Chlorophyll Fluorescence

A set of Python methods for calculating chlorophyll fluorescence inelastic emission and its contribution to remote sensing reflectance (Rrs) in seawater.

Overview

Chlorophyll a (Chl-a) fluorescence occurs when light absorbed by photosynthetic pigments in phytoplankton is re-emitted at longer wavelengths. The primary emission peak is centered at 685 nm, with a secondary peak around 730-740 nm.

The fluorescence emission adds to the elastic backscattering signal and can contribute significantly to remote sensing reflectance in the red wavelengths, particularly in waters with high chlorophyll concentration or under conditions of photoinhibition.

This module implements formulations from:

Gordon (1979) – Original diffuse reflectance augmentation by chlorophyll fluorescence at 685 nm
Maritorena et al. (2000) – Fluorescence quantum yield determinations
Behrenfeld et al. (2009) – Satellite-detected fluorescence and global phytoplankton physiology
Bricaud et al. (1995) – Phytoplankton absorption parameterization
Sathyendranath & Platt (1998) – Inelastic scattering radiative transfer

Key characteristics:

Excitation range: 370-690 nm (absorbed by photosynthetic pigments)
Primary emission: 685 nm (PS II), FWHM ~25 nm
Secondary emission: ~730 nm (PS I), FWHM ~50 nm
Quantum yield: 0.005-0.07 (varies with irradiance and physiology)

Dependencies

The chlorophyll fluorescence implementation depends on the correct_atmosphere package for downwelling irradiance spectra used to weight the excitation wavelengths.

Install via:

pip install git+https://github.com/ocean-colour/correct-atmosphere.git

The package is imported as:

from correct_atmosphere import downwelling

Module Architecture

The chlorophyll fluorescence implementation is organized into two modules:

Low-level methods (bing.rt.chl_fl):: Core physical calculations including emission line shapes, quantum yields, fluorescence coefficients, phase functions, and wavelength redistribution. Also includes the Gordon (1979) / Sathyendranath & Platt (1998) reflectance formulation calc_R_fluorescence and the Fluorescence Line Height calculation.
Rrs methods (bing.rt.rrs):: The integrated function calc_Rrs_fluorescence that integrates the fluorescence contribution to remote sensing reflectance over excitation wavelengths, using the Bricaud parameterization for phytoplankton absorption and a supplied downwelling irradiance spectrum.

Quick Start

The simplest way to compute the fluorescence contribution to Rrs:

import numpy as np
from bing.rt import calc_Rrs_fluorescence
from correct_atmosphere import downwelling

# Emission wavelengths (where fluorescence appears)
wave = np.linspace(650, 750, 101)

# Excitation wavelengths (where light is absorbed)
wave_ex = np.arange(400.0, 681.0, 1.0)

# IOPs at emission and excitation wavelengths (set by user from a model)
a_em  = ...   # Total absorption at emission wavelengths  [m^-1]
bb_em = ...   # Total backscattering at emission wavelengths [m^-1]
a_ex  = ...   # Total absorption at excitation wavelengths [m^-1]
bb_ex = ...   # Total backscattering at excitation wavelengths [m^-1]
aph_ex = ...  # Phytoplankton absorption at excitation wavelengths [m^-1]

# Downwelling irradiance at excitation and (peak) emission
Ed_ex = downwelling.solar_irradiance(wave_ex)
Ed_em = downwelling.solar_irradiance(685.)

# Fluorescence contribution to Rrs
Rrs_fl = calc_Rrs_fluorescence(
    wave, a_em, bb_em,
    a_ex, bb_ex, aph_ex,
    wave_ex, Ed_ex, Ed_em,
    phi_C=0.02,
    double_gaussian=True,
)

# Fluorescence Line Height (FLH) from satellite Rrs (MERIS/OLCI bands)
from bing.rt import calc_fluorescence_line_height
FLH = calc_fluorescence_line_height(
    Rrs_665=0.001, Rrs_680=0.0015, Rrs_709=0.0005)

Top-level Rrs Method

calc_Rrs_fluorescence

calc_Rrs_fluorescence(wavelength, a_em, bb_em, a_ex, bb_ex, aph_ex, wavelength_ex, Ed_ex, Ed_em, mu_d=None, mu_f=None, phi_C=0.02, double_gaussian=True)

Calculate the contribution to Rrs from chlorophyll fluorescence, integrating over excitation wavelengths.

Parameters:

wavelength (float or ndarray) – Emission wavelength(s) \(\lambda\) in nm (typically 650-750 nm).
a_em (float or ndarray) – Total absorption at emission wavelength(s) [m^-1].
bb_em (float or ndarray) – Total backscattering at emission wavelength(s) [m^-1].
a_ex (ndarray) – Total absorption at excitation wavelengths [m^-1].
bb_ex (ndarray) – Total backscattering at excitation wavelengths [m^-1].
aph_ex (ndarray) – Phytoplankton absorption at excitation wavelengths [m^-1].
wavelength_ex (ndarray) – Excitation wavelengths \(\lambda'\) in nm.
Ed_ex (ndarray) – Downwelling irradiance at excitation wavelengths (units cancel in the normalization).
Ed_em (float or ndarray) – Downwelling irradiance at the (peak) emission wavelength.
mu_d (float, optional) – Mean cosine for downwelling irradiance. Default: raman.MU_D_DEFAULT.
mu_f (float, optional) – Mean cosine for fluorescence emission (isotropic = 0.5).
phi_C (float) – Fluorescence quantum yield. Default 0.02.
double_gaussian (bool) – If True, use the double-Gaussian emission line (685 + 730 nm peaks). Default True.

Returns:

Fluorescence contribution to Rrs in sr^-1.

Return type:

ndarray

The function integrates the contribution from each excitation wavelength following Gordon (1979) / Sathyendranath & Platt (1998), shapes the result with the chlorophyll emission line, then converts the subsurface reflectance to above-surface Rrs.

The function supports both single spectra (1D a_ex) and MCMC chains (2D a_ex with shape (nsamples, nwave_ex)).

Note

The upwelling attenuation \(\kappa^F(\lambda)\) and the energy conversion factor \(\lambda' / \lambda\) are evaluated at every emission wavelength \(\lambda\) in the output grid – not pinned to the 685 nm primary peak. This matters for the double-Gaussian model: pure-water absorption at 730 nm is roughly 4x larger than at 685 nm, so the secondary peak is suppressed by the stronger upwelling attenuation there. See dev/ChlFl/double_gaussian.py for the investigation that motivated this choice.

Low-level Methods (`bing.rt.chl_fl`)

Emission Line Shape

emission_line_single_gaussian(wavelength, lambda_center=685.0, sigma=10.6): Single Gaussian emission line \(h_C(\lambda)\), normalized so that \(\int h_C(\lambda)\,d\lambda = 1\).

emission_line_double_gaussian(wavelength, lambda_primary=685.0, sigma_primary=10.6, lambda_secondary=730.0, sigma_secondary=21.2, weight_primary=0.75): Weighted sum of two Gaussians for the PS II (685 nm) and PS I (730 nm) peaks:

\[h_C(\lambda) = W\,G(\lambda;\lambda_1,\sigma_1) + (1-W)\,G(\lambda;\lambda_2,\sigma_2)\]

Quantum Yield Methods

quantum_yield_constant(phi=0.02): Return a constant quantum yield \(\Phi_C\).

quantum_yield_irradiance_dependent(PAR, phi_max=0.07, phi_min=0.01, E_k=100.0): Irradiance-dependent quantum yield (Morrison 2003):

\[\Phi_C(E) = \Phi_{min} + (\Phi_{max} - \Phi_{min}) \frac{E_k}{E + E_k}\]

quantum_yield_depth_profile(depth, K_PAR=0.05, PAR_surface=500.0, phi_max=0.07, phi_min=0.01, E_k=100.0): Depth profile combining Beer-Lambert PAR attenuation with the irradiance-dependent quantum yield.

Absorption Efficiency

absorption_efficiency(wavelength_ex, a_ph): Returns 1.0 within the photosynthetic excitation range (370-690 nm) and 0.0 outside.

absorption_efficiency_weighted(wavelength_ex, a_ph, a_ph_ref=None): Weights by relative phytoplankton absorption \(a_{ph}/a_{ph,\rm ref}\) within the excitation range.

Fluorescence Coefficients

fluorescence_scattering_coeff(a_ph, phi_C=0.02): \(b_C(\lambda') = \Phi_C\,a_{ph}(\lambda')\).

fluorescence_backscattering_coeff(a_ph, phi_C=0.02): \(b_{bC} = 0.5\,b_C = 0.5\,\Phi_C\,a_{ph}\) (isotropic emission).

Phase Function

fluorescence_phase_function(psi): Isotropic phase function \(\tilde{\beta}_C(\psi) = 1/(4\pi)\).

fluorescence_backscatter_fraction(): Returns 0.5 (the isotropic backscatter fraction).

Volume Scattering Function

fluorescence_vsf(wavelength_ex, wavelength_em, psi, a_ph, phi_C=0.02, double_gaussian=False): Combines the scattering coefficient, wavelength redistribution and angular distribution into \(\beta_C(\lambda', \lambda, \psi)\).

Reflectance Contribution

calc_R_fluorescence(a_em, bb_em, a_ex, bb_ex, a_ph_ex, Ed_ratio=1.0, phi_C=0.02, mu_d=0.9, mu_f=0.5)

Fluorescence subsurface reflectance contribution following Sathyendranath & Platt (1998):

\[R^F(\lambda,0) = \frac{E_d(\lambda')}{E_d(\lambda)} \cdot \frac{b_{bF}(\lambda')/\mu_d}{K(\lambda') + \kappa^F(\lambda)}\]

where \(b_{bF} = 0.5\,\Phi_C\,a_{ph}\) is the fluorescence backscattering coefficient, \(K = (a + b_b)/\mu_d\) is the downwelling attenuation, and \(\kappa^F = (a + b_b)/\mu_f\) is the upwelling attenuation for fluorescence.

calc_R_fluorescence_integrated(wavelength_em, wavelength_ex, a_em, bb_em, a_ex, bb_ex, a_ph_ex, Ed, phi_C=0.02, mu_d=0.9, mu_f=0.5, double_gaussian=False): Subsurface fluorescence reflectance integrated over excitation wavelengths using a trapezoidal rule.

Fluorescence Line Height (FLH)

calc_fluorescence_line_height(Rrs_665, Rrs_680, Rrs_709, lambda_665=665.0, lambda_680=680.0, lambda_709=709.0)

Compute the Fluorescence Line Height. The baseline at the fluorescence band is linearly interpolated between 665 nm and 709 nm, and subtracted from the measured Rrs(680).

Reference wavelengths vary by sensor:

MODIS: 667, 678, 748 nm
MERIS / OLCI: 665, 681, 709 nm

calc_normalized_fluorescence_line_height(Rrs_665, Rrs_680, Rrs_709, lambda_665=665.0, lambda_680=680.0, lambda_709=709.0): Normalized FLH (nFLH), defined as the FLH divided by the baseline reflectance.

Convenience Methods

get_emission_spectrum(wavelength_ex, wavelength_em_range=None, n_points=100, double_gaussian=False): Return (wavelength_em, intensity) arrays for plotting the emission line shape.

summary_at_wavelength(wavelength_ex, a_ph, phi_C=0.02): Return a dictionary of fluorescence quantities at a single excitation wavelength.

Physical Constants

Constant	Value	Description
`LAMBDA_FL_PRIMARY`	685 nm	Primary emission wavelength (PS II)
`LAMBDA_FL_SECONDARY`	730 nm	Secondary emission wavelength (PS I)
`SIGMA_FL_PRIMARY`	10.6 nm	Primary peak standard deviation
`SIGMA_FL_SECONDARY`	21.2 nm	Secondary peak standard deviation
`FWHM_FL_PRIMARY`	25 nm	Primary peak FWHM
`FWHM_FL_SECONDARY`	50 nm	Secondary peak FWHM
`WEIGHT_PRIMARY`	0.75	Weight of primary peak in double-Gaussian
`PHI_FL_DEFAULT`	0.02	Default quantum yield (HydroLight value)
`PHI_FL_HIGH_LIGHT`	0.01	Quantum yield at high irradiance
`PHI_FL_LOW_LIGHT`	0.07	Quantum yield at low irradiance
`LAMBDA_EX_MIN`	370 nm	Minimum excitation wavelength
`LAMBDA_EX_MAX`	690 nm	Maximum excitation wavelength

Using Fluorescence in the BING Fitting Workflow

Chlorophyll fluorescence is enabled in BING fits through the parameter configuration object generated by bing.parameters.p_ntuple.gen(). The relevant fields are:

include_Chl_fl – include the fluorescence contribution in the forward model. Default False.
phi_C – fluorescence quantum yield. Default 0.02.
double_gaussian – if True, use the double-Gaussian (685+730 nm) emission line. Default True.

These settings are extracted into a small dictionary, the rt dict, by bing.rt.defs.rt_dict_from_p() (see The Radiative-Transfer Dictionary (rt_dict_from_p)). The rt dict is then passed through the chisq and MCMC fitting machinery to control which radiative-transfer corrections are applied during forward modeling.

Physical Background

Fluorescence Process

Chlorophyll fluorescence occurs through:

Absorption: Light at 370-690 nm absorbed by photosynthetic pigments
Excitation: Chlorophyll molecules raised to higher energy states
Energy dissipation: Photosynthesis, NPQ (heat), or fluorescence
Emission: 685 nm (PS II) and 730 nm (PS I) Gaussian peaks

Quantum Yield Variability

Condition	\(\Phi_C\)	Notes
High light (surface)	0.005-0.01	NPQ reduces yield
Low light (depth)	0.05-0.07	Maximum yield
Typical / average	0.02	HydroLight default
Nutrient stress	Variable	Can increase yield

Emission Spectrum Shape

Single Gaussian (PS II only):

\[h_C(\lambda) = \frac{1}{\sigma\sqrt{2\pi}} \exp\!\left[-\frac{(\lambda - 685)^2}{2\sigma^2}\right]\]

with \(\sigma = 10.6\) nm (FWHM = 25 nm).

Double Gaussian (PS II + PS I):

\[h_C(\lambda) = 0.75 \cdot G(685, 10.6) + 0.25 \cdot G(730, 21.2)\]

Reflectance Formulation

The fluorescence contribution to subsurface reflectance follows the Gordon (1979) / Sathyendranath & Platt (1998) formulation, integrated over excitation wavelengths and shaped by the emission line:

\[R^F(\lambda) = h_C(\lambda) \int \frac{E_d(\lambda')}{E_d(\lambda)}\, \frac{\lambda'}{\lambda}\, \frac{b_{bF}(\lambda')/\mu_d}{K(\lambda') + \kappa^F(\lambda)}\, d\lambda'\]

with \(b_{bF}(\lambda') = 0.5\,\Phi_C\,a_{ph}(\lambda')\), \(K(\lambda') = (a(\lambda') + b_b(\lambda'))/\mu_d\), and \(\kappa^F(\lambda) = (a(\lambda) + b_b(\lambda))/\mu_f\).

The \(\kappa^F(\lambda)\) and \(\lambda' / \lambda\) factors carry explicit emission-wavelength dependence, which is important for the double-Gaussian model: the 730 nm secondary peak sits in a region of much stronger pure-water absorption than the 685 nm primary peak, so it must be evaluated with its own \(\kappa^F\).

Comparison with Raman Scattering

Both chlorophyll fluorescence and Raman scattering are inelastic processes that redistribute light to longer wavelengths.

Property	Fluorescence	Raman
Source	Phytoplankton pigments	Water molecules
Excitation range	370-690 nm	All visible wavelengths
Emission	685, 730 nm (fixed)	Shifted by ~3400 cm^-1
Angular distribution	Isotropic (\(b_b/b = 0.5\))	Nearly isotropic (\(b_b/b = 0.5\))
Magnitude	Depends on Chl, varies widely	Fixed for given water type
Variability	High (physiology-dependent)	Low (temperature / salinity)

Limitations and Notes

Quantum yield uncertainty: \(\Phi_C\) varies significantly with phytoplankton physiology, light history, and nutrient status. The default value of 0.02 is approximate.
PS I contribution: The 730 nm peak is typically smaller than the primary PS II peak, but can be significant in certain conditions.
Absorption model: The Bricaud parameterization represents average phytoplankton. Specific phytoplankton groups may differ.
Excitation irradiance: The integrated Rrs method requires a downwelling irradiance spectrum (provided by correct_atmosphere).
Depth integration: For satellite applications, the signal represents a depth-integrated fluorescence weighted by the light field.

References

[Gordon1979]

Gordon, H.R. (1979). “Diffuse reflectance of the ocean: the theory of its augmentation by chlorophyll a fluorescence at 685 nm,” Appl. Opt. 18, 1161-1166.

[Bricaud1995]

Bricaud, A., Babin, M., Morel, A., and Claustre, H. (1995). “Variability in the chlorophyll-specific absorption coefficients of natural phytoplankton: Analysis and parameterization,” J. Geophys. Res. 100, 13321-13332.

[Maritorena2000]

Maritorena, S., Morel, A., and Gentili, B. (2000). “Determination of the fluorescence quantum yield by oceanic phytoplankton in their natural habitat,” Appl. Opt. 39, 6725-6737.

[SathyendranathPlatt1998]

Sathyendranath, S. and Platt, T. (1998). “Ocean-colour model incorporating transspectral processes,” Appl. Opt. 37, 2216-2227.

[Behrenfeld2009]

Behrenfeld, M.J. et al. (2009). “Satellite-detected fluorescence reveals global physiology of ocean phytoplankton,” Biogeosciences 6, 779-794.

[OOWB]

Ocean Optics Web Book: https://www.oceanopticsbook.info/view/scattering/level-2/chlorophyll-fluorescence