Importancia y evaluación de los miembros superiores

through a linear regression using the HadCRUT3 data. Simple standard meth- ods determine the relationship between the two series over a calibration pe- riod during which they are both available. The climate variable of interest is then deduced from this relationship over the whole reconstruction period (e.g. Juckes et al., 2007). According to the method employed, different ways to carry out this operation can be considered (Jones et al., 2009). In this study, three different approaches of calibration have been used and tested. First, the variance of the proxy record is matched to the one of the instrumental record. We have also considered two different linear regression models, the direct and indirect regression, differing on the choice of the variable considered indepen- dent and the variable considered dependent. These different approaches are explained in the following subsections.

Matching variance

The calibration is performed here as it was done, for instance, by Mann et al. (2008) in the CPS (composite-plus-scale) approach (see also Mann and Jones, 2003; Moberg et al., 2005; D’Arrigo et al., 2006). This scaling approach consists in adjusting the mean and variance of the proxy to the instrumental temper- ature data over the calibration period (here the entire common time period between proxy and instrumental data). Each proxy record is first standard- ized by removing the long-term mean and by dividing it by the standard de- viation. The proxy series is then centered so that the mean of the time series is equal to the mean of the target instrumental series, over the defined pe- riod of overlap, and finally multiplied to have the same standard deviation as the nearest available proxy data. This technique, referred to as “inflating“ in statistical downscaling, is not meaningful according to von Storch (1999),

5.2. Method and data 89

because it does not guarantee that the climatic signal have the right variance, since all local variability is not related to large-scale variability.

Direct regression

The direct regression is another CPS method (e.g. Briffa and Osborn, 2002). In this approach, we are interested in predicting the temperature T from the proxy time series P, and choose the temperature T as the dependent variable and the proxy data P as the independent variable. The corresponding equa- tion is

T=αP+β

where the equation coefficients α and β are determined by least square fit. This approach is known to significantly underestimate the amplitude of the vari- ability and of the trends (e.g. Christiansen, 2011). Indeed, the reconstructed amplitudes are scaled by the correlation between T and P: α=σP,T/σP2, where σP,Tdenotes the covariance and σP2the variance. Then α=ρσT/σP, where ρ is

the correlation between T and P (ρ<1, mostly on the order of 0.2 to 0.6 (Table 5.1)). Then, the variance of the estimated temperature T∗is σ_T2∗ =ρ2σ_T2 < σ_T2

(von Storch et al., 2004).

Indirect regression

Here, the calibration of each proxy against local temperature is done by re- gressing the proxy on the instrumental data and then by inverting the regres- sion slope to predict the local temperature at each proxy location (Moberg, 2012). This is done, for instance, in the so-called LOC (local) method (Chris- tiansen, 2011; Christiansen and Ljungqvist, 2011). The proxy data P is then the dependent variable and the temperature T is the independent variable. The indirect regression equation is

P=γT+δ

where the equation coefficients γ and δ are determined by least square fit. The equation relating a proxy P to the corresponding local temperature T is then

T= (P−δ)/γ

The disadvantage of this method is that, preserving the low-frequency vari- ability, it results in an exaggerated high-frequency variability (Moberg, 2012).

5.2.4

Experimental design

The model LOVECLIM, described in Sections 2.2 and 3.2, is constrained by the proxy-based reconstructions calibrated as explained in Section 5.2.3 us- ing a particle filter. The data assimilation method is applied as in Chapter 4. Please refer to Section 4.3.1 for the details about this technique. The version of the model used here, LOVECLIM1.3, slightly differs from the version used in the 3 previous chapters, the main improvement is in the way the perturba- tions within the ensemble are applied in order to avoid a bias present in the previous approach. The perturbed variable is now the air surface temperature instead of the atmospheric streamfunction, as tests using this methodology have shown that it assumes a more symmetric distribution of the ensemble around its mean compared to the previous one. The experiments starts in 850 AD, the initial conditions being derived as for the simulations presented in Chapter 3. Three different simulations are conducted, differing in the way the assimilated reconstructions are calibrated: matching variance, direct re- gression and indirect regression. They are named VAR, DIR and IND, respec- tively. As the indirect regression calibration method provides a more noisy signal than the direct regression method, a larger proxy error is chosen in this case. This choice is made arbitrarily here. The value of the error is 2.0◦C for IND, compared to 0.5◦C for VAR and DIR. The scaling factor for the error of representativeness is fixed to 1.5. The choice of a different proxy error or scal- ing factor would lead to a different number of particles kept at each analysis step.

Acronym Data assimilation method Proxy series assimilated

NOD None, Ensemble mean None

ONE None, Single simulation None

SPA Particle filter Spatial reconstruction (Mann et al., 2009) OLD Simplified particle filter (Mann et al., 2008) + new set of proxies,

variance matching calibration VAR Particle filter (Mann et al., 2008) + new set of proxies,

variance matching calibration DIR Particle filter (Mann et al., 2008) + new set of proxies,

direct regression calibration IND Particle filter (Mann et al., 2008) + new set of proxies,

indirect regression calibration