Radiative Transfer in Coupled Environmental Systems: An Introduction to Forward and Inverse Modeling

412 Pages·2015·22.275 MB·English

by  Knut Stamnes, Jakob J. Stamnes

Table of Contents
Cover
Related Titles
Title Page
Copyright
Preface
Acknowledgments
Chapter 1: Introduction
1.1 Brief History
1.2 What is Meant by a Coupled System?
1.3 Scope
1.4 Limitations of Scope
Chapter 2: Inherent Optical Properties (IOPs)
2.1 General Definitions
2.2 Examples of Scattering Phase Functions
2.3 Scattering Phase Matrix
2.4 IOPs of a Polydispersion of Particles—Integration over the Size Distribution
2.5 Scattering of an Electromagnetic Wave by Particles
2.6 Absorption and Scattering by Spherical Particles—Mie–Lorenz Theory
2.7 Atmosphere IOPs
2.8 Snow and Ice IOPs
2.9 Water IOPs
2.10 Fresnel Reflectance and Transmittance at a Plane Interface Between Two Coupled
Media
2.11 Surface Roughness Treatment
2.12 Land Surfaces
Problems
Chapter 3: Basic Radiative Transfer Theory
3.1 Derivation of the Radiative Transfer Equation (RTE)
3.2 Radiative Transfer of Unpolarized Radiation in Coupled Systems
3.3 Radiative Transfer of Polarized Radiation in Coupled Systems
3.4 Methods of Solution of the RTE
3.5 Calculation of Weighting Functions—Jacobians
Problems
Chapter 4: Forward Radiative Transfer Modeling
4.1 Quadrature Rule—The Double-Gauss Method
4.2 Discrete Ordinate Equations—Compact Matrix Formulation
4.3 Discrete-Ordinate Solutions
Problems
Chapter 5: The Inverse Problem
5.1 Probability and Rules for Consistent Reasoning
5.2 Parameter Estimation
5.3 Model Selection or Hypothesis Testing
5.4 Assigning Probabilities
5.5 Generic Formulation of the Inverse Problem
5.6 Linear Inverse Problems
5.7 Bayesian Approach to the Inverse Problem
5.8 Ill Posedness or Ill Conditioning
5.9 Nonlinear Inverse Problems
Problems
Chapter 6: Applications
6.1 Principal Component (PC) Analysis
6.2 Simultaneous Retrieval of Total Ozone Column (TOC) Amount and Cloud Effects
6.3 Coupled Atmosphere–Snow–Ice Systems
6.4 Coupled Atmosphere–Water Systems
6.5 Simultaneous Retrieval of Aerosol and Aquatic Parameters
6.6 Polarized RT in a Coupled Atmosphere–Ocean System
6.7 What if MODIS Could Measure Polarization?
Appendix A: Scattering of Electromagnetic Waves
A.1 Absorption and Scattering by a Particle of Arbitrary Shape
A.2 Absorption and Scattering by a Sphere—Mie Theory
Appendix B: Spectral Sampling Strategies
Transmission in an Isolated Spectral Line
The Random Band Model
B.1 The MODTRAN Band Model
B.2 The -Distribution Method
B.3 Spectral Mapping Methods
B.4 Principal Component (PC) Analysis
B.5 Optimal Spectral Sampling
Appendix C: Rough Surface Scattering and Transmission
C.1 Scattering and Emission by Random Rough Surfaces
Appendix D: Boundary Conditions
D.1 The Combined Boundary Condition System
D.2 Top of Upper Slab
D.3 Layer Interface Conditions in the Upper Slab
D.4 Layer Interface Conditions in the Lower Slab
D.5 Bottom Boundary of Lower Slab
References
Index
End User License Agreement
List of Illustrations
Chapter 2: Inherent Optical Properties (IOPs)
Figure 1 Scattering phase functions calculated using a Mie code. (a) For aerosols with
asymmetry factor 0.79275. (b) For clouds with asymmetry factor 0.86114. HG
scattering phase functions [see Eq. (24)] with asymmetry factors equal to those for the
cloud and aerosol particles are shown for comparison. The Rayleigh scattering phase
function [Eq (23)] describing molecular scattering is also shown for comparison.
Figure 2 Rayleigh [Eq. (23)], FF [Eq. (27)], and Petzold scattering phase functions.
Figure 3 Coordinate system for scattering by a volume element at . The points , ,
and are located on the unit sphere. The incident light beam with Stokes vector is
in direction with unit vector , and the scattered beam with Stokes vector
is in direction with unit vector [16].
Figure 4 Volume fractions of brine pockets ( ) and air bubbles ( ) (squares), and
scattering coefficients of brine pockets ( ) and air bubbles ( ) (circles). The two
curves to the left represent air bubbles, and the two curves to the right represent brine
pockets (after Hamre et al. [28] with permission).
Figure 5 Comparisons of IOPs calculated using Mie computations with those obtained
using Eq. (226) (Parameterization 1), which is valid for wavelengths shorter than about
1.2 m [28], and Eq. (230) (Parameterization 2), which is valid also in the near-
infrared region (after Stamnes et al. [123] with permission).
Figure 6 Spectral variation of the coefficients (left) and (right) in Eq. (239).
The solid curves indicate original data provided by Bricaud et al. [131]. The dotted
horizontal lines indicate extrapolated values.
Figure 7 Schematic illustration of the connection between size distribution and
phytoplankton functional types (PFTs). The right panel shows the percentage bio-
volume versus the PSD slope for each of the three different phytoplankton groups.
The particle size range for each phytoplankton group is provided in the text.
Figure 8 Imaginary part of the refractive index of phytoplankton and mineral particles
(redrawn after Babin et al. [130]) with permission.
Figure 9 Absorption coefficients [m ] derived from Mie computations for slope
parameters and 6.0 of the PSD. The absorption coefficient used in the
GSM model is also shown for comparison. (a) CHL = 0.01976 mg m . (b) CHL =
0.1976 mg m . (c) CHL = 1.976 mg m . (d) CHL = 19.76 mg m .
Figure 10 Scattering coefficients [m ] derived from Mie computations for slope
parameters and 6.0 of the PSD. The scattering coefficient used in the
GSM model is also shown for comparison. (a) CHL = 0.004323 mg m . (b) CHL =
0.04323 mg m . (c) CHL = 0.4323 mg m . (d) CHL = 4.323 mg m .
Figure 11 Remote sensing reflectances derived from our CRTM (coupled
atmosphere–ocean radiative transfer model) using our PSD bio-optical model with
and the GSM bio-optical model. (a): CHL = 0.001976 mg m . (b): CHL =
0.1976 mg m . (c): CHL = 1.976 mg m . (d): CHL = 19.76 mg m .
Figure 12 Angles relevant for reflection and refraction of a plane wave at a plane
interface between two dielectric media with different refractive indices.
Figure 13 A beam of light with electric field amplitude and intensity proportional to
is incident in a medium with refractive index upon an interface with a second
medium with refractive index at an angle with the interface normal. The beam is
transmitted through an area of the interface at an angle with the interface normal
into the second medium with intensity proportional to [from Sommersten et al.
[87] with permission].
Figure 14 Illustration of the scattering geometry used to calculate bidirectional
reflection coefficients.
Figure 15 Schematic illustration of a plane wave incident on a rough surface
characterized by a Gaussian random height distribution , where
with mean height . Medium 1 above the surface has
permeability and permittivity , whereas medium 2 below the surface has
permeability and permittivity .
Figure 16 Illustration of tilt angle ( ) and relative azimuth angle ( ) in rough
surface scattering. Here, is the normal to the facet, is the tilt angle, and is the
azimuth angle relative to the glint direction ( ).
Figure 17 Reflectance [Eq. (307)] for a Cox–Munk 1-D Gaussian
BRDF for a solar zenith angle of and a wind speed of 5 m s versus
zenith angle . The relative azimuth angle corresponds to the glint direction
(see Figure 16).
Figure 18 Illustration of the geometry involved in the description of the BRDF. The
phase angle is the supplementary angle of the scattering angle , that is, .
The relative azimuth angle corresponds to the glint direction, and to
the backscattering (hot-spot) direction.
Figure 19 Hapke BRDF [Eq. (318)] for a solar zenith angle of 60 , , ,
, and . The relative azimuth angle corresponds to the glint
direction, and to the backscattering (hot-spot) direction.
Figure 20 RPV BRDF [Eqs. (322)–(325)] for a polar incidence angle of 60 ,
, , , and . The relative azimuth angle corresponds to
the glint direction, and to the backscattering (hot-spot) direction.
Figure 21 Ross–Li BRDF [Eqs. (326)–(331)] for a polar incidence angle of 60 ,
, , and . The relative azimuth angle corresponds to the
glint direction, and to the backscattering (hot-spot) direction.
Chapter 3: Basic Radiative Transfer Theory
Figure 22 Illustration of the doubling concept. Two similar layers, each of optical
thickness , are combined to give the reflection and transmittance matrices of a layer
of twice the optical thickness ([see Eqs. (462) and (463)]).
Figure 23 Illustration of the adding concept. Two dissimilar inhomogeneous layers are
combined (added) to give the reflection and transmission matrices for two layers with
different IOPs. Illumination from the top of the two layers yields Eq. (464) for the
reflection matrix and Eq. (465) for the transmittance matrix. Illumination from the
bottom of the two layers yields Eqs. (466) and (467).
Figure 24 Architecture of a radial basis function neural network (RBF-NN).
Figure 25 Radial basis function neural network (RBF-NN).
Chapter 4: Forward Radiative Transfer Modeling
Figure 26 Schematic illustration of the quadrature adopted for a coupled atmosphere-
water system. The dotted line labeled marks the separation between region II, in
which light is refracted from the atmosphere into the water, and region I of total
reflection in the water, in which the upward directed light beam in the water undergoes
total internal reflection at the water–air interface. The quadrature angles in region II
are connected to those in the atmosphere through Snell's law. Additional quadrature
points are added to represent quadrature angles in region I as indicated. Note that for
“bookkeeping purposes”, in the air corresponds to in the water.
Figure 27 Schematic illustration of two vertically inhomogeneous slabs separated by an
interface across which the refractive index changes abruptly like in an atmosphere–
water system.
Chapter 5: The Inverse Problem
Figure 28 The Gaussian or normal distribution, given by in Eq. (608), is
symmetric about its maximum value at , and has a full-width at half-
maximum (FWHM) of about 2.35 .
Figure 29 Schematic illustration of the geometry of the lighthouse problem.
Figure 30 The Cauchy or Lorentzian distribution, given by Eq. (634), is symmetric with
respect to its maximum at , and has an FWHM of 2 .
Figure 31 The contour in the parameter space along which constant is an
ellipse centered at . The ellipse is characterized by the eigenvalues and
eigenvectors defined by Eq. (646).
Figure 32 Schematic illustration of covariance and correlation. (a) Posterior pdf with
zero covariance (inferred values of and are uncorrelated). (b) Large and negative
covariance ( constant along dotted line). (c) Large and positive covariance (
constant along dotted line).
Figure 33 Change of variables in one dimension. The function maps the point to
.
Figure 34 Change of variables from Cartesian to polar coordinates.
Figure 35 Schematic illustration of the prior pdf (dashed line) and the Gaussian
likelihood function (solid line) for the parameter .
Figure 36 (a) A particular misfit norm, , and its associated model norm,
. (b) A particular misfit norm, , and its associated model norm, .
Chapter 6: Applications
Figure 37 TOA reflectance spectra. (a) Obtained from DISORT,
. (b) Correlation between DISORT and TWOSTR
reflectance spectra. (c) Difference between TWOSTR and DISORT reflectance
spectra. The residuals are plotted as a function of the DISORT reflectance to show
systematic deviations more clearly. After Natraj et al. [105] with permission.
Figure 38 Same as Figure 37 (a), but reconstructed using PC analysis. After Natraj et
al. [105] with permission.
Figure 39 Relation between the radiation modification factor (RMF) defined in Eq.
(901) and cloud optical depth (COD) based on simulated data obtained from the RTM
(after Fan et al. [235] with permission).
Figure 40 Correlations between TOC values derived by the RBF-NN and LUT methods
(a), and relation between COD values derived using the RBF-NN method and RMF
values derived using the LUT method (b) in 2012 (after Fan et al. [235] with
permission).
Figure 41 (a) COD impact on the ratio of the TOC values derived from the RBF-NN
method and OMI (2010–2013). (b) RMF impact on the TOC values derived from the
LUT method and OMI (2010–2013) (after Fan et al. [235] with permission).
Figure 42 Directional hemispherical reflectance of clean snow for 24- m-radius snow
grain from 1) ISIOP computed IOPs and ISBRDF, 2) Mie computed IOPs and DISORT
[236], and 3) ASTER spectral library observations [237] for 10 solar zenith angle for
the visible and near-infrared (a) and infrared (b) spectral regions (adapted from
Stamnes et al. [123]).
Figure 43 Spectral albedo of snow as a function of wavelength. (a) grain size 1000 m,
and impurity concentrations (in top down order) 0, 0.01, 0.1, 1, 5, 10 ppmw. (b) Pure
snow with grain size (in top down order) 50, 100, 200, 500, 1000, and 2000 m.
Figure 44 Comparison of the temporal variations (16-day averages from April 7 to
October 15, 2003) of GLI-derived snow-covered land area and sea-ice-covered area
with those derived from MODIS (land snow) and AMSR (sea ice). Images of the
extents of snow and sea ice cover for the period of April 7–22 from GLI and MODIS +
AMSR are also shown (after Hori et al. [253] with permission).
Figure 45 Sixteen-day average GLI snow surface temperature around the northern polar
region from April 7 to May 8, 2003 (after Hori et al. [253] with permission).
Figure 46 Scatter plot between snow surface temperatures from MODIS and GLI (after
Hori et al. [253] with permission).
Figure 47 Sixteen-day average of the GLI-derived snow grain size of the shallow layer
( ) around the northern polar region from April 7 to May 8, 2003 (after Hori et al.
[253] with permission).
Figure 48 Ratio of the snow grain radius of the top surface ( ) to that of the shallow
layer ( ) for the April 7–22 period. White colored areas are the same as in Figure
45 (after Hori et al. [253] with permission).
Figure 49 Map of snow metamorphism potential around the northern polar region for
the period of April 7–22, 2003, determined from the relation between snow surface
temperature ( ) and snow grain size ( ). Warm (orange) color denotes small grains
under high temperature, whereas cold (blue) color indicates coarse grains under low
temperature (after Hori et al. [253] with permission).
Figure 50 Spatial distribution of melt onset date around the northern polar region in
2003 determined from the relation between snow surface temperature ( ) and snow
grain size ( ). Date is indicated by Julian Day (after Hori et al. [253] with
permission).
Figure 51 Schematic illustration of the atmosphere and ocean with incident solar
irradiance and optical depth increasing downward from at the top of the
atmosphere. The incident polar angle is , which after refraction according to
Snell's law changes into the angle . Since the ocean has a larger refractive index
than the atmosphere, radiation distributed over sr in the atmosphere will be
confined to a cone less than sr in the ocean (region II). Upward radiation in the
ocean with directions in region I will undergo total internal reflection at the ocean–air
interface (adapted from Thomas and Stamnes [18]).
Figure 52 Comparison of irradiance results obtained with C-DISORT and a C-MC
code for RT in a coupled atmosphere-ocean system. The simulations are for an
atmosphere containing only molecular absorption and Rayleigh scattering and for an
ocean having no Rayleigh scattering, only absorption and scattering from a chlorophyll
concentration of 0.02 mg/m , uniformly distributed to a depth of 61 m, below which the
albedo is zero (adapted from Gjerstad et al. [27] with permission).
Figure 53 Schematic illustration of various contributions to the TOA radiance in the
case of a wind-roughened ocean surface. (1) Diffuse downward component reflected
from the ocean surface; (2) direct, ocean-surface reflected beam; (3) beam undergoing
multiple scattering after ocean-surface reflection, and (4) (multiply) scattered beam
reaching the TOA without hitting the ocean surface (adapted from Ottaviani et al. [286]
with permission.)
Figure 54 Sun-normalized sunglint TOA radiance (solid and thin curves) at 490 nm for
an SZA of 15 , along the principal plane of reflection, and relative error incurred by
ignoring multiple scattering along the path from the surface to the TOA (dotted
curves). Each plot contains three representative wind speeds (1, 5, and 10 m/s). The
upper row pertains to small aerosol particles in small amounts ( , left panel) and
larger amounts ( , right panel). The bottom row is similar to the top one, but for
large aerosol particles. The error curves have been thickened within the angular ranges
in which retrievals are attempted (corresponding to in normalized
radiance units) (adapted from Ottaviani et al. [285] with permission).
Figure 55 Retrieved values of four of the five parameters: aerosol optical depth,
bimodal fraction of aerosols, CDM absorption coefficient at 443 nm, and
backscattering coefficient at 443 nm (adapted from Stamnes et al. [292]).
Figure 56 Retrieved chlorophyll concentration for the same SeaWiFS image as in
Figure 55, showing distributions of the other for parameters and residuals (adapted
from Stamnes et al. [292]).
Figure 57 Left four panels: Comparison of reflected polarized radiation components
computed by C-VDISORT (dashed curves) with benchmark results [181] (solid curves)
for a homogeneous layer of nonabsorbing aerosol particles, simulated by putting half of
the aerosol particles (optical depth = 0.1631) in each slab, and setting the refractive
index to 1.0 in both slabs. From top to bottom: Stokes parameters , and the
degree of linear polarization. Right four panels: Same as left four panels except that the
comparison is between C-VDISORT (dashed curves) and C-PMC (solid curves). Red:
; green: ; blue: . Two-hundred discrete-ordinate streams,
photons (after Cohen et al. [84]).
Figure 58 Similar results as in Figure 57 but for the transmitted polarized radiation
components (after Cohen et al. [84]).
Figure 59 Left four panels: Comparisons of reflected polarized radiation components
computed by C-VDISORT (dashed curves) with benchmark results [181] (solid curves)
for a homogeneous layer of nonabsorbing cloud particles, simulated by setting the
refractive index equal to 1.0 in both slabs, and putting half of the cloud particles
(optical depth = 2.5) in each slab. From top to bottom: Stokes parameters , and
the degree of linear polarization. Right four panels: Same as the left four panels except
that the comparisons are between C-VDISORT (dashed curves) and C-PMC (solid
curves). Red: ; green: ; blue: . Two hundred discrete-ordinate
streams, photons (after Cohen et al. [84]).
Figure 60 Results similar to those in Figure 59, but for the transmitted polarized
radiation components. Two hundred discrete-ordinate streams, photons (after
Cohen et al. [84]).
Figure 61 Comparisons between polarized radiation components computed with C-
VDISORT (dashed curves) and C-PMC (solid curves) for the same physical situation
as in Figure 57, except that the refractive index was set equal to 1.0 in the upper slab
and equal to 1.338 in the lower slab. Upper four panels: just above interface. Lower
four panels: just below the interface. Red: ; green: ; blue: .
Sixty-four discrete-ordinate streams in upper slab, 96 streams in lower slab,
photons (after Cohen et al. [84]).
Figure 63 Upper four panels: Similar to Figure 61, but at TOA for nonabsorbing
aerosol particles with an optical depth 0.1631 in each slab. Lower four panels: Similar
to Figure 61, but at TOA for nonabsorbing aerosol particles with an optical depth of
0.3262 in the upper slab, and cloud particles with an optical depth of 5.0 and single-
scattering albedo of 0.9 in the lower slab (after Cohen et al. [84]).
Figure 62 Similar to Figure 61 (aerosol particles with an optical depth of 0.1631 in
each slab) except that we put nonabsorbing aerosol particles of optical depth 0.3262 in
the upper slab, and cloud-like particles with single-scattering albedo of 0.9 and optical
depth of 5.0 in the lower slab. 96 discrete-ordinate streams in upper slab. One-hundred
and twenty-eight streams in lower slab, photons (after Cohen et al. [84]).
Figure 64 Left: Schematic illustration of the two (coarse and fine) aerosol modes.
Middle: Ten different aerosol fractions: 1, 2, 5, 10, 20, 30, 50, 80, 95%, and
eight different relative humidities: RH 50, 70, 75, 80, 85, 90, 95%. Right:
“Continuum” of models obtained by interpolation between the discrete ones.
Figure 65 Comparison of RBF-NN forward model results for with C-VDISORT
results (after Stamnes et al. [293] with permission).