## Sampling and time spacing

The sampling of a climate archive (Fig. 1.13) can influence the de-tectability of extreme events. Table 6.5 lists the notation for this section.

Consider the case that the spacing, d(i), is large compared with the sample duration, D(i), or its diffusion-extended form, D'(i), and also large compared with the persistence time, t. It may then be that the time series fails to record information about an extreme event, Xout(i). This would render the series useless for the purpose of risk analysis. Another case is a hiatus, where d(i*) » D(i*) and d(i*) » t only at a certain point, i*. For fitting an extreme value distribution, the log-likelihood function may be adapted (Coles 2004) to take the absent portion of information into account. For estimating the occurrence rate (Section 6.3.2), it is indicated to "exclude" the hiatus prior to the analysis, that means, to shift artificially the portion of the time series before or after the hiatus. The calculations (kernel estimation, cross-validation) are carried out on those time-transformed data and the hiatus "included" by inserting the time-gap in the results.

A further case is uneven spacing when hiatuses are absent. Block extremes detection for fitting a GEV distribution may then be enhanced by fixing the number of observations, k, per block rather than the length of a block.

This section focuses on still another, "ice core" case, where the sample duration is large and the age-depth relation is strongly nonlinear, leading to large changes in D(i). (In ice cores, which are influenced by compaction, D(i) can exhibit strong trends.) This poses a detection problem for extremes because with D(i) changes also the recording quality downcore (inhomogeneity). Note that the NGRIP sulfate record (Section 6.3.2.8) does not suffer strongly from inhomogeneities of this kind owing to a very high time resolution that allowed to have D(i) w const. We follow Mudelsee (1999) and study the physics of the recording system to derive a data transformation that corrects for the inhomogene-ity.

Suppose that the archive is an ice core with a segmented sampling and the measured variable is, for example, sulfate. The objective is to detect the extremes stemming from an event of short duration (e.g., volcanic

6.4 Sampling and time spacing Table 6.5. Notation for Section 6.4.

i = 1,... ,n |
Index, segment (top: i = 1) |

n |
Number of segments |

k |
Index, extreme events within segment i |

D(i) |
Duration, segment i |

Dout(i)k |
Duration, extreme event k within segment i |

Ntrend (i) |
Number of particles of interest (sulfate) from background, segment i |

Ntrend (i) |
Number of particles not of interest (non-sulfate) from background, |

segment i | |

Nout(i) |
Number of particles of interest (sulfate) from extreme events, seg- |

ment i | |

Nout(i)k |
Number of particles of interest (sulfate) from kth extreme event, |

segment i | |

NVout(i) |
Number of particles not of interest (non-sulfate) from extreme |

events, segment i | |

Fout(i)k |
Flux of particles of interest (sulfate) from kth extreme event, segment i |

-Ftrend (i) |
Flux of particles not of interest (non-sulfate) from background, seg- |

ment i | |

A(i) |
Exposure area to flux of particles of interest (sulfate), segment i |

A(i) |
Exposure area to flux of particles not of interest (non-sulfate), seg- |

ment i | |

X (i) |
Concentration of particles of interest (sulfate), segment i |

Xtrend (i) |
Concentration of particles of interest (sulfate) from background, seg- |

ment i | |

X '(i) |
Transformed concentration of particles of interest (sulfate), segment |

eruption) against the background trend. The sulfate concentration is

Assumption 1. The number of non-sulfate particles from the background dominates,

This is certainly fulfilled because the bulk of the material is water. Then we have

Ntrend(i)

where Xtrend(i) is the time-dependent background sulfate concentration, Xtrend(i) = Ntrend(i) + Ntrend(i) + Nout(i) + Nout(i) ' (6.46)

Consider first the case of a single event recorded in segment i. For a short duration, Dout(i)k=1 ^ D(i), there is input of material (sulfate and non-sulfate) from the event (eruption) added to the background. The flux of sulfate particles from the event is

A(i) ■ Dout(i)fc=i where Nout(i)k=1 is the number of such particles; here Nout(i)k=1 = Nout(i). The area A(i) represents the susceptibility of segment i exposed to the flux of incoming particles, mainly the perpendicular component. The flux of non-sulfate particles from the background is

X(i) X (i) ~ Fout(i)fc=1 ■ A(i) ■ Dout(i)fc=1 (6 ,Q)

X(i) - Xtrend(i) " W*) ■ A(i) ■ D(i) ^ (6.49)

Assumption 2. The susceptibility to incoming sulfate particles is proportional to the susceptibility to incoming non-sulfate particles,

The degree to which this is fulfilled depends on the site and the changes of the type of deposition (Wagenbach et al. 1996; Fischer 1997). For wet deposition, where the particles are scavenged by precipitation, the assumption should be well fulfilled because water and particles are transported to the exposed segment by the same carrier.

[X(i) - Xtrend(i)] D(i) <X Fout(i)k_=1 ' D"^1 . (6.51)

F'trend(i)

This formula for one single event is only approximate. The generalization to several extreme events per segment i is straightforward: approximately,

F'trend(i)

Assumption 3. The background trend is constant over the whole time interval.

This is certainly violated. However, there is confidence that the temporal variability of the numerator of the term on the right-hand side of Eq. (6.52) is clearly larger (event versus non-event) than the variability of the denominator. For example, Wagenbach (1989) reported Ftrend(i) to be in the interval from 0.2 to 0.4 (arbitrary units) for an Alpine ice core site.

Assumption 3 and Eq. (6.52) lead to the transformed values,

which should approximate the climatic signal of the events,

The transformation corrects for the dilution of the extreme values by the background values; the degree of the dilution depends on D(i).

For the practice of extreme value analysis on segmented ice core data (unevenly spaced timescale), a multi-stage approach is indicated. (1) Estimate the trend component (e.g., by using the running median). (2) Transform the data (Eq. 6.54). (3) Estimate trend and variability on the transformed data to select the threshold for detecting the POT data.

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