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Gridding Daily Climate Variables for use in
ENSEMBLES
Malcolm Haylock, Climatic Research Unit
Nynke Hofstra, Mark New, Phil Jones
Overview
• Applications → Scale
• Stochastic or Deterministic
• Methods
• Determining the best method
• Data preprocessing
• Uncertainty
Applications
• Daily P, Tmin, Tmax, SLP, Snow- Precipitation only for now
• Validation of RCMs- What is the true scale of RCMs? - Need to create gridded observations
that are area average
• Analysis of past changes
Stochastic or Deterministic• Stochastic
- obs(x) = z(x) + ε(x)- assume that observed station data are only one of
many possible “realisations” that could have occurred.
- Interpolate using inter-station covariance.• spatial and temporal
- generally don’t reproduce observations (inexact interpolation).
• Deterministic- obs(x) = z(x)- assume that observed station data are the only
possible realisation.- exact interpolation
Why Stochastic?
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 5 10 15 20 25 30 35 40
Distance
Semivariogram
• Variation at the local scale can not be determined using available station network
€
E z(x) − z(x ')[ ] /2
Variogram=E(z(x)-z(x1)](normalised)
Methods
• Kriging
• Thin plate splines
• (Reduced Space) Optimum Interpolation
• Angular Distance Weighting
• Conditional Interpolation
Kriging• Highly developed stochastic method used
extensively in the geosciences.
• obs(x) = z(x) + ε(x), z(x) is an autocorrelated random field calculated as a linear weighted average of surrounding stations.
• Weights determined by statistically modeling the regional variation by fitting an appropriate function to the variogram.
• Variations to handle anisotropy (spatial covariance dependent on orientation), large scale trends and other common problems.
• Statistical model may be different for each day.
Anisotropy
Thin Plate Splines• Stochastic method that fits a surface to the data
using smooth functions of the station separation distance
• Can be considered as a special case of Kriging with a particular class of covariance functions, however these functions are rarely used in Kriging.
• Contains a smoothing parameter which is usually set by cross validation.
• Implicit error estimation by cross validation.
Optimum Interpolation• Stochastic model developed for data assimilation
• Accounts for both spatial and temporal autocorrelation- unlike traditional Kriging and Splines which only
use spatial.- is temporal autocorrelation appropriate for precip.?
• Assumes Gaussian covariance error distribution - one of several models possible in Kriging.
• Reduced Space version uses EOFs to greatly speed calculation and limit dependence on small scale variation.- appropriate for daily precip?
Angular Distance Weighting
• Interpolation of anomalies
• Weight based on distance
and angle
• Stations closest to grid
points have greater weight
• Stations with biggest mean
angle have greater weight
• Elevation not included
• E.g. New et al. 2000, CRU dataset
j
k
l
θ
Grid point
Station
dist
Conditional Interpolation
• So far only interpolation of precipitation
• Interpolation is conditional on synoptic state
• Synoptic state defined with Self Organising Maps
• Interpolation in two steps- Wet or dry target location- If wet: interpolation of magnitude
• Weights regard distance, radial distribution and synoptic state
• Calculation of area mean
• Hewitson and Crane 2005
Selecting the best method(s)
• Cross validation- for all stations, remove the station then
calculate predicted value and evaluate appropriate error statistic (e.g. RMS).
- Assumes predicted value is a point value, but stochastic methods give the expected value and so hopefully the smallest average error.
• Can test models using a region with high station density by omitting stations and comparing with true are average.
Data Preprocessing
• Stochastic methods require Gaussian-distributed data
• Obtain consistency across region by interpolating anomaly from monthly mean (T, SLP) or % of monthly total (P).
• Interpolated results can be applied to previously gridded monthly data that utilise many more stations.
Rainfall Skewnessdaily/month
dry days removed
Uncertainty• Measurement error
• Homogeneity error
• Interpolation error- method• use many methods or best method
- statistical model within method• choose best model but still a
generalisation- station network• cross validation