Rapid update cycling with delayed observations

T. J. Payne

doi:10.1080/16000870.2017.1409061

Rapid update cycling with delayed observations

Tellus

Volume 69 (2017): Issue 1

By: T. J. Payne

Open Access

|Jan 2017

Figures & Tables

Delay in receiving various observation types in the Met Office observation processing system, for observations valid between 9Z on 15 June 2015 and 3Z on 18 June 2015.

Cycling with non-overlapping windows as used at the Met Office. Observations in the dark blue (dark green) region are assimilated for an ‘early cut-off’ run at 9Z (15Z). The extra observations in the lightly shaded regions are assimilated for ‘late cut-off’ runs at 12Z and 18Z.

Rapid update cycling with overlapping windows. Observations are assimilated within one hour of receipt.

Table 1.

Optimal filtering with overlapping windows in linear-Gaussian case.

Table 2.

Number of observations valid at times $k - 6, \dots, k - 1$ , available for analyses performed at $k, k + 2, k + 4$ .

No of obs available with validityAnalysisLagtime:Total no.performed at(hours)k-6k-5k-4k-3k-2k-1of obs availablek088765438k+2288887645k+4488888848

Table 3.

Validity time of the most recent analysis available at times $k, \dots, k + 5$ for lags of 0, 2 and 4 hours.

Validity time of most recentLaganalysis available at:(hours)kk+1k+2k+3k+4k+50k-1k-1k-1k-1k-1k-12k-7k-7k-1k-1k-1k-14k-7k-7k-7k-7k-1k-1

RMS forecast error from most recent available analysis at times 0–5 hours into the next window following an assimilation window $[k - 6, k)$ , for lags 0 (black), 2 (blue) and 4 (green). Dashed lines denote RMS error at times *before* the analysis is performed. Also shown in red is the RMS error in the optimal RUC analysis using the data available at each time. In this example $σ_{q} = 0.182$ .

Table 4.

Summary of methods 0,1,2,3.

Method0123Number of prior stateswhich are analysis statesNN10from previous cycleFormulation of

{\underset{̲}{x}}_{k}^{b}

(37)(37)(36)(35)Formulation of

{\underset{̲}{x}}_{k}^{a}

(16)(33), (34)(33), (34)(33), (34)

Table 5.

The background states for the analysis at $t = 3$ given the analysed states available from $t = 2$ , illustrated with $N = 2$ for Methods 0,1,2,3. After the analysis at $t = 2$ we have analyses $x_{0}^{a}$ , $x_{1}^{a}$ and $x_{2}^{a}$ . The backgrounds $x_{1}^{b}$ , $x_{2}^{b}$ and $x_{3}^{b}$ at $t = 3$ are formed from these analyses as shown in the table.

Time0123Analyses at

t = 2

x_{0}^{a}

x_{1}^{a}

x_{2}^{a}

Backgrounds at

t = 3

x_{1}^{b}

x_{2}^{b}

x_{3}^{b}

Methods 0 & 1

x_{1}^{a}

x_{2}^{a}

M_{2}^{3} x_{2}^{a}

Method 2

x_{1}^{a}

M_{1}^{2} x_{1}^{a}

M_{1}^{3} x_{1}^{a}

Method 3

M_{0}^{1} x_{0}^{a}

M_{0}^{2} x_{0}^{a}

M_{0}^{3} x_{0}^{a}

Table 6.

Values of ${\underset{̲}{M}}_{k}^{(ℓ)}$ , for $N = 2$ , following Table 5.

Method

ℓ = 1

Method

ℓ = 2

Method

ℓ = 3

{\underset{̲}{M}}_{k}^{(ℓ)}

\begin{matrix} 0 & I & 0 \\ 0 & 0 & I \\ 0 & 0 & M_{k}^{k + 1} \end{matrix})

\begin{matrix} 0 & I & 0 \\ 0 & M_{k - 1}^{k} & 0 \\ 0 & M_{k - 1}^{k + 1} & 0 \end{matrix})

\begin{matrix} M_{k - 2}^{k - 1} & 0 & 0 \\ M_{k - 2}^{k} & 0 & 0 \\ M_{k - 2}^{k + 1} & 0 & 0 \end{matrix})

Table 7.

${\underset{̲}{x}}_{j + N}^{a}$ , ${\underset{̲}{B}}_{N + j}^{- 1}$ and ${\underset{̲}{A}}_{N + j}^{- 1}$ for different methods in limit $Q \to \infty$ , for $j \geq N$ .