DIRSIG Water Manual (Version 1.0)
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DIRSIG Water Manual (version 1.0) The content contained in this manual has been pulled primarily from chapters of a dissertation (Goodenough 2007), though some limited content from a validation document (Mobley, et al 1993) is included for reference. A recent conference paper demonstrating on-going validation is also included (Speir 2010). Most of the input syntax relevant to users is presented in the third section (“Chapter 5”). The first two sections provide background information on the radiometry and techniques used for the in- water solutions. A section including limited verification/validation is included in the fourth section and additional information is attached covering the original canonical problem set and a brief introduction to additional validation work. Finally, a list of possibly useful references is provided at the end. Chapter 2 Radiometric Foundation of Photon Mapping Contents 2.1 Introduction . 15 2.2 SI Base Units . 16 2.3 Rays and Photons . 16 2.4 Spectral Notation . 18 2.5 Radiometric Terms . 19 2.5.1 Spectral Flux, Φλ .................................... 19 2.5.2 Spectral Radiant Intensity, Iλ .............................. 19 2.5.3 Spectral Irradiance, Eλ ................................. 20 2.5.4 Spectral Radiance, Lλ .................................. 21 2.6 Volumetric Additions . 24 2.7 Radiative Transfer Equation . 25 2.7.1 Inherent Optical Properties . 25 2.7.2 The Integro-Differential Form of the RTE . 29 2.7.3 Solution of a Linear First Order ODE . 30 2.7.4 The Integral Form of the RTE . 32 2.7.5 Practical Form of the RTE . 32 2.8 Computation of the RTE Components . 34 2.1 Introduction Discussion of radiative transfer requires an efficient language that can be used to describe the concepts un- derlying the physical processes. Two well established languages exist that are focused on the propagation of electro-magnetic radiation. These are radiometry and photometry. The primary difference between the two 15 16 Chapter 2. Radiometric Foundation of Photon Mapping systems is that photometry is concerned with the visible region of the EM spectrum (as detectable by the human eye) while radiometry is applicable for a much broader spectrum (usually limited to the ultraviolet through long infrared region). While the field of spectral water studies is largely interested in the visible region of the spectrum, there is definite benefit in examining wavelengths outside this region (e.g. mapping of suspended sediments using infrared radiation [Li et al., 2003]). For this reason, along with notation sim- plification and consistency with existing work, we choose to use radiometry as the language to describe this work. The sections that follow start by presenting standard Le Systeme` International d’Unites´ (SI) units and then build upon these basic quantities to develop a complex conceptual description of radiative transfer in media. 2.2 SI Base Units All of the terms that we will use to describe the radiometry underlying the photon mapping methods can be derived from three of the seven base SI units plus the two supplementary derived, “dimensionless” SI units relating to angle measurement [Bur. Intl. Poids et Mesures, 1991]. Base Quantity Name Symbol length meter m mass kilogram kg time second s plane angle radian [angle]r or rad solid angle steradian sr The attached diagram (Figure 2.1 and accompanying text (a handout from Taylor [1995]) puts these units into context along with the other base and fundamental derived SI units. Note that the supplementary derived units (steradian and radian) are shown on the lower right-hand side of the diagram. 2.3 Rays and Photons One of the building blocks of modern physics is knowledge of the wave/particle duality of light. This dual nature is essential for explaining the behavior of light in different circumstances. For the purposes of propa- gating light in a synthetic scene, it is convenient to always think of light as bundles of energy. These bundles (or quanta) travel in straight lines (rays). By tracing rays through a scene from detector to light sources and handling all of the interactions along the way, we can build up an image of the scene. This is the essence of ray tracing. We will also use this conceptual view of light as rays/particles to derive the necessary radiometric units that follow. The traditional name for the bundle of energy associated with light is the photon. A photon carries a discrete amount of energy, q, that is related to the wavelength, λ, by hc q = [ joules], (2.1) λ λ 2.3. Rays and Photons 17 Relationships of the SI derived units with special names and symbols and the SI base units Derived units SI BASE UNITS without special SI DERIVED UNITS WITH SPECIAL NAMES AND SYMBOLS names Solid lines indicate multiplication, broken lines indicate division kilogram kg newton (kg·m/s2) pascal (N/m2) gray (J/kg) sievert (J/kg) 3 N Pa Gy Sv MASS m FORCE PRESSURE, ABSORBED DOSE VOLUME STRESS DOSE EQUIVALENT meter m 2 m joule (N·m) watt (J/s) becquerel (1/s) hertz (1/s) LENGTH J W Bq Hz AREA ENERGY, WORK, POWER, ACTIVITY FREQUENCY second QUANTITY OF HEAT HEAT FLOW RATE (OF A RADIONUCLIDE) s m/s TIME VELOCITY katal (mol/s) weber (V·s) henry (Wb/A) tesla (Wb/m2) kat Wb H T 2 mole m/s CATALYTIC MAGNETIC INDUCTANCE MAGNETIC mol ACTIVITY FLUX FLUX DENSITY ACCELERATION AMOUNT OF SUBSTANCE coulomb (A·s) volt (W/A) C V ampere A ELECTRIC POTENTIAL, CHARGE ELECTROMOTIVE ELECTRIC CURRENT FORCE degree (K) farad (C/V) ohm (V/A) siemens (1/!) kelvin Celsius °C F ! S K CELSIUS CAPACITANCE RESISTANCE CONDUCTANCE THERMODYNAMIC TEMPERATURE TEMPERATURE t/°C = T /K – 273.15 candela 2 steradian radian cd lux (lm/m ) lumen (cd·sr) 2 2 (m/m = 1) lx lm sr (m /m = 1) rad LUMINOUS INTENSITY ILLUMINANCE LUMINOUS SOLID ANGLE PLANE ANGLE FLUX The diagram above shows graphically how the 22 SI derived units with special names and symbols are related to the seven SI base units. In the first column, the symbols of the SI base units are shown in rectangles, with the name of the unit shown toward the upper left of the rectangle and the name of the associated base quantity shown in italic type below the rectangle. In the third column the symbols of the derived units with special names are shown in solid circles, with the name of the unit shown toward the upper left of the circle, the name of the associated derived quantity shown in italic type below the circle, and an expression for the derived unit in terms of other units shown toward the upper right in parenthesis. In the second column are shown those derived units without special names [the cubic meter (m3) excepted] that are used in the derivation of the derived units with special names. In the diagram, the derivation of each derived unit is indicated by arrows that bring in units in the numerator (solid lines) and units in the denominator (broken lines), as appropriate. Two SI derived units with special names and symbols, the radian, symbol rad, and the steradian, symbol sr (bottom of the third column of the diagram), are shown without any connections to SI base units – either direct or through other SI derived units. The reason is that in the SI, the quantities plane angle and solid angle are defined in such a way that their dimension is one – they are so-called dimensionless quantities. This means that the coherent SI derived unit for each of these quantities is the number one, symbol 1. That is, because plane angle is expressed as the ratio of two lengths, and solid angle as the ratio of an area and the square of a length, the SI derived unit for plane angle is m/m = 1, and the SI derived unit for solid angle is m2/m2 = 1. To aid understanding, the special name radian with symbol rad is given to the number 1 for use in expressing values of plane angle; and the special name steradian with symbol sr is given to the number 1 for use in expressing values of solid angle. However, one has the option of using or not using these names and symbols in expressions for other SI derived units, as is convenient. The unit “degree Celsius,’’ which is equal to the unit “kelvin,” is used to express Celsius temperature t. In this case, “degree Celsius’’ is a special name used in place of “kelvin.’’ This equality is indicated in the diagram by the symbol K in parenthesis toward the upper right of the °C circle. The equation below “CELSIUS TEMPERATURE’’ relates Celsius temperature t to thermodynamic temperature T. An interval or difference of Celsius temperature can, however, be expressed in kelvins as well as in degrees Celsius. Figure 2.1: Relationships between SI units (unaltered from the NIST Manual [Taylor, 1995] and included for convenience. Usage in this text is consistent with the NIST disclaimer: http://www.nist.gov/public affairs/ disclaim.htm) 18 Chapter 2. Radiometric Foundation of Photon Mapping where h is Planck’s constant (6.6256 10 34 joules sec) and c is the speed of light in a vacuum (2.9979 108 m ). · − · · s This work uses a technique known as photon mapping [Jensen, 2001]. In contrast to real, physical photons, the photons in photon mapping usually represent discrete bundles of power ([watts]), which describes the rate at which photons are emitted from the source or are traveling through a medium. In order to distinguish between the two concepts in this document, physical photons (i.e. ones that have energy given by Equation 2.1) will always be written in italics.