Downscaling in Time. Andrew W. Robertson, IRI. Advanced Training Institute on Climate Variability and Food Security, 12 July 2002

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1 Downscaling in Time Andrew W. Robertson, IRI Advanced Training Institute on Climate Variability and Food Security, 12 July 2002

2 Preliminaries Crop yields are driven by daily weather variations! Current seasonal climate predictions (made with either GCM or empirical models) address only seasonal (3-month) averages Temporal downscaling: deriving weather from climate

3 Weather and climate We observe current weather Try to predict climate: a probabilistic description of weather How often is it likely to rain, when is the rainy season likely to begin, how long are dry spells likely to be? Weather: a particular daily sequence drawn from the population of weather sequences (climate) Probabilistic description is central because weather is unpredictable more than 2 weeks ahead

4 Kenya s Long Rains An example of differentiating weather and climate Northward seasonal migration of precipitation during March May Monthly mean satellite-derived precipitation

5 Daily precipitation variance Daily standard deviation exceeds the monthly mean Sub-monthly r.m.s. satellitederived precipitation

6 Mean atmospheric circulation Monsoonal evolution from dry northeasterlies to wet southeasterlies Monthly-mean wind vectors at 850 hpa

7 A probabilistic description of monsoon evolution Hidden Markov Model Rainfall occurrence depends only on the weather state P(R t S t ) R t 1 S t 1 R t R t+1 S t S t+1 daily rainfall occurrence time series: Markov chain of hidden rainfall states P(S t S t -1 )

8 Daily rainfall states in Kenya 3-state HMM trained on 29 seasons of daily rainfall at 7 stations 10 0 a) State 1 (830 days) b) State 2 (1083 days) c) State 3 (755 days) KEY: p=1 p= Rainfall occurrence probabilities P(R t S t )

9 Estimated state sequence S t 1 S t S t March April May Day in Season - dry state (#3, yellow) tends to occur in March - wet states (#1, green), (#2, blue) tend to occur in April May To get rainfall sequence: P(R t S t )

10 Atmospheric circulation state-composites Vectors: low-level winds Contours: mid-tropospheric descent

11 Pentad analysis of wet/dry spells Okoola (1999) - 700hPa wind vectors during March May Wet Pentads Wet minus Dry Year Dry Pentads

12 Summarizing over Kenya The climatological-mean smooth seasonal evolution is not realized, but rather Erratic switches between flow regimes characterizes monsoon evolution Temporal downscaling is concerned with predicting interannual variations in daily character of the rains and their northward progression

13 Temporal downscaling of Seasonal climate predictions Weather can t be predicted more than two weeks in advance (sensitivity to initial conditions), so we can t hope to be accurate on any particular day For this reason, the date of monsoon onset is highly unpredictable Aim to forecast changes in the probability distribution of daily weather sequences and to quantify their uncertainty Signal-to-noise ratio: Signal: Predictable change (shift?) in the distribution of weather Noise: unpredictable spread in the distribution of weather

14 Some statistics we need to get right 1. Precipitation occurrence Probability of rain Wet/dry spell length Spatial correlations between stations Log-odds ratio (odds of rain at one station vs. rain at another) 2. Precipitation amount Daily histogram

15 Daily Precipitation Occurrence Probabilities Seven Kenyan Daily Precip stations Occurrence during Probability March May March-May 0.5 Simulated by HMM Lodwar Observed

16 Wet/Dry Spell Durations 1 Wet/Dry Spell Durations at Lodwar March-May Probability{spell >=T} WET DRY Solid - Observed Dashed - HMM Spell Duration T (days)

17 Approaches to temporal downscaling 1. Historical analog techniques Use various subsets of past years based on a seasonalmean predictor(s) K-nearest neighbors 2. Stochastic weather generators Parameters estimated from seasonal (or monthly) GCM predictions 3. Statistical transformation of daily GCM output Usually includes a stochastic element Regression based Weather state models, e.g. HMM

18 Historical Analogs Simplest & most widely used approach Take daily sequences of weather observed during past events as possible scenarios for a predicted event An event can be defined according to the threshold of an index, such as Niño-3 SST, or a GCM-predicted seasonal-mean quantity (e.g. regional precip.)

19 Advantage: multivariate structure (both spatial and between-variables) is preserved Disadvantage: may not be many events in the historical record and every event is different (sampling problems)

20 K-Nearest Neighbors Refinement of the analog approach, retaining its advantages and partially solves the sampling problem Past years daily sequences D t are again selected from the historical record according to the value of some (seasonal-mean) predictor x * but here the past year t is resampled according to the distance x t - x *

21 So we select the k nearest neighbors of x * in the historical record, estimate appropriate weights to assign to each, and resample D t accordingly The resulting superensemble of years (each is repeated many times) can then be fed to a crop model

22 Approaches to temporal downscaling 1. Historical analog techniques Use various subsets of past years based on a seasonal-mean predictor(s) K-nearest neighbors 2. Stochastic weather generators Parameters estimated from seasonal (or monthly) GCM predictions 3. Statistical transformation of daily GCM output Usually includes a stochastic element Regression based Weather state models, e.g. HMM

23 Weather generators Use concept of Monte Carlo stochastic simulation Let computer generate a large number of daily sequences using a stochastic model Honor the statistical properties of the historical data of the same weather variables at the site Precipitation frequency and amount; dry-spell length; monsoon onset and end; monsoon break probabilities Daily max and min temperatures, solar radiation Cast seasonal prediction in terms of changes in these statistical properties

24 History Probabilistic modeling of precipitation predates numerical weather prediction, but seminal work in 1960s Markov chain model introduced by Gabriel and Neumann (1962) By 1980s, models extended to be able to treat a suite of variables and could be used in agric. apps. ( WGEN Richardson 1981, ) Large literature: many extensions to the basic model

25 Example of a Weather Generator PREC variable model parameters occurrence Markov chain (order=1) p11, p01 [monthly] (precipitation) amount Gamma distribution _, _ [monthly] SRAD (solar radiation) TMAX (max. temperature) TMIN (min. temperature) AR model: x*(t+1) = Ax*(t) + Be A,B: 3x3 matrices + 3 (wet/dry) (avg s/std's) x m( x) x* = s( x) where: m(x), s(x) depend on day of the year and precipitation occurrence Dubovsky et al. (2001) Need to consider the statistical dependence of the weather variables with each other on the same day, as well as their persistence Markov process used to model persistence Solar radiation and Tmax are likely to be lower on wet days, so precipitation is usually chosen as the driving variable with others conditioned on it Daily-total precipitation is often exactly zero, so most weather generators treat occurrence and amount separately

26 Precipitation occurrence fully defined by the two conditional probabilities ( transition probabilities ) p 01 = Pr{wet day t dry day t-1} (dry --> wet) p 11 = Pr{wet day t wet day t-1} (wet --> wet) p 10 = 1-p 11 p 00 = 1-p 01 (wet --> dry) (dry --> dry) These lag-1 autocorrelations are estimated from historical data

27 This simple model is able to capture persistence, and many statistics of interest can be derived from its transition probabilities: p Frequency of wet days: π = p 01 p 11 p 01 < π < p 11, so simulations yield sequences of wet and dry days that are more persistent than independent draws according to the climatological probability π. Wet- and dry-spell lengths follow a geometric distribution Pr{spell = T} = p(1 p) T 1, T =1,2,3,... The mean and variance of the no. of wet days contained within a string of T consecutive days can also be computed

28 Precipitation Amount Distribution of daily amount is strongly skewed to the right Exponential is simplest reasonable model. It requires only one parameter µ, yet reproduces the strong positive skewness Two-parameter gamma distribution is most popular choice (shape α and scale β) α=1 yields exp distrib, while the extra flexibility improves the fit

29 Simulation Draw a string of random numbers u [0,1]. If day t-1 was dry, then day t is simulated to be wet if u < p 01.. Wilks & Wilby (1999)

30 Multi-site extension Run a series of WG s in parallel Use spatially correlated random numbers Wilks (1998)

31 WGs for downscaling seasonal forecasts Use climatological parameters Then rescale the generated daily values such that their monthly means exactly match the monthly GCM prediction (Hansen) Additive offsets for temperatures & insolation parameters Multiplicative adjustment for precip., with repeated (stochastic) generation to match target

32 Realizations of daily weather in forecast seasonal climate IRI forecasts are in terms of tercile probabilities {p B, p N, p A } Want to make WG parameters depend on these probabilities

33 Forecast parameters can be estimated from the historical record (Briggs & Wilks 1996, Wilks 2002) by: 1. computing their values averaged over the years in each tercile of the local precip record {π (B ), π ( N), π (A ) } 2. and weighting according to the forecast {p B, p N, p A } π (p B, p N, p A, ) = p B π (B) + p N π (N ) +p A π (A ) Similar approach can be taken to condition WG parameters on ENSO phase (Woolhiser et al. 1993)

34 Approaches to temporal downscaling 1. Historical analog techniques Use various subsets of past years based on a seasonal-mean predictor(s) K-nearest neighbors 2. Stochastic weather generators Parameters estimated from seasonal (or monthly) GCM predictions 3. Statistical transformation of daily GCM output Usually includes a stochastic element Regression based Weather state models, e.g. HMM

35 Statistical transformation of daily GCM output (1) Regression based Make an empirical model by regressing daily station variables (T max, T min, precip.) against daily grid-scale atmospheric observations Use resulting empirical relationships in conjunction with GCM predictions perfect-prog. approach which needs to assume that the GCM perfectly simulates the grid-scale

36 E.g. from Wilby et al. (1999), wet day probability for a given day i is downscaled using 3 grid-box predictor variables: surface specific humidity (SH), SLP, 500hPa geopotential height (H), and lag-1 autocorrelation O i = α 0 + α Oi 1 + α SH SH i + α slp slp i + α H H i The α s are estimated using linear least squares regression For a given site and day, a wet day is returned if uniformly-distributed random number o i Similar approach used for precip amount (exp), Tmin, Tmax

37 (2) Weather state models Assume GCM can simulate the large-scale weather state ( perfect prog ) Weather state models attempt to relate local weather to large-scale atmospheric controls Advantage of increased physicality E.g., the non-homogeneous HMM

38 The HMM revisited A non-homogeneous Hidden Markov Model can link weather state-occurrence to atmospheric controls Rainfall occurrence is conditionally dependent on the weather state Transition probabilities modulated by X S t 1 R t 1 X t 1 R t R t+1 S t S t+1 X t X t+1 daily rainfall occurrence time series: Markov chain of hidden rainfall states Daily time series of atmospheric predictor

39 Downscaling with a NHMM A Markov chain of discrete atmospheric states (rather than precip occurrence as in the WG) Extra state layer is identified with physical atmospheric states Dynamical controls influence these states Then we simply take the predicted daily sequences of X from an ensemble of GCM predictions

40 HMM Downscaling over SW Australia Charles, Bates & Hughes (2000) NHMM fitted to 15 yrs ( ) May Oct. daily gauge precip & Reanalysis X X: area-average SLP, north-south SLP gradient, 850hPa dew-point temperature depression 10 yrs of GCM (~500km resolution) & limited area model (LAM) nested within it (125km res) Simulated X from GCM & LAM used to drive the NHMM to downscale precip occurrence at 30 stations

41 Six NHMM weather states Charles, Bates & Hughes (2000) Daily ppt occurrence probabilities & SLP averaged over all classified days in each state

42 Downscaled vs. Obs precip probabilities & log-odds ratio Charles, Bates & Hughes (2000)

43 Downscaled rainfall amounts vs Observations Charles, Bates & Hughes (2000)

44 Rank correlations between stations: Downscaled vs Obs Charles, Bates & Hughes (2000)

45 Issues for future work While many methodologies are well developed, the application to seasonal prediction is very recent These methods have never been systematically compared in the context of seasonal climate prediction! Many rely on empirical cross-scale relationships (perfect prog.), yet in the seasonal prediction context MOS-type corrections should be possible Need to test in conjunction with crop models

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