Bio-optical Models

BING implements various bio-optical models for ocean color remote sensing analysis. This document describes the available models and their mathematical formulations.

Overview

Bio-optical models in BING relate Inherent Optical Properties (IOPs) to Remote Sensing Reflectance (Rrs):

\[R_{rs}(\lambda) = f \cdot \frac{b_b(\lambda)}{a(\lambda) + b_b(\lambda)}\]

where: - \(a(\lambda)\) is the total absorption coefficient - \(b_b(\lambda)\) is the total backscattering coefficient - \(f\) is a factor depending on the solar zenith angle and viewing geometry

Model Components

Total Absorption

The total absorption coefficient is decomposed as:

\[a(\lambda) = a_w(\lambda) + a_{nw}(\lambda)\]

where: - \(a_w(\lambda)\) is pure water absorption - \(a_{nw}(\lambda)\) is non-water absorption

Non-water Absorption Models

Exponential Model (Bricaud)

\[a_{ph}(\lambda) = A_{ph} \cdot [Chl]^{E_{ph}} \cdot a_{ph}^*(\lambda)\]

where \(a_{ph}^*(\lambda)\) is the chlorophyll-specific absorption coefficient.

CDOM Absorption

\[a_{dg}(\lambda) = a_{dg}(\lambda_0) \cdot \exp[-S_{dg}(\lambda - \lambda_0)]\]

where \(S_{dg}\) is the spectral slope.

Total Backscattering

\[b_b(\lambda) = b_{bw}(\lambda) + b_{bnw}(\lambda)\]

where: - \(b_{bw}(\lambda)\) is water molecular backscattering - \(b_{bnw}(\lambda)\) is particulate backscattering

Non-water Backscattering Models

Power Law Model

\[b_{bnw}(\lambda) = b_{bnw}(\lambda_0) \cdot \left(\frac{\lambda_0}{\lambda}\right)^Y\]

where \(Y\) is the spectral slope parameter.

Lee Model

Based on Lee et al. (2002):

\[b_{bp}(\lambda) = b_{bp}(443) \cdot \left(\frac{443}{\lambda}\right)^Y\]

Implemented Models

Absorption Models (anw)

class bing.models.anw

ExpBricaud

Exponential Bricaud model for phytoplankton absorption

Parameters:

A_ph: Amplitude coefficient
E_ph: Exponent for chlorophyll dependency
S_dg: CDOM spectral slope

GIOP

Generalized IOP model absorption

Parameters:

a_dg_443: CDOM absorption at 443 nm
a_ph_443: Phytoplankton absorption at 443 nm
S_dg: CDOM spectral slope

GSM

Garver-Siegel-Maritorena model absorption

Parameters:

Chl: Chlorophyll concentration
a_dg_443: CDOM absorption at 443 nm

QAA

Quasi-Analytical Algorithm absorption

Parameters:

a_total_ref: Total absorption at reference wavelength
S: Spectral slope

Backscattering Models (bbnw)

class bing.models.bbnw

PowerLaw

Power law backscattering model

Parameters:

b_bp_ref: Particulate backscattering at reference wavelength
Y: Spectral slope

Lee

Lee et al. (2002) backscattering model

Parameters:

b_bp_443: Particulate backscattering at 443 nm
Y: Spectral slope (default: 1.0)

GSM

GSM backscattering model

Parameters:

b_bp_443: Particulate backscattering at 443 nm

Constant

Wavelength-independent backscattering

Parameters:

b_bp: Constant backscattering value

Model Initialization

Initialize models using the utilities module:

from bing.models import utils as model_utils
import numpy as np

# Define wavelengths
wavelengths = np.arange(400, 701, 5)

# Initialize absorption and backscattering models
model_names = ['ExpBricaud', 'PowerLaw']
models = model_utils.init(model_names, wavelengths)

# Access individual models
anw_model = models[0]  # Absorption model
bbnw_model = models[1]  # Backscattering model

Model Configuration

Setting Chlorophyll

For models that use chlorophyll:

# Set chlorophyll concentration
anw_model.set_aph(Chl=1.5)  # mg/m³

Custom Parameters

Override default parameters:

from bing.models.anw import ExpBricaud

# Custom initialization
model = ExpBricaud(
    wave=wavelengths,
    A_ph=0.05,
    E_ph=0.65,
    S_dg=0.015
)

Model Evaluation

Forward Modeling

Calculate Rrs from model parameters:

from bing import rt as bing_rt

# Set parameters
params = [0.01, 0.65, 0.015, 0.001, 1.2]  # Example parameters

# Evaluate models
anw = anw_model.eval(params[:3])
bbnw = bbnw_model.eval(params[3:])

# Add water contributions
aw = absorption.a_water(wavelengths)
bbw = scattering.b_water(wavelengths) * 0.5

# Total IOPs
a_total = aw + anw
bb_total = bbw + bbnw

# Calculate Rrs
Rrs = bing_rt.calc_Rrs(a_total, bb_total)

Model Comparison

Compare different model combinations:

from bing.parameters import standard

# Different model combinations
configs = [
    standard.expb_pow(),    # ExpBricaud + PowerLaw
    standard.giop(),         # GIOP + Lee
    standard.gsm_gsm(),      # GSM + GSM
]

for config in configs:
    models = model_utils.init(config.model_names, wavelengths)
    # Perform fitting and analysis...

Performance Considerations

Model Selection Guidelines

ExpBricaud + PowerLaw: Good for general cases, especially with chlorophyll data
GIOP: NASA’s operational model, well-validated
GSM: Empirical model, good for Case 1 waters
QAA: Semi-analytical, good for clear waters

Computational Efficiency

Simple models (Constant, PowerLaw): Fast evaluation
Complex models (GSM, QAA): More computational overhead
MCMC fitting: Use appropriate number of steps based on model complexity

Model Validation

Residual Analysis

from bing import evaluate

# Calculate residuals
residuals = (measured_Rrs - modeled_Rrs) / uncertainty

# Statistical metrics
rmse = np.sqrt(np.mean(residuals**2))
bias = np.mean(residuals)

print(f"RMSE: {rmse:.4f}")
print(f"Bias: {bias:.4f}")

Cross-Validation

from sklearn.model_selection import KFold

# K-fold cross-validation
kf = KFold(n_splits=5, shuffle=True)

for train_idx, test_idx in kf.split(data):
    train_data = data[train_idx]
    test_data = data[test_idx]

    # Fit model on training data
    # Evaluate on test data

Advanced Topics

Custom Model Implementation

Create custom models by subclassing base classes:

from bing.models.base import BaseModel

class CustomAbsorption(BaseModel):
    def __init__(self, wave, **kwargs):
        super().__init__(wave, **kwargs)
        self.n_params = 3  # Number of parameters

    def eval(self, params):
        # Custom evaluation logic
        a0, slope, offset = params
        return a0 * np.exp(-slope * (self.wave - 443)) + offset

Model Coupling

Couple absorption and backscattering models:

# Coupled parameterization
def coupled_model(params, wave):
    # Shared parameters
    Chl = params[0]

    # Absorption depends on Chl
    anw = 0.05 * Chl**0.65 * absorption_spectrum(wave)

    # Backscattering also depends on Chl
    bbnw = 0.001 * Chl**0.5 * backscatter_spectrum(wave)

    return anw, bbnw

References

Bricaud et al. (1995) - Phytoplankton absorption
Lee et al. (2002) - Backscattering models
Maritorena et al. (2002) - GSM algorithm
Werdell et al. (2013) - GIOP framework
Lee et al. (2002) - QAA algorithm