Lecture 2 Introduction to Inherent Optical Properties (IOPs) and Radiative Transfer Apparent Optical Properties (AOPs) C. Roesler 3 July 2007

Readings Mobley (Light and Water) –section 1.4 Solid Angles –section 3.1 IOPs –section 3.2 AOPs –section 5.10 Divergence Law

Inherent Optical Properties inherent to the water dependent upon the composition and concentrations of the particulate and dissolved substances and the water itself independent of external properties such as the light field (i.e. should be the same if measured in situ or in discrete sample)

What are the IOPs? Absorption, a –Color –Darkens Scattering, b –Clarity –Brightens

What are the IOPs? Absorption, a Scattering, b Beam attenuation, c (transmission) a + b = c The IOPs tell us something (e.g. concentration, composition) about the particulate and dissolved substances in the aquatic medium; how we measure them determines what we can resolve

Review of IOP Theory oo Incident Radiant Flux No attenuation Transmitted Radiant Flux tt

Conservation of radiant flux  a Absorbed Radiant Flux  b Scattered Radiant Flux  o = tt oo Incident Radiant Flux Transmitted Radiant Flux

Beam Attenuation Theory Attenuance C = fraction of incident radiant flux attenuated  C = tt aa bb oo

Beam Attenuation Theory tt aa bb c = fractional attenuance per unit distance  c =  c  x =  c x =  c (m -1 ) = oo xx  0 c dx = xx  c(x-0) =  c x =

Beam Attenuation: The Measurement Reality aa  c = (-1/x)  ln(  t /  o ) Detected flux (  t ) measurement must exclude scattered flux detector oo tt bb x source

Beam Attenuation The Measurement Reality oo tt aa bb x  c = (-1/x)  ln(  t /  o ) The size of the detector acceptance angle (FOV) determines the retrieved value of c source The larger the detector acceptance angle,

Ex. transmissometer/c-meter FOV% b detected o <1 0.7 o ~ o ~ o ~ o ~18 Roesler and Boss 2004

Absorption Theory tt aa a = fractional absorptance per unit distance  a =  a (m -1 ) = oo xx  A = bb need to measure

Absorption Measurement Reality  a = (-1/x)  ln[(  t +  b )/  o ] Detected flux measurement must include scattered flux oo tt aa bb undetected scattered flux must be corrected for x sourcedetector diffuser

Currently the only commercially available absorption meter: WETLabs ac9

AC9 25 cm absorption tube optimized to collect scattered flux attenuation tube optimized to reject scattered flux

How much scattered light is detected? Theoretically Detector FOV ~ 40 o ~90% of scattered flux 10% of scatter can ~abs loss is in side and backward directions must correct for scattering losses

Scattering, b, has an angular dependence, E ↑  =d  1 dS  d  dSdr  o What is d  ? And b =  4   d  which is described by the volume scattering function,  (  ) = power per unit steradian emanating from a volume illuminated by irradiance = d  1 1 d  dV E dr dV = dS dr E =  /dS [  mol photon m -2 s -1 ] dS sin  d  d  =  1 d   o  drd 

So we can calculate scattering, b, from the volume scattering function,  b =  4   d  We define the phase function:   b ~ =  0   0  sin  d  d   22 b f = 2   0  sin  d  and b b = 2    /2  sin  d   /2  If there is azimuthal symmetry = 2   0  sin  d  

Summary of the IOPs Note: c = a + b  is not solid angle in this case  ≡ b/c single scattering albedo is related to a, r if 1) all scattered light detected 2) optical path = geometric path is related to c, r if 1) no scattered light detected toto r oo Then b = c - a

Apparent Optical Properties Derived from radiometric parameters Depend upon the light field Depend upon the IOPs Ratios or gradients of radiometric parameters “Easy” to measure Difficult to interpret

Apparent Optical Properties What is the color and brightness of the ocean? How does sunlight penetrate the ocean? How does the angular distribution of light vary in the ocean?

AOPs: Average Cosines Ratios of radiometric parameters L (  ) [  mol photons m -2 s -1 sr -1 ] E d =        L (  ) cos  E od =        L (  )   d =  u = E u. E ou  = E d - E u. E o

Example of isotropic light field L(  ) constant for all   d = E d =        L (  ) cos  sin  d  E od        L (  ) sin  d  d  and then the math happens… (i.e. you try it!)  d = in terms of degrees  = average cosines are not unique quantities

AOPs: Reflectance Ratios of radiometric parameters L (  ) [  mol photons m -2 s -1 sr -1 ] E d =        L (  ) cos  d  E od =        L (  ) d  R = E u. Irradiance E d Reflectance R RS = L u. Remote Sensing or E d Radiance Reflectance

Roesler et al 2003 Reflectance Ranges Roesler and Perry

AOPs: diffuse attenuation coefficients Radiometric Gradients L (  ) [  mol photons m -2 s -1 sr -1 ] E d =        L (  ) cos  d  E od =        L (  ) d  dE(z) = -K(z) E(z) dz E z K(z) dz = -1 dE E(z) z z K z = -ln(E(z o )) + ln(E(z))if K was constant with z K = -1 ln(E(z o )/E(z)) z E(z) = E(z o ) e -Kz

AOPs: diffuse attenuation coefficients Why would K vary with depth? generated from HL by Curt Zaneveld et al Opt Exp

AOPs: diffuse attenuation coefficients NOTE c  K c  beam attenuation K  diffuse attenuation < z K d1 K d2 E d1 E d2 constant c

Radiative Transfer Equation relates the IOPs to the AOPs

Radiative Transfer Equation Consider the radiance, L(  ), as it varies along a path r through the ocean, at a depth of z d L(  ), what processes affect it? dr absorption along path r scattering out of path r dz = scattering into path r

Radiative Transfer Equation Consider the radiance, L(  ), as it varies along a path r through the ocean, at a depth of z d L(  ), what processes affect it? dr cos  d L(  ) = -a L(z,  ) -b L(z,  ) +    (z,  ’,  ’ )L(  ’  ’ )d  ’ dz If there are sources of light (e.g. fluorescence, raman scattering, bioluminescence), that is included too: a( 1,z) L( 1,z,  ’  ’ ) →(quantum efficiency) → L( 2,z,  )

0 ∫ 2    ∫  [ 0    ∫   (z,  ’,  ’ ) sin  d  d  ] L(  ’  ’ ) sin  d  ’  d  ’ = An example of the utility of RTE cos  d L(  ) = -a L(z,  ) -b L(z,  ) +    (z,  ’,  ’ )L(  ’  ’ )d  ’ dz Divergence Law (see Mobley 5.10) Integrate the equation over all solid angles (4  ), d  dĒ = dz 0 ∫ 2    ∫  -c L(z,  ) sin  d  d  = 0 ∫ 2  0 ∫  d L(  ) cos  sin  d  d  = dz d(E d – E u ) = dz [ 0    ∫   (z,  ’,  ’ ) sin  d  d  ] 0 ∫ 2    ∫  [ 0    ∫   (z,  ’,  ’ ) L(  ’  ’ ) sin  d  ’  d  ’ ] sin  d  d  =

An example of the utility of RTE cos  d L(  ) = -a L(z,  ) -b L(z,  ) +    (z,  ’,  ’ )L(  ’  ’ )  ’ dz dĒ = -c E o + b E o dz -1 dĒ = a Eo Ē dz Ē a = K Ē  Gershun’s Equation K Ē = a.  divide both sides by E substitute AOPs

Readings for Lecture 3 (Absorption) Mobley –section 3.3 Optically significant constituents –section 3.7 Absorption