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

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 2Collin, Lecture 10Emmanuel, Lecture 18

Curt, Lecture 6

Emmanuel, Lecture 8

Curt, Lecture 9Collin, Lectures 2 & 10

Collin, Lecture 2Mary Jane, Lecture 7Curt, Lecture 12

Collin, Lectures 3 & 4Mary 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, bb, 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

Ed and Eu

HydroLight runs: Chl = 1.0 mg Chl/m3, etcSun at 0, 30, 60 deg in clear sky, and solid overcast

Ed and Eu at 555 nm

1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

1.00E+01

0 10 20 30 40 50

depth [m]

Ed

or

Eu

[W

/m^

2 n

m]

Ed(0)

Eu(0)

Ed(30)

Eu(30)

Ed(60)

Eu(60)

Ed(ovcst)

Eu(ovcst)

Factor of 10 variation for different sky conditions

Note: Ed and Eu 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

R(z,λ) Eu(z,λ)

Ed(z,λ)

z

R = Eu/Ed

HydroLight runs: Chl = 1.0 mg Chl/m3, etcSun at 0, 30, 60 deg in clear sky, and solid overcast

R = Eu/Ed at 555 nm

0.01

0.015

0.02

0.025

0 10 20 30 40 50

depth [m]

R

R(0)

R(30)

R(60)

R(ovcst)

± 30% variation for different sky conditions

approaching same value

R = Eu/Ed

HydroLight runs: Chl = 0.1,1, 10 mg Chl/m3

Sun at 0, 30, 60 deg in clear sky

R(Chl,sun) at 5 m

0

0.02

0.04

0.06

400 450 500 550 600 650 700

wavelength [nm]

R =

Eu

/Ed

R(0.1,30))

R(1,30)

R(10,30)

R(1,0)

R(1,60)

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, Lw

total upwelling radiance in air (above the surface) = water-leaving radiance + surface-reflected radiance

Ed()

Lr(,,)

Lt(,,)

Lw(,,)

Lu(,,)

Lu(,,) = Lw(,,) + Lr(,,)

An instrument measures Lu (in air), but Lw is what tells us what is going on in the water. It isn’t easy to figure out how much of Lu is due to Lw.

Remote-sensing Reflectance Rrs

sea surface

Often work with the nadir-viewing Rrs, i.e.,with the radiance that is heading straight up from the sea surface ( = 0)

Rrs(in air,θ,φ, λ) Lw(in air,θ,φ,λ)

Ed(in air,λ)[sr 1]

The fundamental quantity used in ocean color remote sensing

Rrs(θ,φ,λ) =

upwelling water-leaving radiancedownwelling plane irradiance

Example Rrs HydroLight runs: Chl = 0.1,1, 10 mg Chl/m3

Sun at 0, 30, 60 deg in clear sky

Rrs shows very little dependence on sun angle and strong dependence on the water IOPs—a very good AOP

0.00E+00

2.00E-03

4.00E-03

6.00E-03

8.00E-03

1.00E-02

1.20E-02

400 450 500 550 600 650 700

wavelength [nm]

Rrs

[1/

sr]

Rrs(0.1,0)

Rrs(0.1,30)

Rrs(0.01,60)

Rrs(1,0)

Rrs(1,30)

Rrs(1,60)

Rrs(10,0)

Rrs(10,30)

Rrs(10,60)

Chl = 0.1

Chl = 1

Chl = 10

Ed and Eu at 555 nm

1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

1.00E+01

0 10 20 30 40 50

depth [m]

Ed

or

Eu

[W

/m^

2 n

m]

Ed(0)

Eu(0)

Ed(30)

Eu(30)

Ed(60)

Eu(60)

Ed(ovcst)

Eu(ovcst)

slopes are very similar

Ed and Eu

Magnitude changes are due to sun angle; slope is determined by water IOPs.

This suggests trying...Kd(z,)dlnEd(z,)dz1Ed(z,)dEd(z,)dzKd(z,)dlnEd(z,)dz1Ed(z,)dEd(z,)dzThe 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 Eu, Eo, L(,), etc, and define lots of different K functions: Ku, Ko, KL(,), etc.

How similar are the different K’s?

0.120

0.130

0.140

0.150

0.160

0.170

0.180

0.190

0.200

0 5 10 15 20

depth [m]

K(4

40

nm

) [

1/m

]

Kd

Ku

Ko

Knet

KLu

HydroLight: homogeneous water with Chl = 2 mg/m3, 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 Ed(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 Kd from 5 m downward, and get a fairly constant Kd value below 5 m• You then use Ed(z) = Ed(0)exp(-Kd z) and the computed Kd from 5 m downward to extrapolate Ed(5 m) back to the surface

How accurate is this Ed(0) likely to be?

How similar are the different K’s?HydroLight: homogeneous water with Chl = 2 mg/m3, 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.

0.120

0.130

0.140

0.150

0.160

0.170

0.180

0.190

0.200

0 10 20 30 40 50 60 70 80 90 100

depth [m]

K(4

40

nm

) [

1/m

]

Kd

Ku

Ko

Knet

KLu

k = 0.174 1/m

Virtues and Vices of K’sVirtues:• K’s are defined as rates of change with depth, so don’t need absolutely calibrated instruments• Kd is very strongly influenced by absorption, so correlates with chlorophyll concentration• about 90% of water-leaving radiance comes from a depth of 1/Kd (H. Gordon)• radiative transfer theory provides connections between K’s and IOPs and other AOPs (recall Gershun’s equation: a = Knet )

Vices:• not constant with depth, even in homogeneous water• greatest variation is near the surface• difficult to compute derivatives with noisy data

Average (Mean) CosinesThe average cosine of the downwelling radiance distribution. Notation: d denotes all downward directions (also written 2d).d(z,)dL(z,,,)cosddL(z,,,)dEd(z,)Eod(z,)d(z,)dL(z,,,)cosddL(z,,,)dEd(z,)Eod(z,)Ditto for upwelling radiance, and total radiance:u(z,)Eu(z,)Eou(z,)u(z,)Eu(z,)Eou(z,)and (z,)Ed(z,)Eu(z,)Eo(z,)Note: Eo = Eod + Eou, butdudu

Mean Cosines

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 10 20 30 40 50 60 70 80 90 100

depth [m]

mu

_d

, mu

_u

, or

mu

Mud(0.5)

Muu(0.5)

Mu(0.5)

Mud(0.8)

Muu(0.8)

Mu(0.8)

highly absorbing water: b = a, so o = 0.5, vshighly 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

0

0.2

0.4

0.6

0.8

0 5 10 15 20 25 30 35 40 45 50

depth [m]

a a

nd

b [

1/m

]

a

b

0

0.05

0.1

0.15

0.2

0.25

0 5 10 15 20 25 30 35 40 45 50

depth [m]

R [

n.d

.] o

r K

d a

nd

KL

u

[1/m

] R=Eu/Ed

Kd

KLu

0.3

0.4

0.5

0.6

0.7

0.8

0 10 20 30 40 50

depth [m]

mu

_d

an

d m

u_

u

Mud

Muu

HydroLight run with Chl = 0.5 mg/m3 background and Chl = 2.5 mg/m3 max at 20 m

Note how well Kd and KLu correlate with the IOPs, but R isn’t much affected. Why?

d and u are not strongly affected

The Real World: Inhomogeneous Water

0

0.2

0.4

0.6

0.8

0 5 10 15 20 25 30 35 40 45 50

depth [m]

a a

nd

b [

1/m

]

a

b

0

0.05

0.1

0.15

0.2

0.25

0 5 10 15 20 25 30 35 40 45 50

depth [m]

R [

n.d

.] o

r K

d a

nd

KL

u

[1/m

] R=Eu/Ed

Kd

KLu

0.3

0.4

0.5

0.6

0.7

0.8

0 10 20 30 40 50

depth [m]

mu

_d

an

d m

u_

u

Mud

Muu

HydroLight run with Chl = 0.5 mg/m3 background and Chl = 2.5 mg/m3 max at 20 m

Note how well Kd and KLu 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 aR bb/a

Explain These AOPs

-1.00E-01

-5.00E-02

0.00E+00

5.00E-02

1.00E-01

1.50E-01

0 5 10 15

depth [m]

K [

1/m

]

Kd

Ku

Ko

Knet

KLu

0

0.05

0.1

0.15

0.2

0 5 10 15

depth [m]

R R=Eu/Ed

0.4

0.5

0.6

0.7

0.8

0 5 10 15

depth [m]

mu

_d

an

d m

u_

u

Mud

Muu

What does it mean for Ku and KLu 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 Rb = 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(,) = Lo = a constant

Relations Among IOPs and AOPsManipulating the radiative transfer equation gives various relations among IOPs and AOPs, which are exact if there are no internal sources, e.g.aKddcaKddcThese relations sometimes provide useful sanity checks on data.

aKneta(1R)KdRKuaKneta(1R)KdRKuRewritten forms of Gershun’s eq.RKda/dKua/uddz(EdEu)aEo,ddz(EdEu)aEo,

The Asymptotic Radiance DistributionAs 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 DistributionConsequence: 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 Kd, Ku, Ko, KL(,) 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.

0.120

0.130

0.140

0.150

0.160

0.170

0.180

0.190

0.200

0 10 20 30 40 50 60 70 80 90 100

depth [m]

K(4

40

nm

) [

1/m

]

Kd

Ku

Ko

Knet

KLu

k = 0.174 1/m

The Asymptotic Radiance DistributionWhat 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 509-511 for the details)

The Asymptotic Radiance DistributionHow 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 degblack sky

heavily overcast sky

More Terminologydiffuse 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 TerminologyLambertian 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 TerminologyLambertian 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