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Data assimilation to improve retrievals of
land surface parameters
Tristan Quaife (University of Reading, UK)
December 2012
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MODIS:
Moderate Resolution Imaging Spectrometer
• Two in orbit (onboard
Terra and Aqua)
• Since 2000
• Daily global coverage
• 250m 1km resolution
• Various spectral channels
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Some definitions - BRF
• Bidirectional reflectance factor
– The ratio of the radiance reflected into a finite solid
angle and the radiance reflected into the same angle
(under the same illumination conditions) from a perfect
Lambertian reflector
– This is what is reported as “reflectance” from passive
optical sensors (MODIS, Landsat, LISS etc) which
observe the top of atmosphere radiance
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Some definitions - BRDF
• Bidirectional reflectance distribution function
– ratio of incremental radiance, dLe, leaving surface
through an infinitesimal solid angle in direction (v, v),
to incremental irradiance, dEi, from illumination direction
’(i, i)
1
)(
),()',( sr
dE
dLBRDF
i
e
Ω'
Ω'ΩΩΩ
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BRDF example
Modelled barley reflectance, v from –50o to
0o (left to right, top to bottom).
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BRDF example
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MODIS BRDF sampling
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Some definitions – spectral albedo
• Spectral albedo
– The integral of the BRDF, at one wavelength, over the
viewing hemisphere with respect to the illumination
conditions. Two commonly used forms:
2
,1
ΩΩΩ ;Ω dBRDF2DHR =
222
,1
2;2 ΩΩΩΩΩΩ ddBRDFdBHR =
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Some definitions – albedo
• Albedo (or “broadband” albedo)
– The spectral albedo integrated over the desired
wavelength range:
– p(λ) is the proportion of incoming radiation at the given
wavelength and α(λ) is the spectral albedo
dp
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From data to information...
• The key to understanding the remote sensing
signal is to understand how the target modifies the
EMR distribution
• To begin with, assume the canopy is filled with
randomly positioned ‘plate’ absorbers (i.e.
leaves)...
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0
Depth
, z
1-D plane-parallel semi-infinite turbid medium
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0
Depth
, z Ω δm
I0
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0
Depth
, z
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1-D Scalar Radiative Transfer Equation
• for a plane parallel medium (air) embedded with a
low density of small scatterers
• change in specific Intensity (Radiance) I(z,) at
depth z in direction with respect to z:
• Requires either numerical techniques or fairly
broad assumptions to solve
( )( )zJzI
z
zIse ,),(
,W+W-=
¶
W¶km
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A [relatively] simple semi-discrete solution Gobron, N., B. Pinty, M. M. Verstraete, and Y. Govaerts (1997), A semidiscrete model for
the scattering of light by vegetation, J. Geophys. Res., 102(D8), 9431–9446,
doi:10.1029/96JD04013.
Single scattered
soil reflectance
Single scattered
canopy reflectance
Multiply scattered
reflectance (requires numerical solution)
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Clumping
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Shadowing
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Geometric optics
Shaded
crown Illuminated
crown
Illuminated
soil
Shaded
soil
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Operational models
• Probably the minimum model required to be
general enough to work for most vegetation types
requires some element of radiative transfer and
geometric optics
• However such models are typically very difficult
and/or time consuming to invert
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Kernel driven BRDF model
ISOTROPIC VOLUMETRIC GEOMETRIC
+ +
surface BRDF =
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f = kernel weight K = kernel value n = number of kernels
λ = wavelength ρ = BRF Ω = view geometry Ω' = illumination geometry
Kernel driven BRDF model
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Dominant scattering map
(near infra red):
Red = Geo
Green = Iso
Blue = Vol
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K is known for any given geometry, so if f can be determined this enables: •Calculation of integral terms (e.g. albedo) •Normalisation of data to common geometries •Higher level data processing (e.g. burn scars)
Kernel driven BRDF model
ρ=Kf y=Hx
H is known for any given geometry, so if y can be determined this enables: •Calculation of integral terms (e.g. albedo) •Normalisation of data to common geometries •Higher level data processing (e.g. burn scars)
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Kernel driven BRDF model - albedo
• Where a is a vector of (pre-computed) integrals of
the BRDF kernels
• N.B. α is a spectral albedo in this case
α=ax
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Formulation of H
H =
Kiso(Ω1; Ωʹ1) Kvol(Ω1; Ωʹ1) Kgeo(Ω1; Ωʹ1)
Kiso(Ω2; Ωʹ2) Kvol(Ω2; Ωʹ2) Kgeo(Ω2; Ωʹ2)
Kiso(Ω3; Ωʹ3) Kvol(Ω3; Ωʹ3) Kgeo(Ω3; Ωʹ3)
Kiso(Ω4; Ωʹ4) Kvol(Ω4; Ωʹ4) Kgeo(Ω4; Ωʹ4)
Kiso(Ω5; Ωʹ5) Kvol(Ω5; Ωʹ5) Kgeo(Ω5; Ωʹ5)
⁞ ⁞ ⁞
Kiso(Ωn; Ωʹn) Kvol(Ωn; Ωʹn) Kgeo(Ωn; Ωʹn)
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Formulation of y
y =
ρ(Ω1; Ωʹ1; λ)
ρ(Ω2; Ωʹ2; λ)
ρ(Ω3; Ωʹ3; λ)
ρ(Ω4; Ωʹ4; λ)
ρ(Ω5; Ωʹ5; λ)
⁞
ρ(Ωn; Ωʹn; λ)
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Formulation of x
x =
fiso(λ)
fvol(λ)
fgeo(λ)
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x = (HTR-1H)-1HTR-1y
• Formulation used for the MODIS BRDF/albedo product (MCD43)
• Requires an 16 day window
Least squares (over determined case)
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Observation errors
R = rI = r 0 0 …
0 r 0 …
0 0 r …
⁞ ⁞ ⁞
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MODIS band 2: NIR
MODIS data product (MOD43)
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MODIS data product (MOD43)
MODIS band 2: NIR
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MODIS data product (MOD43)
Backup inversion algorithm used
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MODIS data product (MOD43)
No inversion!
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Potential improvements
• Eliminate use of the backup algorithm
• Produce daily estimates of kernel weights
– to improve timing of events
– best estimates where no data
• Reduce noise in parameters
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Data assimilation
• A key weakness of the NASA algorithm is that the
inversions are not constrained in anyway by the
preceding inversion results
• This is a clear opportunity to apply data
assimilation techniques…
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Kalman filter
• Additional requirements:
– Dynamic model (M)
– Model covariance (P) and forcing (Q)
– Estimate of initial state
xa = x + K(y - Hx)
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Forsimplicity…
M = I = 1 0 0
0 1 0
0 0 1
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Forsimplicity…
P = pI = p 0 0
0 p 0
0 0 p
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Peculiarities of the system 1
• Each observation is a linear combination of all
state vector elements, and…
• …the dynamic model is not really dynamic
• Consequently small numbers of observations in
the analysis window will produce odd state
estimates (which still fit the observations well)
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Peculiarities of the system 2
• The values of the state elements are unitless and
not readily interpreted in terms of things we can
observe on the ground
– Sometimes qualitative analysis used
• Consequently validation of the state vector values
is more-or-less impossible and we have to resort
to comparing y and Hxa
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Example – reflectance data
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Example – retrieved kernel weights
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Example – predictions of BRF & albedo
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Example – predicted vs. observed BRF
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Example – retrieved kernel weights
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Example – predictions of BRF & albedo
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Example – predictions of BRF & albedo
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Example – predicted vs. observed BRF
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Inthepracticalyouwill…
• Experiment with changing:
– P, Q, R
– The size of the analysis window
– The spectral channel & sensor
• Also there are data from two regions:
– Spain (nice, cloud free data)
– Siberia (sparse, cloudy data)
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Quaife and Lewis (2010) Temporal constraints on linear BRDF model parameters. IEEE TGRS
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Quaife and Lewis (2010) Temporal constraints on linear BRDF model parameters. IEEE TGRS
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EOLDAS
• European Space Agency Project to improve data
retrievals and inter-sensor calibration
• Weak constraint variational DA scheme using the
following cost function:
http://www.eoldas.info/
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EOLDAS examples
Lewis P, Gomez-Dans J, Kaminski T, Settle J, Quaife T, Gobron N,
Styles J & Berger M (2012), An Earth Observation Land Data
Assimilation System (EOLDAS), Remote Sensing of Environment.
Leaf Area
Index
Chlorophyll
Time
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