at the seed of light (c) (CCRS 3). Based on quantum mechanics theory, electromagnetic radiation is described as a set of hotons. Each hoton contains a certain amount of energy that causes it to behave like a wave or a article known as wave-article duality. The energy of a hoton is measured by using Planck-Einstein equation. E h f...() E Energy (J) h Planck s constant (6.66 1-34 Js -1 ) f Wave s frequency (H) It can be concluded from equation () that the increase of hoton energy is directly roortional to wave s frequency; and the frequency itself is inversely roortional to wavelength (equation(1)). Given the relationshi between wavelength and frequency, the energy of a hoton can also be exressed in the equation where the energy of a hoton is inversely roortional to wavelength. Figure Regions of electromagnetic sectrum (htt://science-edu.larc.nasa.gov)..1. Blackbody Radiation Every substance that has temerature above absolute ero ( K) emits electromagnetic radiation. The emitted radiation reresents energy conversion from thermal energy into electromagnetic energy, also known as thermal radiation. Moreover, radiating substance also absorbs electromagnetic radiation to maintain its thermal equilibrium. If it absorbs and emits radiation at all wavelength, a substance is classified as a blackbody. The curves shown in Figure 3 reresent the emitted radiation of a blackbody at various temeratures; it shifts toward shorter wavelength and greater radiation intensity as temerature increases. Mathematically, the curve shae is given by Planck s law of blackbody radiation.
3 under the curve as shown in Figure 3. As exressed by Stefan-Boltmann law, total radiation emitted by a blackbody er unit surface area is directly roortional to the four ower of its absolute temerature. 4 F T...(6) F = Total radiation er unit surface area (Wm - ) = Stefan-Boltmann constant (5.674 x1-8 W m - K -4 ) T = Absolute temerature (K) Figure 3 Emitted intensity of blackbody radiation and maximum wavelength at various temeratures (Salby 1996). hc B 5 1 hc / kt e...(4) 1 B = Blackbody radiation (Wm - ) h = Planck s constant (6.66 1-34 Js -1 ) c = Seed of light (3 1 8 ms -1 ) = Wavelength (m) k = Boltmann constant (1.38 1 3 JK -1 ) T = Absolute Temerature (K) It follows that the eak wavelength of maximum radiation of a blackbody is inversely roortional to its absolute temerature when exressed as a function of a wavelength as stated in Wien s dislacement law. b max.....(5) T max = Peak wavelength (m) b = Wien s dislacement constant (.897 1-3 mk) T = Absolute Temerature (K) The standard amount of radiaton emmited by a blackbody is derived by integrating the Planck function over the entire wavelength domain from to, reresented by the area A true blackbody does not exist in nature. The amount of radiation emitted by a substance would not be the same as the absorbed radiation as it deends on its emissivity. The emissivity, ranged from to 1; indicates a ratio of emitted radiation by a substance to radiation emitted by a blackbody at the same temerature. A blackbody would have an emissivity equal to one while any other substance would be less than that. Generally, the more reflective a substance is, the lower its emissivity where the Stefan- Boltmann law is exressed as: 4 F T.....(7) Emissivity of a substance Most satellite sensors are based on these fundamental laws to distinguish objects by their sectral differences. It is also the same reason why certain satellite sensors are comrised in multisectral bands including the visible and infrared ortions of electromagnetic sectrum to detect changes and interaction in the atmoshere, oceans, and the Earth s surface. With secific techniques and algorithms, further rocessing on the information is used to meet end-user needs in various uroses..1.3 Bowtie Effect The Moderate Resolution Imaging Sectroradiometer (MODIS) instrument acquires image over a scan range of -55 o to +55 o at 75 km height. The length of each swath is 1 km at nadir and gradually becomes larger to the edge of the scene due to its large viewing angle (Gome et al. 4).
4 Consequently, distortion occurs on the image as shown in Figure 4, known as the bowtie effect. The bowtie effect in MODIS data haens when the sensor scanning angle reach to 15 o, as the angle increases the effect becomes more obvious (Wen 8)..1.4 Surface Temerature The surface temerature is considered as the temerature of the outer arts of an object. Whereas in satellite-based remote sensing; it is identified as the average temerature of a surface which is stored in a ixel. The amount of surface temerature that can be reached by an object is associated with its wavelength. Based on the blackbody radiation and as exressed by Planck s law, the amount of energy emitted by a substance deends mainly on its temerature. Then, by inverting Planck s function, it is ossible to estimate surface temerature from brightness temeratures (Dash et al. ). T hc...(8) 5 hc k ln 1 B Figure 4 An examle of bowtie effect in MODIS data (htt://www.sat.dundee.ac.uk/). The bowtie effect causes a dulicated and sometimes trilicated view of the earth s surface. Overlaing data need to be removed before alying any further rocess to obtain accurate results. Various algorithms have been develoed to remove bowtie effect.mainly there are two kinds of them; ehemeris method and non-ehemeris method (Wen 8). In Ehemeris method, geograhic grid will be generated and the data will be matched to it based on its geograhical coordinates with the bowtie effect eliminated at the same time. On the other hand, in non-ehemeris method, the elimination of bowtie effect is done by removing the overlaed arts between neighbouring stris (Junhui 4). Surface temerature is one of the key arameters in the hysics of land surface rocesses which indicate surface-atmoshere interactions and energy fluxes between the atmoshere and the ground either on the regional or global scales (Sellers et al. 1988). It can be estimated from measuring thermal radiance which coming from the land surface by using satellite sensor. After aroriate aggregation and arameteriation, the surface temerature retrieved from satellite data may be used to validate and imrove the global meteorological rediction (Price 198). The surface-leaving radiance is measured by satellite sensors in different sectral channel, most of the sensors are ranged in the infrared region since the temerature of every object on earth is higher than absolute temerature ( K). The measurements are influenced by surface characteristics such as emmisivity and its geometric form (Voogt and Oke 3) and also the distribution of Figure 5 Morhology of the MODIS bowtie effect (Gome et al. 4).
5 temerature and emmisivity within a ixel and the sectral channel of measurement (Becker and Li 1995). Given that remote sensing of surface temerature are based on Planck s law which relates to radiative energy emitted by a true blackbody, Planck s function is multilied by its emmisivity for natural objects which are non-blackbodies (Dash et al. ).. Dynamic Processes in the Atmoshere The ultimate cause of Earth s winds is solar radiation. The amount of solar radiation received at the to of the atmoshere is not the same as the amount of radiation received at the Earth s surface. When it travels through the atmoshere to the Earth s surface, some of the incoming radiation is absorbed, reflected or scattered in all directions by clouds, atmosheric gases, vaours, and dust articles. These events caused the amount of solar radiation that reaches the Earth s surface are not the same in all laces. As the result, it heats the Earth s surface differently. Uneven heating of the Earth s surface then causes differences in air ressure at various locations that makes the wind blow from one oint to the other as a fluid system in the atmoshere...1 Fundamental Forces In the study of atmosheric dynamics, a fluid is treated as continuous medium (continuum) in which a oint is a volume element that is very small comared to the total fluid volume but still contains a very large number of molecules (Holton 4). Atmosheric motions are rimarily caused by ressure gradient force along with the other fundamental forces such as gravitational force, and viscous force (friction). These forces, together with the aarent forces like centrifugal force and Coriolis force are the driving forces of the atmosheric motions. a. Pressure Gradient Force The horiontal ressure gradient force in the equation of motion is the most vital art of dynamics that governs the forcing of the atmoshere (Parish et al. 7). The force is a result of satial differences in atmosheric ressure that causes air to move from high ressure regions to low ressure regions, resulting in wind from local to global scales. Assuming that an infinitesimal arcel of air has volume δx δy δ is centered at the oint x, y, ; where the ressure is increasing from left to right as shown in Figure 6; ressure gradient force can be exressed mathematically in Taylor series exansion as: x higher order terms x Figure 6 The x comonent of the ressure gradient force acting on a infinitesimal arcel of air. Neglecting the higher order terms in this exansion, the ressure force acting on the right and left sides of the arcel can be defined as: x F Ax y x x F Bx y x Resectively, the net horiontal force acting on the arcel is the sum of F Ax and F Bx F X F Ax F Bx xy x Since the mass (m) of the differential volume of the arcel is simly density () times volume; m ρδx ρδy ρδ. Hence, the ressure gradient force er unit mass in x comonent is: F x 1 m x Similarly, the ressure gradient force er unit mass for y and comonents are: F y 1 m y F 1 m
6 In three dimensional Cartesian coordinate directions, total ressure gradient force er unit mass is: F 1.....(9) m The ressure gradient force is erendicular to isobars. The greater the difference of ressure level between isobars, the stronger the ressure gradient force gets. Closely saced isobars indicate stronger ressure gradient force than widely saced one. The acceleration of air arcels deends mostly on the ressure gradient force; strong force will lead to high wind velocity while weak force will have resulted in low wind velocity. Since the motion of wind can occur in any direction, its vector can be broken down into vertical and horiontal comonents. Logically, as the atmosheric ressure decreases exonentially with increasing height, the air would rush off into sace due to the vertical comonent of ressure gradient force. But instead of blowing the wind straight u, the wind is flowing horiontally in most cases because the air articles are being ulled down toward the earth due to gravity which rogressively slows down or even stos the flow that occurs vertically. b. Gravitational Force The air arcels in the atmoshere are made of molecules of different gases which are influenced by Earth s gravitational field. Newton s law of universal gravitation states that every article in the universe attracts each other with a force that directly roortional to the roduct of their masses and inversely roortional to the square of the distance between them. radius (Salby 1996). Between two interacting objects, the gravitational force will get stronger as the mass of either objects increase and will get weaker as the searation distance between the objects increase. Thus, if M is the mass of Earth and m is a mass of an air arcel, the downward force exerted on an air arcel due to Earth s gravitational attraction is: F g m GM g* rˆ.....(11) r As in dynamic meteorology, the height above mean sea level is used as a vertical coordinate. Then, by neglecting the bulges at the equator and assuming the Earth is in erfectly sherical shae; equation (11) can be rewritten as: GM g * g* rˆ ( a ) 1 a g * a...(1) = Gravitational force at mean sea level (kg m s - ) = Height of an air arcel (m) = Radius of the Earth (m) F g GMm ˆ.....(1) r r F = Gravitational force (kg m s - ) g G = Gravitational constant (6.673x1-11 m 3 kg -1 s - ) M = First mass (kg) m = Second mass (kg) r = Distance between masses (m) Comared to Earth, the atmoshere is very thin. Its mass is concentrated and stratified vertically in the lowest 1 km of the Earth s surface; which is less than 1% of the lanet s Figure 7 An air arcel under the influence of Earth s gravitational force. Desite the fact that small variations of gravitational force with altitude are sometimes considered; for meterological alications ( a), these variations in the atmoshere are normally ignored so that gravitational force is simly treated as a constant g* g *) (
7 c. Viscous Force As the air arcels are moving within the atmoshere, it movement leads to friction close to the Earth s surface. This friction forced the air arcel to slow down and also changes its direction. There are at least two tyes of friction occur in the atmoshere; one that occurs between two surfaces as in between the atmoshere and the Earth s surface; and molecular friction between air molecules called viscosity. Friction caused by viscosity is much less significant than the one caused between two surfaces. Molecular viscosity is negligible for the atmoshere below 1 km due to very weak viscous force; it is only matters in a thin layer within a few centimeters of the Earth s surface where vertical shear is very large (Holton 4). Figure 8 Vertical shearing stress on a volume of fluid element in the x direction. The viscous force acting in the x direction of δx δy δvolume in Figure 8 is given by the net difference of stresses over y and directions to the x comonent of the force er unit volume. To obtain the force er unit mass caused by vertical shear in the x direction, it is divided by the mass ρ δx δy δ of the element: Fr 1 x 1 u If the dynamic viscosity coefficient,, is constant, the above equation may be simlified to v ( u / ), where v / is the kinematic viscosity coefficient. Since the wind movement may vary in all directions, it is written in three Cartesian coordinate directions as: u u u F rx v v u x y v v v F ry v v x y w w w F r v v w x y.. Forces in a Rotating Reference Frame As it well known, Earth-based observer is considered to be in a non-inertial frame since the Earth is sinning on its axis. Therefore, Newton s second law of motion has to be modified to describe the atmosheric motion in a rotating coordinate system. In site of small angular velocity of Earth s rotation, the effects caused by rotation of the reference frame are negligible; however, on some atmosheric motions at certain sace and time scales, the effect of aarent forces is imortant and must be accounted for (Lynch and Cassano 6). a. Centrifugal Force As an object is moving in a circular motion with radius r, its direction is continuously changing so that its velocity does not remain constant. To maintain its circular ath, a net force; called the centrietal force is directed toward the curvature of the ath. Thus, the centrietal acceleration is given by the rate of change of angular velocity. ace r.....(13) a ce = Centrietal acceleration (kg m s - ) = Angular velocity (rad s -1 ) r = Radius of curvature (m) Newton s third law of motion states that for every action there is an equal and oosite reaction. The centrifugal force is the equal counterart force that is exerted in the oosite direction of the centrietal force; it is an aarent force invoked to make Newton s second law of motion work in a rotating frame. The magnitude of centrifugal force acting on a body of mass m is given by: F sf ma ce m r.....(14)
8 Viewed from a rotating frame of reference, the centrifugal force roduced by Earth s rotation is a result of the square of Earth s angular velocity () times the osition vector R from the Earth s axis rotation. Closer to the equator, the centrifugal force roduced by Earth s rotation is stronger than at the oles where it reach the minimum. Similar with the gravitational force, the centrifugal force act on the center of mass of the object. The resultant of these body forces is known as gravity. g g * R.....(15) Since the Earth is not a erfect shere, g * directed slightly away from the Earth s center instead of ointing directly to it excet at the equator and the oles where its surface are bulged out and flattened. Therefore, as the gravity, denoted g, is the resultant of both gravitational force and centrifugal force, the value of Earth s gravity would vary at different lace. b. Coriolis Force The Coriolis force is an aarent force that can only exists on any moving object situated on a rotating frame of reference; hence it disaears in a non accelerating inertial frame of reference. As the Earth rotates, the Coriolis force influences the wind movement from its intended ath, causing it to undergo curved motion. The Coriolis force acts in a direction erendicular to the object s motion. It deflects wind to the right in the Northern Hemishere, and to the left in the Southern Hemishere. The Coriolis arameter is defined as: f sin( )......(16) f = Coriolis arameter = Earth s angular velocity = Latitude The strength of Coriolis force varies with latitude. The magnitude of Coriolis force er unit mass acting on a horiontally moving air arcel is equal to the roduct of the Coriolis arameter and arcel s velocity. The effect gets stronger as latitude increases, it reaches maximum at the oles and becomes ero at the equator. Likewise, at the same velocity, the wind that blows closer to the oles will be deflected more than the wind that blows near the equator. In onal and meridional comonents, the deflected motion caused by horiontal acceleration erendicular to the motion s ath is given by Du v sin( ) fv Dt Co Dv u sin( ) fu Dt Co The acceleration caused by Coriolis force can be rearranged by combining the horiontal comonents in the vectorial form as in the below equation where V reresents horiontal velocity and kˆ indicates vertical unit vector defined in ositive uward direction. DV f kˆ V.....(17) Dt Co Though Coriolis force deflects the wind direction, it does not affect the wind seed. However, the effect of Coriolis force for masses that move over small distances is negligible; it is only significant over longer distances and larger regions. As the wind seed increases, the Coriolis force increases and greater deflection will occurs; hence, a strong wind blow will be deflected far from its intended ath rather than slowly blowing wind...3 Structure of the Static Atmoshere Thermodynamic rocesses that occur in the atmoshere involve the transfer of energy; the heat roduced by thermal and mechanical rocesses then leads to changes in weather. All gases, including the mixture of gases in the atmoshere, are found closely aroximate to the state of an ideal gas. The gas roerties like ressure, temerature, mass and its volume are related to each other and determine the state of the gas. V m RT.....(18) = Pressure (Pa) V = Volume (m 3 ) m = Mass (kg) R = Gas Constant (J kg -1 K -1 ) T = Absolute temerature (K)
9 The ideal gas constant, R, has different values for each articular gas or a mixture of gases. For dry air arcel that contains no water vaor, R = 87 J kg -1 K -1. Since density = d m/ V the ideal gas equation can be rewritten as: RT.....(19) In different case, the equation can also be modified by eliminating air density and relacing it with secific volume,, it is defined as the ratio of gas volume to its mass or simly the inverse of air density. The standard unit of secific volume is commonly exressed in m 3 kg -1, deicting the volume occuied by one unit of mass at a given temerature and ressure. RT.....() In general, the thermodynamics roerties of an air mass determine the articular weather condition in the atmoshere over the area in which the air mass covers. As an air mass travels from one lace to another, it is being exosed to new environments and its thermodynamics roerties may change gradually over time. These changes is then used as a fundamental to understand different atmosheric henomenon ranging from the smallest cloud microhysical rocesses to the general circulation of the atmoshere (Wallace and Hobbs 6). a. The Hydrostatic Equation The ressure of air in the atmoshere at any height is determined by the force er unit area exerted by the weight of air influenced by gravity that acts on its surface. Thus, as the atmosheric ressure decreases with increasing height, there will be an uward motion caused by the ressure gradient force. Alied to an atmoshere at rest, the uward ressure gradient force is oosed by the downward ull of gravity. When there is a balance between these two forces, the atmoshere is in the state of hydrostatic equilibrium.for atmoshere in hydrostatic equilibrium, when forces are in balance, there is no net vertical force acting on it. Thus, there is no vertical acceleration occurs. The hydrostatic balance, as illustrated in Figure 9 is mathematically exressed as d g d.....(1) Most of the time, the atmoshere aroximates hydrostatic balance; however, this balance is not achieved for an intense small-scale system such as tornadoes and thunderstorms where the air raidly accelerates in a vertical manner (Ahrens 4). If the ressure of a fixed oint on the Earth at height is (), then the hydrostatic equation to an infinite height is given by ( ) ( ) d g d.....() Since () =, the ressure at height, (), is equal to the weight of air and is the result of gravity force that acts on the air above its level. Hence, the air ressure at the mean sea level, (), would be 113.5 hpa or 1.13 1 5 Pa; also known as 1 atmoshere (1 atm). b. Geootential As gravity is a conservative force, the work done by gravity does not deend on its ath and is identically ero (Lynch and Cassano 6). Since the work done by gravity is equal to ero, it is reresented as a gradient of a function; called geootential. The geootential is defined as the required work that must be done against gravity to raise a unit mass of air from sea level to a given height (Salby 1996). ( ) g( ) d.....(3) Figure 9 Balance of vertical forces of the atmoshere in the state of hydrostatic balance. The use of geootential in momentum equations has the advantage to avoid using gas density, thus creating simler equations. It is useful for most atmosheric alications since direct measurement of air density can be extremely difficult. If gravity were constant, a surface of geootential at all laces would be
1 at the same height. But, since the value of gravity vary at different laces; the altitude of geootential surface would also be different. The geootential height is defined as ( ) 1 Z g g gd...(4) The concet of geootential height is useful for aroximating geometric height of a constant-ressure surface in the atmoshere. The thickness of two surface levels can be easily found by calculating the difference between geootential heights. c. Isobaric Coordinate System For many alications associated with the governing dynamics of the atmosheric motions, it is very common to transform the equations of motion from height (Cartesian) coordinates (x, y, ) to isobaric coordinates (x, y, ). As what it shown in hydrostatic equation, ressure is related to geometric height by a single-valued monotonic function. Pressure decreases monotonically with height that its surfaces never intersect; hence instead of using height, ressure can be used as an alternative coordinate system. Figure 1 Sloe of ressure surfaces in the x, lane. As illustrated in Figure 1, the ressure difference between two lines of AB and BC are articularly the same. Judging from the similarities it can be stated that: The subscrits are used to indicate variables which are being held constant during differentiation. Taking the limits of x, x x x and by using the hydrostatic balance equation to substitute the variables, it can be obtained that 1 x g x x...( 5) Based on this outcome, the horiontal comonent of ressure gradient force can be rewritten as F x 1 m x x F y 1 m y y Given that ressure acts as the vertical coordinate, it aears that density is no longer required for comuting the ressure gradient force; which is a great advantage as it simlifies the equation and indirectly facilitates the observation. Thus, in isobaric coordinate system, the horiontal ressure gradient force on a surface of constant ressure is determined by the gradient of geootential...4 Horiontal Momentum Equations In general, air motion comes from a balanced flow which is rofoundly affected by the the force of ressure gradient, Coriolis force and friction (viscous) force. For largescale movement of air in the atmoshere, viscosity is sufficiently small that frictions near the Earth s surface could be neglected. Therefore, the horiontal momentum equation in height coordinates is given in the vectorial form as ( ( x1 x) x1 ( ) ( 1 ) 1 x x Dv h Dt ˆ fk v h 1 h...(6) x x x Where v h ui vj is a horiontal velocity vector. The condition where the forces acting on a arcel of air is in equilibrium with each other is considered as