EUMETSAT

1. EUMETSAT Monitoring weather and climate from space Anne O’Carroll Three-way statistics SST from polar orbiters, CMS, Lannion, 5-6 March 2013

2. Outline • Introduction and theoretical basis • Representativity error • Application and examples • Future applications and issues

3. Overview • The collocation of three independent sets of observations enables the standard deviation of error to be derived for each observation type. • Method used initially for wind at KNMI/OSI-SAF (e.g. Stoffelen, 1998); now used for waves at ECMWF (e.g. Janssen et al, 2008); SST (e.g. O’Carroll et al, 2008); and sea-ice drift (e.g. Hwang and Lavergne et al, 2010); etc.... Drifting buoys AATSR AMSR-E

4. Theoretical basis Expression for the error in observation x: xi = xT + bi + εi [1] So for 3 sets of collocated observations of types i=1,2,3: x1 = xT + b1 + ε1 [2] x2 = xT + b2 + ε2 x3 = xT + b3 + ε3 Consider the three sets of observations as three sets of pairs: xi – xj = bi – bj + εi - εj [3]

5. Derivation For an ensemble of observations the mean difference is: bij = x̄i – x̄j = bi – bj [4] As the variance between i and j is expressed as: Vij = (xij – bij)^2 (insert [3] xi – xj = bi – bj+ εi – εjand [4]) Vij = ((x̄i – x̄j + εi – εj) – bij)^2 (insert [4]) Vij = ((x̄i – x̄j + εi – εj) – x̄i – x̄j )^2 (as bij = x̄i – x̄j = bi – bj = b̄i – b̄j + εi – εj) Then: Vij = ((x̄i – x̄j + εi – εj) – (b̄i – b̄j + εi – εj))^2 Vij = ((x̄i – x̄j) – (b̄i – b̄j))^2 (insert [3] barred) Vij = ε̄i^2 – ε̄j^2 - 2 ε̄iε̄j [5]

6. Consideration of covariances Covariance ij = rij σi σj where r is the correlation coefficient between the two random variables i and j So if the standard error in i σi is equal to the mean random error in i ε̄i then: Covariance ij = rij σi σj = σij = ε̄ij =ε̄iε̄j [6] Inserting [6] into [5] and as σi =ε̄i then: Vij = σi ^2 –σj^2 - 2rij σi σj [7]

7. Assumptions • Covariances between the observations are uncorrelated, i.e. Covij = rij σi σj = 0 • Therefore the correlation coefficient rij = 0 • We are assuming that the observations are independent and therefore it is fair to say that the uncertainties between the observations are uncorrelated. • Covariances of uncertainties of representativity on the space-time scale are negligible. • Test that assumptions are valid through variation of collocation criteria and spatial and temporal scales.

8. Uncertainty on each observation type Given assumptions, equation 7 becomes: Vij = σi ^2 –σj^2 [8] Therefore enabling σ1, σ2 and σ3to be derived for three sets of collocated observations. Rearranging simultaneous equations from [7] gives: [9] σ1^2 = ½(V12 + V31 – V23) + r12 σ1 σ2 + r31 σ3 σ1 – r23 σ2 σ3 σ2^2 = ½(V12 + V31 – V23) + r12 σ1 σ2 + r31 σ3 σ1 – r23 σ2 σ3 σ3^2 = ½(V12 + V31 – V23) + r12 σ1 σ2 + r31 σ3 σ1 – r23 σ2 σ3

9. Representativity error Given three independent data sources, there can still be representativity errors based on the spatial matchup and temporal matchup process. • What matchup criteria are needed in order to minimise the representativity uncertainties? • Time difference +- 1 hour? • Buoy location within satellite pixel? • What is the effect on the representativity uncertainty when these criteria are relaxed?

10. Estimation of representativity uncertainty Example: • Collocated drifting buoy (1), IASI (2), and AVHRR (3) observations • Buoy +- 2hours of IASI/AVHRR observations • Buoy in central quarter of IASI IFOV; buoy within AVHRR pixel -> Assume temporal criteria small enough for negligible error contribution (night-only). Will test assumption by varying time in other examples. -> Assume negligible representativity uncertainty between AVHRR and IASI (co-located in space and time), therefore r23 term is zero Rearranging [9]:- σ1 – 2r12 σ1 σ2 – 2r31 σ1 σ2 = ½(V12 + V31 - V23) [10] σ2 – 2r12 σ1 σ2 – 2r31 σ1 σ2 = ½(V23 + V12 - V31) σ3 – 2r12 σ1 σ2 – 2r31 σ1 σ2 = ½(V31 + V23 – V12)

11. Check difference in collocation time (July to November 2012) Two experiments, with criteria as before plus AVHRR confidence >=3 and time difference between buoy and IASI/AVHRR is: • +-2 hours • +-1 hour So effect of tightening the time constraint: ∆σ(IASI) = +0.017K, ∆σ (AVHRR) = -0.008K, ∆σ (buoy) = -0.008K -> Small differences but will proceed with +-1 hour time-constraint

12. Check difference in spatial homogeneity of IASI IFOV (July to November 2012) Two experiments, with criteria as before plus AVHRR confidence >=3 and +-1 hour (3-sigma stats): • Without AVHRR standard deviation test. • Standard deviation of AVHRR SSTs in 21x21 box < 0.4 to 0.2K. So effect of reducing homogeneity of scene: ∆σ(IASI) = -0.017K, ∆σ (AVHRR) = -0.019K, ∆σ (buoy) = -0.019K -> Representativity errors due to spatial variations approaching 0.02K -> Uncertainties from both r12 and r31

13. Mean differences versus standard deviation in box • Night-time, +- 1hour • IASI/AVHRR standard deviations decrease under 0.3K • More fluctuations of buoy standard deviations than IASI/AVHRR (<0.5K). • Anomalous buoy observation contributing to higher stdev and bias at 0.1K standard deviation.

14. Mean differences versus box cover • More fluctuation in buoy comparisons, and higher standard deviations for buoy comparisons in clearer boxes. • All biases decrease for clearer boxes.

15. 10-degree standard deviation of errors Lowest errors in mid Oceans for all observation types. Some cases of low IASI errors where high AVHRR errors and vice versa.

16. 3-way analyses (Apr 2010 to May 2011) • Monthly global IASI errors (all QL) 0.22K to 0.42K • Monthly errors reduce follow PPF5 upgrade in September 2010 • Overall global errors: IASI = 0.27K AVHRR = 0.14K drifting buoy = 0.20K

17. Practical issues and future applications • Need to ensure large enough sample to do analysis (time-series, global maps). • Needs to be careful consideration of which observations to combine in terms of temporal and spatial collocations. • Multi-matchup datasets provide invaluable resources for investigating the global uncertainties of each observation type. • Datasets with 4+ observation types will provide even more scope for investigating and ensuring representativity errors are negligible. • Would be interesting to investigate more different spatial lengths of collocations e.g. Different buoy matchup criteria, transects using radiometers...