Ocean Optics Inversion Algorithm

Transcription

1 Ocean Optics Inversion Algorithm N. J. McCormick 1 and Eric Rehm 2 1 University of Washington Department of Mechanical Engineering Seattle, WA mccor@u.washington.edu 2 University of Washington Applied Physics Laboratory and School of Oceanography Seattle, WA erehm@u.washington.edu ICTT-22: September 15, 211

2 Outline Introduction to analytic algorithms Ocean optical properties The new algorithm Tests of the algorithm Conclusions

3 Motivation Rehm PhD project: Characterize constituents of ocean waters from wavelength-dependent optical measurements Tasks: Perform unique set of in-water light field measurements (i.e., NAB8) for an iterative optical inversion method against a limited data set of water optical properties acquired simultaneously with different detectors Compare iterative optical inversion method results against those of limited data set The problem: His iterative inversion method sometimes did not converge to a global minimum

4 General objectives of (semi-)analytic inversion algorithms To obtain an initial estimate sometimes needed for convergence to a global minimum estimate from an iterative algorithm To minimize or eliminate iterative forward problem calculations (e.g., steepest descent, Levenberg-Marquart, etc.) for obtaining approximate optical property estimates To retain the physical insights from analytical methods (e.g., eigenfunction method) that provide an opportunity for analytic error estimation techniques

5 Brief history of 1-D analytic algorithms (1-D is good enough for ocean optics) (One wavelength with notation suppressed, depth z and 1 µ 1, ϕ 2π) Algorithms for full angle-dependent angular fluxes L(z, µ, ϕ): Developed in by Case, Kuščer, Siewert, McCormick, Sanchez, Larsen, and others But L(z, µ, ϕ) impractical to measure in an ocean environment

6 Low-order angular moments for ocean optics Algorithms for hemispherical fluxes and partial currents: Developed in 199s by Sanchez, McCormick and students Tao and Leathers Downward scalar flux : E d (z) = Upward scalar flux : E u (z) = Downward partial current : E d (z) = Upward partial current : E u (z) = 2π 1 2π 1 2π 1 2π 1 L(z, µ, ϕ)dϕdµ L(z, µ, ϕ)dϕdµ µl(z, µ, ϕ)dϕdµ But E d (z) and E u (z) impractical to measure in an ocean environment µ L(z, µ, ϕ)dϕdµ

7 Constraints on our new analytic algorithm 1. Rehm had depth-dependent data in spatially uniform waters only for Downward partial current : E d (z) = 2π 1 µl(z, µ, ϕ)dϕdµ Vertically upward angular flux : L u (z) = L(z, 1, ϕ = ) 2. To eliminate surface illumination magnitude, need ratio r rs (z) = Vertically upward angular flux Downward partial current = L u(z) E d (z) with which only one optical property can be determined

8 Objective of new algorithm Use measurements of E d (z), dominated by absorption, and L u (z), dominated by scattering, to see if they are sufficient to characterize an optical property in an ocean environment

9 Ocean optical properties What optical properties is the light field most sensitive to? Rehm had data for r rs (z) and also limited data for Absorption coefficient a Backscattering coefficient b b, as computed from the scattering phase function β(µ, µ) β(µ, µ) = 2π β(µ, ϕ, µ, ϕ = )dϕ with b b = b β(µ, µ)dµ 1

10 Typical angular scattering function For average oceanic particles (Petzold 1972), most scattering is in the forward direction. (Source: C.D. Mobley, private comm.)

11 Implications of ocean properties 1. Need to develop algorithm to determine b b /a 2. Assume isotropic plus delta forward phase function with single adjustable factor F : [ ( β(µ, µ) = 1 bb 2 F b ) ] b δ(µ µ) F where b b = b b /b is backscattering fraction 3. Apply to measurement data not near the surface so asymptotic transport approximation is applicable

12 The algorithm 1. Substitute phase function in one-speed, plane geometry equation. Get effective optical depth τ and albedo ϖ [ τ = a 1 + b ] b/a z and ϖ = b b/a F b b /a + F 2. Multiply transport equation for µ by µl(τ, µ) and one for µ by µl(τ, µ), integrate both over 1 µ 1, and subtract results ϖ d dτ µ 2 L(τ, µ)l(τ, µ)dµ = µl(τ, µ)dµ 1 1 L(τ, µ )dµ

13 The algorithm, cont d 3. Multiply transport equation by unity and integrate over 1 µ 1 to obtain 1 1 L(τ, µ)dµ = (1 ϖ) 1 d dτ 1 1 µl(τ, µ)dµ and substitute to get [ d 1 4 µ 2 L(τ, µ)l(τ, µ)dµ ϖ ( 1 ) 2 ] µl(τ, µ)dµ = dτ 1 ϖ 1 and convert τ to measurement depth z m

14 The algorithm, cont d 4. Divide by [E d (z m )/2π] 2 and substitute to get algorithm for b b /(af ): b b /a F [ 1 1 2π 16π 2 1 ϖ 1 ϖ = b b/a F µl(z m, µ) E d (z m ) for each measurement depth z m 2 µl(z m, µ) dµ] = E d (z m ) µl(z m, µ) dµ. E d (z m )

15 The algorithm, cont d 5. A big problem: no data for L(z m, µ) and L(z m, µ), µ 1, so define two new radiometric functions Λ(z m ) = 2π 1 Ω(z m ) = 16π 2 1 µ L(z m, µ) dµ = E u(z m ) L u (z m ) L u (z m ), µ 2 L(z m, µ) E d (z m ) L(z m, µ) dµ, L u (z m ) to get final algorithm in terms of measured r rs (z m ) b b /a F = Ω(z m )r rs (z m ) [1 Λ(z m )r rs (z m )] 2 Ω(z as )r rs (z m ) [1 Λ(z as )r rs (z m )] 2

16 The algorithm, cont d 6. Compute asymptotic Ω(z as ) and Λ(z as ) using isotropic-scattering classic asymptotic eigenmode L(τ, µ) φ(ν, µ) exp( τ/ν ) = ϖν /2 ν µ exp( τ/ν ) for asymptotic eigenvalue ν to get: Λ(z as ) = 2π(ν + 1) 1 µ(ν + µ) 1 dµ Ω(z as ) = 16π2 (ν + 1) 1 µ2 (ν 2 µ2 ) 1 dµ 1 µ(ν µ) 1 dµ

17 The algorithm, cont d 7. Determine factor F from Ω(z as ) and Λ(z as ) and r rs (z m ) obtained by Ecolight c calculations by minimizing cost function g F = G T F G F b b /a F Ω(z as )r rs (z m ) [1 Λ(z as )r rs (z m )] 2 G F = 1 ϖν ( ) 2 ln ν + 1 ν 1 (b b /a) (b b /a) measured and parameterize F as a function of b b /a with F = p 1 (b b /a) + p 2

18 Tests of the algorithm With computer generated data sets: 63 different simulations (7 different chlorophyll concentrations, 3 depths, 3 incident illumination directions with no simulated noise or natural variability) and selected phase functions β FF (training set) β Petzold (validation set) With Rehm s 28 North Atlantic Bloom (NAB8) data for L u (z), E d (z) a(z) b b (z) estimated from measurements at two wavelengths

19 Test of algorithm: synthetic data F tuned with training set: L u, E d using (β FF ) 6 5 β FF β Petzold Absolute retrieval error for b b /a with: Training set (β FF ) Validation set (β Petzold ) Retrieval Absolute Error (%) Chlorophyll (mg m -3 ) Conclusion: It appears delta-isotropic phase function is not bad

20 NAB8 vertical IOPs and radiometric data b b (488) (m -1 ) (A) L u (µw cm -2 sr -1 ), E d (µw cm -2 ) (B) 1 a(488) b b (488) b b /a(488) 1 L u (488) E d (488) Depth (m) Depth (m) a (m -1 ), b b /a (dimensionless)

21 Determination of F from NAB8 field data F(b b /a, 488) (A) F(b b /a, λ) nm 44 nm 488 nm 51 nm 532 nm (B) b b /a b b /a λ (nm) p 1 p 2 R

22 b b /a validation from NAB8 field data b b /a est (488) (A) (D) Relative Absolute Error (%) Relative Abs. Error (%) Relative Error (%) b b /a meas (488) (B) (C) k = number of profiles used for F Depth (m)

23 Use in global optimization algorithms 1 8 Inside IOP Subspace Outside IOP Subspace Relative error, a(49) {%) Local minima (error > 5%): start (open circles), end (solid dots). Global minimum (error < 5%): start (unmarked gray dots) Using analytical algorithm: start points inside gray box

24 Conclusions One-parameter delta-isotropic phase function acceptable F factor is spectrally dependent Algorithm works OK approaching asymptotic regime with best results in deeper waters Algorithm gives useful estimate for starting and constraining iterative algorithms Algorithm doesn t work well for wavelengths where inelastic (Raman) scattering important

25 Future work? Extend algorithm to multigroup theory so can treat inelastic scattering for wavelengths >55 nm or so b b /a is the optical property typically obtained from satellite data so develop algorithm for near-surface measurements by computing approximate Λ( + ) and Ω( + ) factors with singular eigenfunction half-space albedo problem angular flux

26 Want more details? If accepted, see Rehm and McCormick, Optics Express (211 or 212) Ask a question

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28 Raman scattering wavelength redistribution function (Source: Mobley, Light and Water, 1994)