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High-resolution rainfall field re-construction based upon Kriging and local singularity analysis
Li-Pen Wang1,2 and Christian Onof1
1 Imperial College London, UK 2 Katholieke Universiteit Leuven, Belgium
Hydrofractals '13, Kos Island (Greece), 17th-19th October 2013
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Contents
1. Introduction
2. Local singularity analysis
3. Bayesian radar-raingauge data merging and its integration with local singularity analysis
– Methodologies
– Preliminary result
4. Conclusions & on-going research
• Appendix: the impact of local singularity analysis on radar-raingauge error (or bias) field construction
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1. INTRODUCTION
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Why we need to adjust radar rainfall data?
0
5
10
15
20
25
00:05 01:05 02:05 03:05 04:05 05:05 06:05 07:05
Rai
n D
ep
th (
mm
)
Time (5 min)
Cumulative Rain Depth (23/08/2010 event)@Beal RG
Beal_RG Radar 1km
Beal HS raingauge rainfall depth accumulations: 23/08/2010 event
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Sensors commonly used for estimation of rainfall at catchment scales
RAINGAUGE RADAR
Accuracy
Coverage, spatial characterisation of rainfall field
Raingauge Weather Radar
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2. LOCAL SINGULARITY ANALYSIS
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Local singularity analysis decomposes a geo-value into a singular and a non-singular components
1 1
Mass density
Non-singularity component:The Background magnitude that does not change as scale varies
The “singularity” component, of which the value varies at different scales according to local singularity exponent, termed α(x)
1
2
3
ε1
ε2
ε3
ρ1
ρ3
ρ2
α = 2, no singularity exists
α > 2, local depletion
α < 2, local enrichment
α ≠ 2, singularity exists
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As compared to the original radar (RD) field, the Non-Singular (NS) one is smoother and more symmetric
20110526 1525: Original RD 20110526 1525: Non-Singular RD
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A A’
E E’
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The degree of “smoothing” is in particular strong at the locations where more local extreme magnitudes are seen
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The multifractal spectrum of a Non-Singular field is narrower and the α values are less diverse (concentrating around 2, i.e. non singular)
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3. BAYESIAN DATA MERGING AND ITS INTEGRATION WITH LOCAL SINGULARITY ANALYSIS
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Principle of radar-raingauge data merging technique
interpolation comparison
error (or bias) field construction/fitting
adjustment
output
a) b)
c) d)
e) f)
g)
RG data Radar data
(Todini, 2001; Ehret et al., 2008)
Block-Kriging Interpolation:
- RG field yG
- estimation error covariance CεG
Kalman Filter: - Construct the error field ε and
its covariance Cε - Derive the “Kalman Gain” based
upon CεG and Cε
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d)
Block-Kriging interpolation
Singularity extraction
BK rain gauge field Non-Singular (NS) radar field
Local singularity
(α) field
Error field fitting
Comparison (error field construction)
e)
f)
g)
Adjustment
Singularity recovery
Reconstructed field
h)
Kalman Filter
Integration of local singularity analysis
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PRELIMINARY RESULTS
Reconstruction of a 2009 summer storm crossing Central London area
• This storm led to flooding in North-West London
• The water company of the area wants to reconstruct this storm in order to improve the design of the sewer system (they are interested in appropriately estimating the return period of the storm)
• Original radar QPEs underestimate rainfall depths: when inputting the radar QPEs into the hydraulic model of the area, no flooding is observed.
• The Bayesian merging led to smoothening of the convective cells initially observed in the radar images (although the radar estimates were inaccurate, the shape of the convective cells was properly captured by it)
• Local Singularity Analysis was applied with the aim of better preserving the intense precipitation areas during the Bayesian merging
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Deployment of rain gauges, backgrounded by radar rainfall accumulations over the event period
56
48
40
32
24
16
8
4
0
Rainfall Depth (mm)
B A
168 000
172 000
176 000
180 000
184 000
188 000
192 000
196 000
516 000 520 000 524 000 528 000 532 000 536 000 540 000 544 000
No
rth
ing
(m)
Easting (m)
MIDAS (1-hour) LGfL Nearby (30 min) LGfL SURR (30 min)
Point A Point B (Maida Value tube stn) EA RGs (15 min)
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Images at each step of the Bayesian data merging with/without local singularity analysis
Non-singular Radar
Non-singular Merged
Nimrod (Original) Block-Kriged RGs Bayesian Merged
Reconstructed (Singularity-sensitive Merged)
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Nimrod (Original) Block-Kriged RGs Merged
Quantile-quantile plots at each step of the Bayesian data merging with/without local singularity analysis
Non-singular Radar
Non-singular Merged Reconstructed
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Merged radar rainfall estimates with local singularity analysis are visually more realistic and show better temporal continuity
16:55 GMT 17:00 GMT 17:05 GMT
Bayesian Merging
Reconstructed: Local Singularity + Bayesian Merging
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Comparison of the merged radar rainfall accumulations and rates against independent EA raingauge records
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Conclusions & on-going research
• Local singularity analysis enables the decomposition of a geo-datum into 2 components: a local singularity exponent and a non-singular value, where the latter is of better normality than the original geo-datum. This facilitates the merging process since the existing merging techniques are developed mostly based upon the 1st and/or 2nd (statistical) moment approximation.
• It can be observed that local singularity analysis enables a visually more realistic and less smooth merged rainfall field; this is because the proposed methodology can recover the valuable singularity information that were smoothed off in the conventional merging process.
• In our case study, both the original and the “singularity-sensitive” Bayesian data merging techniques were found to effectively reduce the cumulative radar rainfall bias (as compared to the raingauge records), but the latter can better capture the local peaks in instantaneous rainfall rate profiles.
• The proposed methodology is now being used to re-construct a number of storm events observed in Edinburgh (UK) during the Summer of 2011 and for which high density raingauge rainfall and sewer flow data are available.
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THANK YOU FOR YOUR ATTENTION Li-Pen Wang: [email protected]
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Appendix
The impact of local singularity analysis on radar-raingauge error (or bias) field construction
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Obtaining merged estimates using Kalman filter
yMerged = yRD + K (yG – yRD)
K = Cε (Cε + CεG)-1
Kalman Gain is the function of the covariances of estimation errors (uncertainty) of BK interpolation (yG) and RD-BK bias (yG – yRD). If Cε >> CεG, the BK-RG estimates (yG) are trusted more than RD estimates (yRD); If Cε << CεG, the RD estimates (yRD) are trusted more than BK-RG estimates (yG).
The magnitude of Cε and its estimation will largely affect the quality of data merging!
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Radar data at 4 different time steps are used to assess the impact of singularity extraction on covariance of radar-raingauge errors (bias)
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Magnitude of the covariance of BK-RD errors can be largely decreased in “extreme” cases.
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Coefficient of Variation (CV) of variogram/covariance estimation
Range
Sill
h
γ (h)
Mean (μ) of the semi-variances at this distance
Standard deviation (σ) of the semi-variances at this distance
Coefficient of Variation (CV) = σ / μ
C(h) = C(0) – γ(h)
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The estimation uncertainty of variogram estimation of radar-raingauge errors can be reduced in “extreme” cases.