Apparent Optical Properties (AOPs) Curtis D. Mobley University of Maine, 2007 (ref: Light and Water, Chapters 3 & 5)

Once again… Curt, Lecture 1 Collin, Lecture 2 Collin, Lecture 10 Emmanuel, Lecture 18 Curt, Lecture 6 Emmanuel, Lecture 8 Curt, Lecture 9 Collin, Lectures 2 & 10 Collin, Lecture 2 Mary Jane, Lecture 7 Curt, Lecture 12 Collin, Lectures 3 & 4 Mary Jane, Lecture 5

Light Properties: measure the radiance as a function of location, time, direction, wavelength, L(x,y,z,t, , ,  ), and you know everything there is to know about the light field. You don’t need to measure irradiances, PAR, etc. Material Properties: measure the absorption coefficient a(x,y,z,t,  ) and the volume scattering function β(x,y,z,t, ,  ), and you know everything there is to know about how the material affects light. You don’t need to measure b, b b, etc. Nothing else (AOPs in particular) is needed. In a Perfect World

L(x,y,z,t, , ,  ) is too difficult and time consuming to measure on a routine basis, and you don’t need all of the information contained in L, so therefore measure irradiances, PAR, etc. (ditto for VSF….) Reality Can we find simpler measures of the light field than the radiance, which are also useful for describing the optical characteristics of a water body (i.e., what is in the water)? Idea

Depend on the directional structure of the ambient light field (i.e., on the radiance) Depend on the absorption and scattering properties of the water body (via the radiance) Display enough regular features and stability to be useful for describing a water body A good AOP depends weakly on the external environment (sky condition, surface waves) and strongly on the water IOPs AOPs

Depend on the directional structure of the ambient light field (i.e., on the radiance) Depend on the absorption and scattering properties of the water body (via the radiance) Display enough regular features and stability to be useful for describing a water body A good AOP depends weakly on the external environment (sky condition, surface waves) and strongly on the water IOPs AOPs DON’T FORGET THIS PART: IRRADIANCES ARE NOT AOPs!!

AOPs are almost always Ratios of radiometric variables or Normalized depth derivatives of radiometric variables (is the Secchi depth an exception?) AOPs

E d and E u HydroLight runs: Chl = 1.0 mg Chl/m 3, etc Sun at 0, 30, 60 deg in clear sky, and solid overcast Factor of 10 variation for different sky conditions Note: E d and E u depend on the radiance and on the abs and scat properties of the water, but they depend strongly on incident lighting, so not useful for characterizing a water body. Again: irradiances are NOT AOPs!

Irradiance Reflectance R Ratio of upwelling plane irradiance to downwelling plane irradiance z

R = E u /E d HydroLight runs: Chl = 1.0 mg Chl/m 3, etc Sun at 0, 30, 60 deg in clear sky, and solid overcast ± 30% variation for different sky conditions approaching same value

R = E u /E d HydroLight runs: Chl = 0.1,1, 10 mg Chl/m 3 Sun at 0, 30, 60 deg in clear sky Variability due to Chl concentration Variability due to sun angle R depends weakly on the external environment and strongly on the water IOPs

Water-leaving Radiance, L w total upwelling radiance in air (above the surface) = water-leaving radiance + surface-reflected radiance Ed()Ed() Lr(,,)Lr(,,) Lt(,,)Lt(,,) Lw(,,)Lw(,,) Lu(,,)Lu(,,) L u ( , ,  ) = L w ( , ,  ) + L r ( , ,  ) An instrument measures L u (in air), but L w is what tells us what is going on in the water. It isn’t easy to figure out how much of L u is due to L w.

Remote-sensing Reflectance R rs sea surface Often work with the nadir-viewing R rs, i.e.,with the radiance that is heading straight up from the sea surface (  = 0) The fundamental quantity used in ocean color remote sensing R rs (θ,φ,λ) = upwelling water-leaving radiance downwelling plane irradiance 

Example R rs HydroLight runs: Chl = 0.1,1, 10 mg Chl/m 3 Sun at 0, 30, 60 deg in clear sky R rs shows very little dependence on sun angle and strong dependence on the water IOPs—a very good AOP Chl = 0.1 Chl = 1 Chl = 10

slopes are very similar E d and E u Magnitude changes are due to sun angle; slope is determined by water IOPs.

This suggests trying... The depth derivative (slope) on a log-linear plot as an AOP. This leads to the diffuse attenuation coefficient for downwelling plane irradiance: We can do the same for E u, E o, L( ,  ), etc, and define lots of different K functions: K u, K o, K L ( ,  ), etc.

How similar are the different K’s? HydroLight: homogeneous water with Chl = 2 mg/m 3, sun at 45 deg, 440 nm, etc. NOTE: The K’s depend on depth, even though the water is homogeneous, and they are most different near the surface (where the light field is changing because of boundary effects) May not see this if don’t have irradiances very near the surface

Something to Think About Suppose you measure E d (z) but the data are very noisy in the first few meters because of wave focusing, or bubbles, or… so you discard the data from the upper 5 meters You then compute K d from 5 m downward, and get a fairly constant K d value below 5 m You then use E d (z) = E d (0)exp(-K d z) and the computed K d from 5 m downward to extrapolate E d (5 m) back to the surface How accurate is this E d (0) likely to be?

How similar are the different K’s? HydroLight: homogeneous water with Chl = 2 mg/m 3, sun at 45 deg, 440 nm, etc. NOTE: the K’s all approach the same value as you go deeper: the asymptotic diffuse attenuation coefficient, k ,which is an IOP. k  = /m

Virtues and Vices of K’s Virtues: K’s are defined as rates of change with depth, so don’t need absolutely calibrated instruments K d is very strongly influenced by absorption, so correlates with chlorophyll concentration about 90% of water-leaving radiance comes from a depth of 1/K d (H. Gordon) radiative transfer theory provides connections between K’s and IOPs and other AOPs (recall Gershun’s equation: a = K net  ) Vices: not constant with depth, even in homogeneous water greatest variation is near the surface difficult to compute derivatives with noisy data

Average (Mean) Cosines The average cosine of the downwelling radiance distribution. Notation:  d denotes all downward directions (also written 2  d ). Ditto for upwelling radiance, and total radiance: and Note: E o = E od + E ou, but

Mean Cosines highly absorbing water: b = a, so  o = 0.5, vs highly scattering water: b = 4a, so  o = 0.8 Note: the highly scattering water approaches asymptotic values quicker than the highly absorbing water.

The Real World: Inhomogeneous Water HydroLight run with Chl = 0.5 mg/m 3 background and Chl = 2.5 mg/m 3 max at 20 m Note how well K d and K Lu correlate with the IOPs, but R isn’t much affected. Why?  d and  u are not strongly affected

The Real World: Inhomogeneous Water HydroLight run with Chl = 0.5 mg/m 3 background and Chl = 2.5 mg/m 3 max at 20 m Note how well K d and K Lu correlate with the IOPs, but R isn’t much affected. Why? What would happen to K and R if there were a layer of highly scattering but non-absorbing particles in the water? K  a R  b b /a

Explain These AOPs What does it mean for K u and K Lu to become negative? What does  u = 0.5 say about the upwelling radiance distribution at 15 m?

The Answer The water IOPs were homogeneous, but there was a Lambertian bottom at 15 m, which had a reflectance of R b = 0.15 Lambertian means the reflected radiance is the same in all directions (L is isotropic) Exercise: compute  d,  u, and  for an isotropic radiance distribution: L( ,  ) = L o = a constant

Relations Among IOPs and AOPs Manipulating the radiative transfer equation gives various relations among IOPs and AOPs, which are exact if there are no internal sources, e.g. These relations sometimes provide useful sanity checks on data. Rewritten forms of Gershun’s eq.

The Asymptotic Radiance Distribution As you go very deep into homogeneous water, the radiance distribution approaches the form L(z, ,  ) --> L  (  ) exp (-k  z) as z -->  where L  (  ) is the asymptotic radiance distribution and k  is the asymptotic diffuse attenuation coefficient. L  (  ) and k  are determined solely by the IOPs of the water; they are independent of the surface boundary conditions (sun angle, etc). Note at all  (azimuthal) dependence in the near-surface L(z, ,  ) is lost. Light and Water Chapter 9 tells you how to compute L  (  ) and k  given the IOPs (and it ain’t simple)

The Asymptotic Radiance Distribution Consequence: Because the asymptotic radiance decays with depth like exp (-k  z), independently of direction and boundary conditions, all irradiances must also decay at the same rate. This means that K d, K u, K o, K L ( ,  ) all approach the same value as z --> . Likewise, R and the average cosines approach values determined only by the IOPs. Thus all AOPs become IOPs at very large depths. k  = /m

The Asymptotic Radiance Distribution What does L  (  ) look like? Close to L  (  ) sun’s position not discernable at 66 m excellent agreement between measurements and HydroLight (but see L&W p for the details)

The Asymptotic Radiance Distribution How fast you approach the asymptotic values (i.e., how deep is deep enough) depends on the IOPs and boundary conditions: high scattering:  o = 0.8 high absorption:  o = 0.2  sun = 57 deg black sky heavily overcast sky

More Terminology diffuse vs isotropic radiance distributions: “diffuse” means you can’t tell where the sun is “isotropic” means the radiance is the same in all directions isotropic is always diffuse, but diffuse usually isn’t isotropic not diffuse: the sun’s direction is still detectable diffuse: I can tell up from down, but not where the sun is isotropic

More Terminology Lambertian surfaces can be described (by sloppy people) by “The surface reflects light isotropically” “The surface reflects light in a cosine pattern” It can’t be both, so which is it?

More Terminology Lambertian surfaces can be described (by sloppy people) by “The surface reflects light isotropically” “The surface reflects light in a cosine pattern” It can’t be both, so which is it? To be precise: Each point of a Lambertian surface reflects radiance (or individual photons) in a cosine pattern, therefore The measured radiance reflected by a Lambertian surface is isotropic   # surface points seen by radiometer  1/cos 

Bubble Effects on Rrs Lava Falls, Grand Canyon, photo by Curt “no guts, no glory” Mobley