Benjamin Lang$\ast$ Institut für Geographie, Universität Augsburg

Version from June 22, 2015

$\ast$benjamin.lang@geo.uni-augsburg.de

### 1 Introduction

Teleconnections like the North Atlantic Oscillation (NAO) denote recurring and persistent, large-scale patterns of pressure and circulation anomalies that spans vast geographical areas. Since they have a strong inﬂuence on the weather activity of aﬀected regions, their predictability on inter-annual to decadal time scales is of great interest. The VADYtele plug-in evaluates the predictability of indices of well-known teleconnections, like the North Atlantic Oscillation (NAO), the Paciﬁc North American Pattern (PNA) or the Southern Oscillation Index (SOI).

In section 2, the methods concerning the calculation of the teleconnection indices and the applied skill scores are described. Sections 3 and 4 explain the input and the output, respectively, of the VADYtele plug-in.

### 2 Methods

#### 2.1 Teleconnections

Several teleconnection indices from both the Northern Hemisphere (e. g. NAO, PNA) and the Southern Hemisphere (e. g. SOI, SAM) can be analysed concerning their predictability on inter-annual to decadal time scales (cf. Table 1). They include diﬀerent calculation methods – zonal means, grid point values or (Rotated) Principal Component Analysis ((R)PCA) – but all of them are based on monthly / seasonally aggregated anomaly ﬁelds of sea level pressure (psl) or geopotential heights (zg).

Please note the indicated references for detailed information about the calculation methods of the particular indices.

 Index (abbr.) Index Calculation Reference EA East Atlantic Pattern (after CPC) RPCA Barnston and Livezey [1987] EAWR East Atlantic / Western Russia Pattern (after CPC) RPCA Barnston and Livezey [1987] EPNP East Paciﬁc / North Paciﬁc Pattern (after CPC) RPCA Barnston and Livezey [1987] NAO-CPC North Atlantic Oscillation (after CPC) RPCA Barnston and Livezey [1987] NAO-HUR North Atlantic Oscillation (after Hurrell) PCA Hurrell [1995] PNA Paciﬁc / North American Pattern (after CPC) RPCA Barnston and Livezey [1987] POL Polar / Eurasia Pattern (after CPC) RPCA Barnston and Livezey [1987] SAM Southern Annular Mode Zonal means Nan and Li [2003] SCAND Scandinavia Pattern (after CPC) RPCA Barnston and Livezey [1987] SOI Southern Oscillation Index Grid point values Trenberth [1984] TNH Tropical / Northern Hemisphere Pattern (after CPC) RPCA Barnston and Livezey [1987] WP West Paciﬁc Pattern (after CPC) RPCA Barnston and Livezey [1987]
Table 1: Overview of the available teleconnection indices.

#### 2.2 Skill scores

##### 2.2.1 Correlation (CORR)

The Pearson product-moment correlation coeﬃcient is calculated between the observation and the forecast ($COR{R}_{fc,\tau }$) or the reference ($COR{R}_{ref,\tau }$) for a speciﬁc lead time $\tau$.

Figure 1 shows selected results for CORR.

##### 2.2.2 Mean Squared Error Skill Score (MSESS)

The Mean Squared Error Skill Score (MSESS) is based on the Mean Squared Error (MSE), which is calculated for the diﬀerent lead years $\tau$ of the forecast ($MS{E}_{fc,\tau }$ ; shown in (1)) and the reference ($MS{E}_{ref,\tau }$):

 $MS{E}_{fc,\tau }=\frac{1}{I}\sum _{i=1}^{I}{\left({F}_{\tau ,i}-{O}_{t\left(\tau ,i\right)}\right)}^{2}$ (1)

${F}_{\tau ,i}$ is the mean forecast derived from the ensemble for initialisation $i$ for a speciﬁc forecast lead time $\tau$ and ${O}_{t\left(\tau ,i\right)}$ the observation for time $t\left(\tau ,i\right)$, corresponding to the time of initialisation $i$ and forecast lead time $\tau$.

Finally, MSESS is calculated [Goddard et al.2013] by relating forecast to reference:

 $MSES{S}_{\tau }=1-\frac{MS{E}_{fc,\tau }}{MS{E}_{ref,\tau }}$ (2)

Figure 2 shows selected results for MSESS.

##### 2.2.3 Ranked Probability Skill Score (RPSS)

The Ranked Probability Skill Score (RPSS) is based on the Ranked Probability Score (RPS), which is calculated for the diﬀerent lead years $\tau$ of the forecast ($RP{S}_{fc,\tau }$ ; shown in 3) and the reference ($RP{S}_{ref,\tau }$):

 $RP{S}_{fc,\tau }=\frac{1}{I}\sum _{i=1}^{I}\sum _{k=1}^{K}{\left({F}_{\tau ,i,k}-{O}_{t\left(\tau ,i\right),k}\right)}^{2}$ (3)

${F}_{\tau ,i,k}$ is the cumulative probability derived from the ensemble for initialisation $i$ within class $k$ (with three equiprobable classes (below normal, normal, above normal), i.e. $K=3$) for a speciﬁc forecast lead time $\tau$, that is the fraction of ensemble members forecasting the occurrence of class $k$ or lower. ${O}_{t\left(\tau ,i\right),k}$ is the cumulative probability of class $k$ from observations for the time $t\left(\tau ,i\right)$, corresponding to the time of initialisation $i$ and forecast lead time $\tau$ and eﬀectively the Heaviside step function with ${O}_{t\left(\tau ,i\right),k}=0$ if a value within a class higher than $k$ is observed or else ${O}_{t\left(\tau ,i\right),k}=1$ [Kruschke et al.2014].

Since it is biased for ﬁnite ensemble size $m$, the VADYtele plug-in calculates the RPS with hypothetical ensemble size $M$ as estimation of $RP{S}_{fc,\tau ,m}$ [Ferro et al.2008]:

 $RP{S}_{fc,\tau ,M}=RP{S}_{fc,\tau ,m}-\frac{M-{m}_{i}}{M\left({m}_{i}-1\right)I}\sum _{i=1}^{I}\sum _{k=1}^{K}{F}_{\tau ,i,k}\left(1-{F}_{\tau ,i,k}\right)$ (4)

Therefore, the comparison of forecasts and references with diﬀerent ensemble sizes is possible and RPSS can be calculated as:

 $RPS{S}_{\tau }=1-\frac{RP{S}_{fc,\tau }}{RP{S}_{ref,\tau }}$ (5)

Figure 3 shows selected results for RPSS.

### 3 Input

This sections describes the various options for the plug-in. For this purpose, table 2 lists and explains all possible options. You have to choose i.a. the FORECAST (“1”), REFERENCE (“2”) and OBSERVATION of your choice (Please note: REFERENCE and OBSERVATION are remapped to the resolution of the FORECAST).

 Output Output directory mandatory default: /scratch/\$user/evaluation_system/output/vadytele/ Index Teleconnection index for evaluation mandatory Project1 FORECAST project, e.g. ”baseline0”, ”baseline1”, ”prototype”. mandatory Product1 FORECAST product, e.g. ”output2”, ”output1”. mandatory Institute1 FORECAST institute of experiment, e.g. ”mpi-m”. mandatory Model1 FORECAST model of experiment, e.g. ”mpi-esm-lr”. mandatory Experiment1 Preﬁx for FORECAST experiment, e.g. ”decs4e” or ”decadal”. mandatory Ensemblemembers1 FORECAST ensemble members as list, e.g. ”r1i1p1,r2i1p1,r3i1p1” mandatory Project2 REFERENCE project, e.g. ”baseline0”, ”baseline1”, ”prototype” mandatory Product2 REFERENCE product, e.g. ”output2”, ”output1”. mandatory Institute2 REFERENCE institute of experiment, e.g. ”mpi-m”. mandatory Model2 REFERENCE model of experiment, e.g. ”mpi-esm-lr”. mandatory Experiment2 Preﬁx for REFERENCE experiment, e.g. ”decs4e” or ”decadal”. mandatory Ensemblemembers2 REFERENCE ensemble members as list, e.g. ”r1i1p1,r2i1p1,r3i1p1” mandatory Season Season, e.g. ”MAM”, ”JJA”, or a comma separated list with the month(s) of interest, i.e. ”2,4,6,7” mandatory Observation Observation, e.g. ”eraint” or ”merra” mandatory Aggregation Level of aggregation: ”monthly” or ”seasonal” mandatory Integrative Set TRUE for integrative calculation (with all lead years) of the selected index (default=False, i.e. index is calculated separately for all lead years. mandatory

### 4 Output

The processed ﬁles can be found in the selected Output folder for every lead year $\tau$. It contains i. a. plots of the teleconnections patterns (ﬁgure 4), the teleconnection index (ﬁgure 5) and the rank histogram (ﬁgure 6).

The output also encompasses raw data ﬁles of the teleconnection index and the skill scores (CORR, MSESS, RPSS; given as netCDF and ASCII ﬁles, respectively), which can be used for further analysis (cf. ﬁgure 7).

### References

A.G. Barnston and R.E. Livezey. Classiﬁcation, Seasonality and Persistence of Low-Frequency Atmospheric Circulation Patterns. Mon. Weather Rev., 115: 1083–1126, 1987. doi: 10.1175/1520-0493(1987)115.

C.A.T. Ferro, D.S. Richardson, and A.P. Weigel. On the eﬀect of ensemble size on the discrete and continuous ranked probability scores. Meteorol. Appl., 15:19–24, 2008. doi: 10.1002/met.45.

L. Goddard, A. Kumar, A. Solomon, D. Smith, G. Boer, P. Gonzalez, V. Kharin, W. Merryﬁeld, C. Deser, S. J. Mason, B. P. Kirtman, R. Msadek, R. Sutton, E. Hawkins, T. Fricker, G. Hegerl, C. A. T. Ferro, D. B. Stephenson, G. A. Meehl, T. Stockdale, R. Burgman, A. M. Greene, Y. Kushnir, M. Newman, J. Carton, I. Fukumori, and T. Delworth. A veriﬁcation framework for interannual-to-decadal predictions experiments. Clim. Dyn., 40:245–272, 2013. doi: 10.1007/s00382-012-1481-2.

J.W. Hurrell. Decadal Trends in the North Atlantic Oscillation: Regional Temperatures and Precipitation. Science, 269:676–679, 1995. doi: 10.1126/science.269.5224.676.

T. Kruschke, H.W. Rust, C. Kadow, G.C. Leckebusch, and U. Ulbrich. Evaluating decadal predictions of northern hemispheric cyclone frequencies. Tellus A, 66, 22830, 2014. doi: 10.3402/tellusa.v66.22830.

S. Nan and J. Li. The relationship between the summer precipitation in the Yangtze River valley and the boreal spring Southern Hemisphere annular mode. Geophys. Res. Lett., 30:2266–2269, 2003. doi: 10.1029/2003GL018381.

K.E. Trenberth. Signal Versus Noise in the Southern Oscillation. Mon. Weather Rev., 112:326–332, 1984. doi: 10.1175/1520-0493(1984)112.