Statistics as a means to construct knowledge in climate and related sciences -- a discourse -- Hans von Storch Institute for Coastal Research GKSS, Germany Stats seminar U HH + IMPRS, 4. May 2005

The basic approach … … is to combine systematically empirical knowledge („data“) with dynamical knowledge („models“) in order to determine characteristic parameters (“inference”) consistency of models and data (“testing”)

The knowledge represented by data and models are both uncertain. This uncertainty makes us to resort to statistical concepts.

The resulting additional knowledge is best guesses of numbers (ideally together with confidence intervals) evaluation of the consistency of theoretical concepts with observational evidence. These new knowledge claims are based on the amount of available data. In general: If more data are available, the confidence in the numbers increases, but the consistency of the concepts decreases.

In general, the problem may be conceptualized by the state space formalism, with - a state space equation, e.g., Ψ t+1 = F(Ψ t, α, η) + ε(M) with the state variable Ψ t, external parameters η and internal parameters α. The term ε is a random component, which supposedly represents the uncertainty of the model M. - an observation equation x t = B(Ψ t ) + δ(B) with the observable x, and the random component δ.

Examples: 1.Best fit 2.Extreme value 3.PIPs and POPs 4.Downscaling 5.Detection and attribution 6.Determination of parameters 7.Analysis

What have we learned? Always the same question – namely:

1. Best fit

2. Extreme values Long memory? Bunde et al., 2004: Return intervals of rare events in records with long-term persistence. Physica A 342, Distribution P q (r) of return times between consecutive extreme values r. R q is the expected value. Synthetic example with k = annual water levels of the Nile

Significance: Extremes are not uniformly distributed in time, as described by a Poisson process, but appear in clusters. Expected waiting time for next exceedance event conditional upon length of previous waiting time r 0. Synthetic examples with k = annual water levels of the Nile Bunde et al., 2004: Return intervals of rare events in records with long-term persistence. Physica A 342,

3. PIPS … State space equation in low- dimensional subspace Observational equation in high- dimensional space. Parameters P, α determined such that … and POPs Special form Ψ, λ complex numbers; (M) describes the damped rotation in a 2-dimensional space spanned by complex eigenvectors of E(x t+1 x t T ) E(x t x t T ) -1. All eigenvectors form P T.

Example: POP of MJO Real and imaginary part of spatial pattern in equatorial velocity potential at 200 hPa 10-day forecast using state space equation in 2-d space von Storch, H. and J.S. Xu, 1990: Principal Oscillation Pattern Analysis of the Tropical 30- to 60- day Oscillation: Part I: Definition of an Index and its Prediction. - Climate Dyn. 4,

4. Downscaling The state space is simulated by ”reality” or by GCMs. The observation equation relates large- scale variables, which are supposedly well observed (analysed) or simulated, to variables with relevant impact for clients.

Example: snow drops Maak, K. and H. von Storch, 1997: Statistical downscaling of monthly mean air temperature to the beginning of the flowering of Galanthus nivalis L. in Northern Germany. - Intern. J. Biometeor. 41, 5-12 Large scale state: JFM mean temperature anomaly Flowering date anomaly of snow drop (galanthus nivalis)

5. Detection and attribution The state space dynamics is given by the assumption that the complete state of the atmosphere may be given by The “patterns” g k represent the influence of a series of external influences, while ε represents the internal variability of the climate system. Ψ describes the full 3-d dynamics of the climate system. The observation equation is formulated in a parameter space (A), and the state variable is projected on a space of observed variables (L[ψ] ) Here, L is the projector of the full space on the space of observed (and considered) variables, and g r,ad is the adjoint pattern of g k in the reduced space. Detection means to test the null hypothesis while attribution means the assessment that A k is consistent with a k. (i.e. A k lies in a suitable small confidence “interval” of a k )

Detection and attribution (cont’d) Attribution diagram for observed 50-year trends in JJA mean temperature. The ellipsoids enclose non- rejection regions for testing the null hypothesis that the 2- dimensional vector of signal amplitudes estimated from observations has the same distribution as the corresponding signal amplitudes estimated from the simulated trends in the greenhouse gas, greenhouse gas plus aerosol and solar forcing experiments. Courtesy G. Hegerl. Zwiers, F.W., 1999: The detection of climate change. In: H. von Storch and G. Flöser (Eds.): Anthropogenic Climate Change. Springer Verlag, , ISBN

6. Determination of parameters In general, when many observations are available, optimal parameters α may be determined by finding those α which minimize the functional

Example: Determination of parameters – oceanic dissipation M 2 tidal dissipation rates, estimated by combining Topex/Poseidon altimeter data with a hydrodynamical tide models. The solid lines encircle high dissipation areas in the deep ocean From Egbert and Ray [32] Egbert GD, Ray RD (2000) Significant dissipation of tidal energy in the deep ocean inferred from satellite altimeter data. Nature 45:

7. Analysis Skillful estimates of the unknown field Ψ t are obtained by integrating the state-space equations and the observation equation forward in time:

Example: spectral nudging in RCMs State space equation: RCM Observable x t : large-scale features, provided by analyses or GCM output. Correction step: nudging large-scales in spectral domain Percentile-percentile diagram of local wind at an ocean location as recorded by a local buoy and as simulated in a RCM constrained by lateral control only, and constrained by spectral nudging

The purpose of statistics is … to specify pre-defined „models“ of reality by fitting characteristic numbers to observational evidence.  developing and extending models and theories to analyze states and changes by interpreting empirical evidence in light of a pre-specified model.  monitoring weather (and climate) to test theories and models as to whether they are valid in light of the empirical evidence.  falsifying theories and models