Remote sensing data from satellite with high temporal resolution typically have lower spatial resolution, with one pixel often spanning over a square kilometer. The signal recorded by such satellite at a pixel is typically a mixture of reflectance from different types of land covers within the pixel, resulting in a mixed pixel. In this talk we introduce a couple of parametric and nonparametric statistical approaches to deal with the un-mixing problem which integrate information from multiple sources, and present some preliminary results applying the methodology to data from the SMOS (Soil Moisture and Ocean Salinity) mission and the OCO-2 (Orbiting Carbon Observatory 2) mission, which motivated this research.

1. Statistical Approaches for Un-Mixing Problem and
Application to Satellite Remote Sensing Data
Zhengyuan Zhu
Joint work with Remote Sensing Sptail Y working group
May 14, 2018
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2. Spatial Y Working Group Active Members
Jenny Brynjarsdottir (Case Western University)
Xinyue Chang (Iowa State University)
Maggie Johnson (SAMSI, NCSU)
Jon Hobbs (JPL)
Colin Lewis-Beck (Iowa State University)
Anirban Mondal (Case Western University)
Joon Jin Song (Baylor University)
Zhengyuan Zhu (Iowa State University)
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3. 1 Introduction: OCO-2 and SMOS data
2 Parametric un-mixing
3 Non-parametric un-mixing
4 Gap-filling Hyperspectral OCO-2 data
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4. Motivation: Improve the retrieval of OCO-2 Data
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latitude
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5. OCO-2 Data
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meanradiancefunction
pixel
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6. Spatial X vs Spatial Y
Spatial X: Model the retrieval variable (X) as spatial process to
improve the retrivel.
Spatial Y: Develop spatial-temporal functional model for the
hyper-spectral reflectance data (Y) to improve retrieval by
Gap-filling the missing data, i.e., impute Y and provide uncertainty of
the imputation.
Smoothing to reduce the noise in Y.
Un-mixing the pixel with mixed land cover (i.e., transition between
water and land).
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7. Focus on Un-mixing problem
What is un-mixing?
Disregarding atmospheric effects, the signal recorded by the remote
sensing instrument at a pixel is typically a mixture of light scattered by
substances located in the field of view.
Unmixing refers to any process that separates the pixel measurments
from an image into a collection of endmembers and a set of fractional
abundances.
Endmembers: Represent the pure materials present in the image
Abundances: Represent the percentage of each endmember that is
present in the pixel.
The group decided to focus on the un-mixing problem, which will be
the focus of this report.
At the end we also report some preliminary results on gap-filling for
OCO-2.
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8. Another (Simpler) Motivating Example: SMOS Data
2011 - 2017 on 30 footprints (45 km diameter) in northwest Iowa
Between 75 - 85 % of footprints devoted to annual crops
Satellite makes a pass approximately every 12 - 36 hours at the
midlatitudes
Approximately 60% of each footprint is planted in corn, and 40%
soybean
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9. SMOS Data
Pix 25 2015 Pix 25 2017
Pix 6 2015 Pix 6 2017
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Figure 1: Two representative SMOS pixels.
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10. Motivating Problem
Crop-specific peak is of interest in agriculture.
SMOS measurements are largely a mixture of corn and soybean signal
in Iowa.
Can we develop a model to un-mix SMOS satellite signal into two
crop types: corn and soybean?
Ground based auxiliary data available to inform the signature of each
crop type
Objective: Reconstruct crop-specific curves using the model and
quantify measures of uncertainty for each crop signal
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11. Auxiliary Data
Field level data based on experiments in Mead, Nebraska. Shape of corn
and soybean is based on growing degree days (GDD).
Figure 2: Hornbuckle et al. (2016)
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12. 1 Introduction: OCO-2 and SMOS data
2 Parametric un-mixing
3 Non-parametric un-mixing
4 Gap-filling Hyperspectral OCO-2 data
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13. Data Model: Parametric approach
We translate the SMOS time scale to GDD using ground based
temperature data and crop planting dates.
We model τ as a function of GDD from using an asymmetric Gaussian
(AG) function, a frequently used function to model nonlinear vegetation
data.
fAG
θ (d) =



β + (η − β)exp −(d−δ)2
2σ2
1
d ≤ δ
β + (η − β)exp −(d−δ)2
2σ2
2
d > δ
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14. Unmixing Model
The Cropland Data Layer from the NASS provides the proportion of corn
and soybean within each region. Let the proportion for the two dominate
crops by defined as: πc (corn), πb (soybean). We assume a common
intercept (γ) to capture the remaining ground vegetation that is constant
over time.
Let r = 1 . . . R denote pixel regions, s = 1 . . . S seasons, and d = 1 . . . D
denote GDD. For a given curve Xrs(d),
xrs(d) = γrs + πcrsfAG
θ1
(d) + πbrsfAG
θ2
(d) + rs(d) rs(d)
ind.
∼ N(0, σrs)
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15. Unmixing Model
Assuming the curves are exchangeable, and the signatures for corn and
soybean are independent, we an model the parameters hierarchically. Let
µj = (µηj , µδj
, µσ1j, µσ2j) denote the vector of means for the jth mixture
component. We will model these parameters as coming from a
multivariate normal distribution and work with the transformed vector
µθj
= (logit(µηj ), log(µδj
), log(µσ1j), log(µσ2j))
θ1rs
ind.
∼ N4(µθ1 , Σ1)
θ2rs
ind.
∼ N4(µθ2 , Σ2)
logit(γrs)
ind.
∼ N(µγ, τγ)
We put informative priors on the parameter vectors based on MLE
estimates of the AG parameters derived from the Nebraska field data.
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16. Preliminary Results
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Figure 3: Pointwise 95% CI bands (for the mean) for pixels 1, 5, 10, and 20 in
2015.
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17. Preliminary Results
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Figure 4: Median of mixture posterior (red), and 95% CI bands for 3 separate
crops: other (brown), corn (yellow), and soybean (green) for pixels 1, 5, 10, and
20 in 2015.
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18. 1 Introduction: OCO-2 and SMOS data
2 Parametric un-mixing
3 Non-parametric un-mixing
4 Gap-filling Hyperspectral OCO-2 data
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19. Data Model: Non-parametric approach
For a given curve Ys,k(t),
s = 1, · · · , S: locations, k = 1, · · · , K: years (replicates), t = 1, · · · , T,
and j = 1, · · · , J: mixing components
Ys,k(t) =
J
j=1
πs,k,j
L
l=1
βjlBl(t) +
J
j
s,k,j(t),
where s,k,j(t) ∼ N(0, σ2).
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20. Likelihood
Y SKT×1
=
(Y11(t1), · · · , Y1K(t1), · · · , Y11(tT ), · · · , Y1K(tT ), · · · , YSK(tT ))T
βJL×1
= (β11, · · · , β1L, β21, · · · , β2L, · · · , βJL)T
XSKT×JL
=


π111B1(t1) · · · π111BL(t1) · · · π11J B1(t1) · · · π11J BL(t
... · · ·
... · · ·
... · · ·
...
πSK1B1(tT ) · · · πSK1BL(tT ) · · · πSKJ B1(tT ) · · · πSKJ BL(t
L(β, σ2; y) ∝ (σ2)−SKT/2 exp{−
1
2σ2
(y − Xβ)T (y − Xβ)}
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21. Prior & Posterior
β ∼ N(0, τI)
σ2 ∼ IG(a, b)
p(β, σ2; y) ∝ L(β, σ2; y)p(β)p(σ2)
Bayesian regression problem
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22. SMOS Data
Convert Ys,k(t) to functional data with 8(=L) B-spline basis functions
πsjk are assumed known.
Corn + Soybean =85% and other vegetation=15%
60% corn and 40% soybean for the non ”other’ crops
πs,k,1 = 0.51(Corn)
πs,k,2 = 0.34(Soybean)
πs,k,3 = 0.15(Other) for all s and k.
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23. Prior Specifiction
There is a gold standard for two curves (corn & soy): Nebraska Data
Convert these data to functional with 8 B-spline basis functions
Compute mean and standard deviation of the estimated coefficients
βjl ∼ N(meanjl, 3 × SDjl)
Without informative prior, the last component (Other) is badly
behaved. Assume that this is a constant
πs,k,j
L
l=1
β3lBl(t) = β0
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24. Posterior Result
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doy
Tau corn
soy
others
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25. 1 Introduction: OCO-2 and SMOS data
2 Parametric un-mixing
3 Non-parametric un-mixing
4 Gap-filling Hyperspectral OCO-2 data
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26. Model for OCO-2 data
FPCA
r(wj, si) = f(wj, si) + ij
f(wj, si) = µ(wj) +
∞
k=1
ξk(si)φk(wj)
Assume spatial independence and local stationarity to estimate φk(wj). ij
is assumed to be uncorrelated with mean 0 and variance σ2(g(si)) where
g(si) is a function of location si. First 1, 2, 3 PCs explain 98%, 99.9%,
99.9996% of variation respectively.
Kriging
Model each principle component score ξk(s) (k = 1, 2, · · · , K), as a
stationary random field for spatial prediction.
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27. Measurement error in OCO-2
It turns out measurement error depends on the sub-track (1-8)
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28. OCO-2 Gap Filling Using Spatial Functional Kriging
Create artificial gap of up to 12 pixels to see how the methodology
performs on predicting radiance in the gap and compare the prediction
with the observed value. e1 =
1
m
m
j=1
(f(wj, s0) − r(wj, s0))2
r2(wj, s0)
e2 =
1
m
m
j=1
(f(wj, s0) − r(wj, s0))2
(r(wj, s0) − µ(wj))2
.
2 4 6 8 10 12
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number of gaps
e1
2 4 6 8 10 12
0.00.10.20.30.4
number of gaps
e2
It is pretty safe to fill the gap of up to 6 pixels, then the prediction error
starts to pick up.
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29. Land vs.Water
Radiance on pure land and pure water pixels are different and has linear
relationship:
0e+00 2e+19 4e+19 6e+19 8e+19 1e+20
0.0e+004.0e+198.0e+191.2e+20
land
water
For a mixed pixel si, can assume
r(wj, si) = αfL(wj, si) + (1 − α)fW (wj, si) + ij.
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30. Future Work
Complet the work of parametric approach for un-mixing of SMOS
data
Further develop the non-parametric approach for un-mixing of SMOS
data, address issues such as identifiability, introduce spatail
dependence, more efficient computation, etc.
Spatial-temporal modeling of Hyperspectral data: non-stationarity,
include information in time, Hyperspectral un-mixing, etc.
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