Reconstruction of Lamb weather type series back to the eighteenth century

Abstract

The Lamb weather type series is a subjective catalogue of daily atmospheric patterns and flow directions over the British Isles, covering the period 1861–1996. Based on synoptic maps, meteorologists have empirically classified surface pressure patterns over this area, which is a key area for the progression of Atlantic storm tracks towards Europe. We apply this classification to a set of daily pressure series from a few stations from western Europe, in order to reconstruct and to extend this daily weather type series back to 1781. We describe a statistical framework which provides, for each day, the weather types consistent enough with the observed pressure pattern, and their respective probability. Overall, this technique can correctly reconstruct almost 75% of the Lamb daily types, when simplified to the seven main weather types. The weather type series are described and compared to the original series for the winter season only. Since the low frequency variability of synoptic conditions is directly related to the North Atlantic Oscillation (NAO), we derive from the weather type series an NAO index for winter. An interesting feature is a larger multidecadal variability during the nineteenth century than during the twentieth century.

Appendix: details of the classification

• 1. Flow chart of the classification (example with the objective catalogue)

  

• 2. Calculation of the centroids We combine the daily series of pressure measured at the available stations (Table 2) with the daily WTs of the Lamb catalogue (‘subjective’ catalogue herein), or of the Jones et al. catalogue (‘objective’ catalogue herein), to calculate the average normalised pressure at each station and for each WT (so-called ‘centroids’, Fig. 2).

• 3. Explained variance (EV) The efficiency of the chosen centroids (seven main Lamb WTs) to statistically describe the pressure variability recorded at the stations is quantified with the standard metric of the explained variance (EV):

  $${\text{EV}}={\text{BSS}}/{\text{TSS}}=1 - {\text{WSS}}/{\text{TSS}},$$

  (1)

  with BSS the ‘between-types-sum-of-squares’, WSS the ‘within-types-sum-of-squares’ and TSS the ‘total-sum-of-squares’ (e.g., Beck and Philipp 2010). For these calculations, unclassified days are not accounted for (accounting for them slightly decreases EV by 1–2%). Note that EV depends on the number of WTs considered, hence the values of EV calculated with 7 types are not directly comparable to the ones obtained with 26 types, or with other classification techniques (Beck and Philipp 2010).

• 4. Distance between the daily pressure pattern and centroids For each day with pressure data at all stations (otherwise the day is considered as ‘unclassified’), we calculate the Mahalanobis distance D_i between the measured pressure values and the centroids of the corresponding month for the WT i as the sum over the s stations (in matrix form):

  $$D_{i}^{2}={\left( {{P_s} - {{\bar {P}}_i}} \right)^T} \cdot Cov_{i}^{{ - 1}} \cdot \left( {{P_s} - {{\bar {P}}_i}} \right),$$

  (2)

  where P_s is the (s × 1) vector of pressure values measured at the stations for each day, ${\bar {P}_i}$ is the (s × 1) vector of centroids (average values) for the weather type i, and Cov_i is the (s × s) covariance matrix between the pressure values measured at the stations calculated separately for each WT i (as defined in the catalogue).

• 5. Probability distribution of the distance We use this distance D_i as a metric of the probability distribution of distance to each centroid i. Since the pressure associated to each WT is approximately normally distributed, we assume that the square of the distance D_i² follows a chi-squared distribution (e.g., Wilks 2011), that is,

  $$f\left( {{{\text{D}}_{\text{i}}}^{{\text{2}}}/{\text{ WT }}=i} \right){\text{ }}=f{{\chi}}_{\text{s}}^{{\text{2}}},$$

  (3)

  where f(D_i²/WT = i) is the probability density of D_i² for the weather type i, and fχ_s² is the probability density of the chi-squared distribution with s degrees of freedom (the number of stations).

• 6. Classification consistency For each day we evaluate whether the observed pattern of pressure does correctly project onto one of the given seven sets of centroids, and so can be classified. For this, among the seven distances D_i between the observed pressure pattern and the seven WTs, we evaluate whether the shortest distance D_i is short enough to attribute this day to the corresponding WT i. The formal test is the following. The null hypothesis H₀ to reject is that, for each day, the minimum distance Di is too large to attribute this day to any WT. Note that by considering the minimum distance for each day, we do not make an a priori hypothesis on the matching WT. We accept a risk of wrongly rejecting H₀ of α = 1%. We have to compare the minimum value of the given day with the threshold value corresponding to the 1% of the empirical distribution of the minimum values for all days of the catalogue. For the objective catalogue over the 1861–1996 period, the value which leaves 1% of one-sided, right-tail, probability is 20.1, and is used as the threshold value to reject H₀. (Note that this empirical distribution follows quite closely a Chi square distribution with 6 degrees of freedom.)

• 7. Classification of each day If the above test is passed, that is, if there is at least one acceptable WT matching the observed pressure pattern of day j, then we want to calculate the probabilities of the different WTs to match this pressure pattern, P_j, that is, the conditional probabilities of WT i, given P_j : P(WT_j = i/P_j). If day j has to be classified with only one WT, we use the ‘maximum a posteriori rule’ to find the WT i which maximizes the probability of the WT i given the pressure pattern P_j.

• 8. Joint probabilities of WTs For each day j considered as reasonably close to the Lamb WTs, we would like to estimate the probability of each WT i to match the observed pattern of pressure P_j, that is, P(WT_j = i / P_j).

  Using Bayes formula and the law of total probability, we write that:

  $P\left( {W{T_j}=i/{P_j}} \right)=\frac{{f\left( {{P_j}/W{T_j}=i} \right).P\left( {W{T_j}=i} \right)}}{{f\left( {{P_j}} \right)}}$, and

  $$P\left( {W{T_j}=i/{P_j}} \right)=\frac{{f\left( {{P_j}/W{T_j}=i} \right).P\left( {W{T_j}=i} \right)}}{{\mathop \sum \nolimits_{k} f\left( {{P_j}/W{T_j}=k} \right).P\left( {W{T_j}=k} \right)}}$$

  with k describing all WTs.

  The probability of any WT i, P(WT_j = i), is taken as the frequency of this WT over the whole catalogue, F_i.

  Last, we assume that the squared distance of the pressure pattern to the centroids follows a Chi square distribution with s degrees of freedom (with s the number of stations), so that:

  $$f({{\text{P}}_{\text{j}}}/{\text{ W}}{{\text{T}}_{\text{j}}}=i)\,=\,f{{\chi}_s}^{2}({{\text{D}}_{\text{i}}}^{{\text{2}}}/{\text{ W}}{{\text{T}}_{\text{j}}}=i).$$

  Hence we calculate the probabilities P(WT_j = i/P_j) as:

  $$P\left( {W{T_j}=i/{P_j}} \right)=\frac{{{f_{\chi _{s}^{2}}}\left( {D_{i}^{2}/W{T_j}=i} \right).{F_i}}}{{\mathop \sum \nolimits_{k} {f_{\chi _{s}^{2}}}\left( {D_{k}^{2}/W{T_j}=k} \right).{F_k}}}.$$

  (4)

• 9. Skill score of the reconstruction The skill score can be calculated as (e.g., Nicholls 1980):

  $$S=\frac{{{\text{Correct}} - {\text{Climatology}}}}{{{\text{Total}} - {\text{Climatology}}}}$$

  or in the proportion form:

  $$S=\left( {\frac{{{\text{Correct}}}}{{{\text{Total}}}} - \frac{{{\text{Climatology}}}}{{{\text{Total}}}}} \right)/\left( {1 - \frac{{{\text{Climatology}}}}{{{\text{Total}}}}} \right),$$

  (5)

  where ‘Correct’ is the number of correctly classified days in our reconstruction with respect to the calibration catalogue, ‘Climatology’ is the number of correctly classified days in a random series of weather types based on their average frequencies, and ‘Total’ the total number of days compared. The skill score is positive if the technique of classification does a better job than classifying days based on their climatology. We estimate the probability that, for any day, the reconstructed weather type WT matches the weather type WTo of the calibration catalogue by chance as:

  $$P\left( {WT=WTo} \right)=\mathop \sum \limits_{{i=1}}^{{n+1}} P\left( {WT=i} \right) \cdot P\left( {WTo=i} \right),$$

  (6)

  where i is one of the n WTs (plus the unclassified type). We estimate the probability of any WT i, P(WT = i) or P(WTo = i), as its average frequency in the reconstructed series or in the calibration catalogue, respectively (Table 3). Since we also calculated the matching score by considering the most likely two WTs, WT₁ and WT₂, we write the combined probability by chance as:

  $$P\left( {W{T_1}=WTo\mathop \cup \nolimits^{} W{T_2}=WTo} \right)=\mathop \sum \limits_{{i=1}}^{{n+1}} \left( {P\left( {W{T_1}=i} \right) \cdot P\left( {WTo=i} \right)+P\left( {W{T_2}=i} \right) \cdot P\left( {WTo=i} \right)} \right)$$

  and because WT₁ and WT₂ are different, this reads:

  $$P\left( {W{T_1}=WTo\mathop \cup \nolimits^{} W{T_2}=WTo} \right)=2 \times \mathop \sum \limits_{{i=1}}^{{n+1}} P\left( {W{T_1}=i} \right) \cdot P\left( {WTo=i} \right)$$

  (7)

This ‘climatology’ probability is found to be 18% for one WT (i.e., P(WT = WTo)), and 36% for two WTs (i.e., P(WT₁ = WTo OR WT₂ = WTo)), throughout the different reconstructions.