two GV sites of the TRMM , Houston, TX. (HSTN) and ... Data. Motivations. Mathematics of Rainfall Images. (GSM) Estimati
Motivations ○
Mathematics of Rainfall Images ○○○○
(GSM) Estimation and Fusion ○○○○
Sparse Downscaling and Retrieval ○○○○○○
Adaptive Fusion and Sparse Estimation of Multi-sensor Precipitation
Ardeshir Mohammad Ebtehaj 1,2 1
Efi Foufoula-Georgiou 1
Department of Civil Engineering 2 School of Mathematics
Summary ○
Motivations ○
Mathematics of Rainfall Images ○○○○
(GSM) Estimation and Fusion ○○○○
Sparse Downscaling and Retrieval ○○○○○○
Summary ○
Outline
❶ Motivations and Goals Multi-sensor Multi-scale estimation, i.e., Downscaling, Fusion and Retrieval of Precipitation
❷ Mathematical Structure of Precipitation Images Statistics of Precipitation in Transform Domain, Sparsity in the Wavelet Domain
❸ Non-linear Estimation and Multi-sensor Precipitation Fusion
(Bayesian Approach)
Gaussian Scale Mixture, Localized Estimation and Fusion in the Wavelet Domain ❹ Downscaling and Multi-sensor Retrieval via Sparse Representation Sparse Inverse Estimator, Downscaling (SPaD) and Retrieval
Motivations •
Mathematics of Rainfall Images ○○○○
(GSM) Estimation and Fusion ○○○○
Sparse Downscaling and Retrieval ○○○○○○
Summary ○
Motivations and Goals
o State-of-the-art mathematical development for downscaling, multi-sensor Fusion and retrieval?
Motivations •
Mathematics of Rainfall Images •○○○
(GSM) Estimation and Fusion ○○○○
Sparse Downscaling and Retrieval ○○○○○○
Summary ○
Rainfall in Transform Domains, Why?
o Transform Domain Representation 𝐱=
𝑛
𝐜𝑖 𝜙𝑖
𝑖=1
e.g.
𝜙𝑖 =
1 𝑁
𝑒 𝑖𝑘𝑛/𝑁
Compact
𝐱 = 𝜙1 |𝜙2 | … |𝜙𝑛 𝐜 = 𝚽𝐜
o Why Transform Domain?
General:
Efficiency (e.g. Diagonalizing convolution in the Fourier)
Lower dimension due to sparsity of transform coefficients (wavelet)
Decorrelation Capacity (wavelet and Fourier)
Particular: Multiscale representation Tractable a priori information (Sparsity!)
Motivations ○
Mathematics of Rainfall Images ••○○
(GSM) Estimation and Fusion ○○○○
Sparse Downscaling and Retrieval ○○○○○○
Summary ○
Data HSTN MELB
14 12 10 8 6 4 2 0
Dataset MELB_19980217_131700
a)
HSTN_19981113_000200
b)
200 independent reflectivity images over two GV sites of the TRMM , Houston, TX. (HSTN) and Melbourne, FL. (MELB)
HSTN_19981113_000200
c)
MELB_20040926_045000
Data structure
d)
• NEXRAD Level III base Reflectivity • 2A25 Orbital near-surface Reflectivity • A Wide range of Strom Types.
Motivations ○
Mathematics of Rainfall Images •••○
(GSM) Estimation and Fusion ○○○○
Sparse Downscaling and Retrieval ○○○○○○
Summary ○
Sparsity in the Wavelet Domain 4
5
a) a)
x 10 x 10 5
4
45 dBZ 45 dBZ
b) b)
4
(V)
(V)
(D)
c)
4
(D)
Dim: Dim: 2[450,600]; Range: [0,45][0,45] 2[450,600]; Range:
3
2
Frequency
(H)
Frequency
(H)
c)
1
0 -30
3
Top
40
40
35
35
30
30
25
25
20
20
15
15
10
10
5
5
0
0
2
20% 1
0 -20 -30
-10 -20
-100
wH
010
wH
20 10
30 20
30
Dim: Dim: [450,600]; Range: [0,45][0,45] [450,600]; Range:
Sparsity of precipitation images in the wavelet domain (a) Sparsity in the wavelet domain (b) Symmetric Distribution with thicker tail than the Gaussian case
(c) Reconstruction by only top 20% of the coefficients (99.8% of the energy is preserved)
Motivations ○
Mathematics of Rainfall Images ••••
(GSM) Estimation and Fusion ○○○○
Sparse Downscaling and Retrieval ○○○○○○
Summary ○
Decorrelation in the Wavelet Domain a)
Horizontal
b)
Vertical
c)
Diagonal
d)
e)
3
5
5
5
10
10
10
2
15
15
15
1.5
20
20
20
25
25
25
2.5
1 0.5 size: [544, 624]
size: [544,624]
5
10
15
20
25
5
10
15
20
25
Local covariance characterization Local covariance of the rainfall wavelet coefficients for a neighborhood of size 5x5 (diagonally dominant)!
Why wavelet for precipitation images? o A well behaved Symmetric Probability Model o Decorrelation capacity ( Karhunen Loéve-like expansion)
o Covariance matrices can be computed locally for neighborhood of coefficients o Only few coefficients explain the rainfall energy o Exploiting of linear estimation theory in an adaptive way !
5
10
15
20
25
Motivations •
Mathematics of Rainfall Images ••••
(GSM) Estimation and Fusion •○○○
Sparse Downscaling and Retrieval ○○○○○○
Summary ○
Gaussian Scale Mixture (GSM) a Probability Model
Generalized Gaussian Distribution GGD spans a symmetric probability continuum ranging from a Dirac delta (𝜶 → 𝟎) to the uniform density (𝜶 → ∞) .
𝑥 𝑓𝑥 𝑥, 𝜃 ∝ 𝐞𝐱𝐩 − 𝑠
𝛼
Problem! A Gaussian Scale Mixture probability model
0 GSM Simulation Fitted GG density
Gaussian Density
A large family of elliptically symmetric density functions with thicker tail than the Gaussian
-10
𝒅 =𝑑 𝑧𝒖 𝚺𝒅 = 𝔼 𝑧 𝚺 𝒖
log(p)
case (e.g., Laplace, Double exponential, etc.)
-5
𝚺𝒅|𝒛 = 𝑧𝚺𝒖
Log-normal Multiplier -15 -15
-10
-5
0 m=1
e.g., Andrews and Mallows (1974), Wainwright et al., (2001)
dH
5
10
15
Motivations ○
Mathematics of Rainfall Images ○○○○
(GSM) Estimation and Fusion ••○○
Sparse Downscaling and Retrieval ○○○○○○
Summary ○
GSM Optimal Estimation (theory)
min 𝐱 − 𝐲 𝐲
2 2
Spatial Domain x
𝐞 = 𝕍𝒂𝒓[𝐱|𝐲]
𝐱=
𝚺𝐱𝐲 𝐱, 𝐲 𝔼[𝐱𝐲] 𝐲= 𝐲 = 𝐲 𝐲, 𝐲 𝔼[𝐲 2 ] 𝚺𝐲𝟐
𝐲 = 𝐻𝐱 + 𝑣 𝐱 −𝒎𝐱 = 𝚺𝐱 𝐻T 𝐻𝚺𝐱 𝐻𝑇 + 𝚺𝑣 𝐻=𝐼 𝒎𝐱 = 𝒎𝐲 = 0
Wavelet Domain Wavelet coefficients of a neighborhood
𝐲=𝐝+𝑣
( 𝐲 −𝒎𝐲 )
𝐱 = 𝚺 𝐱 𝚺 𝐱 + 𝚺𝑣
−𝟏
𝐲
GSM Wiener
𝐝 = 𝑧𝚺𝐮 𝑧𝚺𝐮 + 𝚺𝑣
𝐝= 𝑧𝐮
−1
−𝟏 𝐲
Maximum A Posteriori (MAP) estimation of z 𝑧𝑀𝐴𝑃 = 𝐚𝐫𝐠 𝐦𝐚𝐱 log 𝒑 𝒚 𝑧 + log 𝒑(𝑧) 𝑧
log 𝑧𝑀𝐴𝑃 + 3𝜎𝑧2 2 1 + 𝑧𝑀𝐴𝑃 𝜎𝑧2 2
𝑁
𝑛=1
2 𝑧𝑀𝐴𝑃 − 𝜆−1 𝑛 (𝑣𝑛 − 1) =0 2 𝑧𝑀𝐴𝑃 + 𝜆−1 𝑛
where: 𝑆𝑆 𝑇 = Σ𝑣 Λ, Q = 𝐞𝐢𝐠 𝑆 −1 Σ𝒖 𝑆 −𝑇 𝑣𝑛 ∈ V = 𝑄 𝑇 𝑆 −1 𝒚 Portilla et al., 2003, Ebtehaj A.M., and E. Foufoula-Georgiou 2011b
Motivations ○
Mathematics of Rainfall Images ○○○○
(GSM) Estimation and Fusion •••○
Sparse Downscaling and Retrieval ○○○○○○
GSM Optimal Estimation (1D-Results)
ID implementation of the GSM Wiener (a) A heavy tail noisy signal (similar to rainfall images in marginal statistics) (b) Standard Gaussian Filtering results (c) GSM Wiener filter
Summary ○
Motivations ○
Mathematics of Rainfall Images ○○○○
(GSM) Estimation and Fusion ••••
Sparse Downscaling and Retrieval ○○○○○○
Summary ○
GSM Framework for Fusion of Precipitation Images
Filtering and Fusion in the wavelet domain
Inverse WT
Lateral Projection Filtered low-pass
Filtered high-pass
TRMM-NEX Error Covaraince 𝚺𝐯𝟑
Filtered high-pass SRE Product [1 1] km
a)
Fused Products Size:[224, 784]; Range:[0, 49] dBZ GSM-Fusion [1 1] km
c)
TRMM Error Covariance 𝚺𝐯𝟏
NEXRAD Error Covariance 𝚺𝐯𝟐
Size:[224, 784]; Range:[0, 50] dBZ 50
Vertical Filtering rain-gauge error
Reflectivity [dBZ]
c)
Reduced Uncertainty!
GSM -Fusion SRE
40
NEXRAD TRM M
30 20 10 0
0
100
200 300 400 E. Foufoula-Georgiou 500 600 7002011b800 Ebtehaj A.M., and
Motivations •
Mathematics of Rainfall Images ••••
(GSM) Estimation and Fusion ••••
Sparse Downscaling and Retrieval •○○○○○
Summary ○
Sparse Approximation and Compressive Sensing
Problem
1
𝐲 = A𝐱 + 𝐞
0.5
where,
𝐲 ∈ ℝ𝑛 (n=120 !) 𝐱 ∈ ℝ𝑚 (m=512) 𝐀 ∈ ℝ𝑛×𝑚 𝐀 : Gaussian Orthogonal Ensemble
x
0 -0.5
-1 0
50
100
150
200
250
300
350
400
450
500
node
L2 Recovery
x
2
0.5
𝐦𝐢𝐧 𝐱 0
-0.5 0
50
100
150
200
250
300
350
400
450
𝟐
s. t.
2