## 1. Introduction

Extreme indices have recently been used by a greater part of the climatological community to assess the impacts of extreme events on our society (e.g., Klein Tank et al. 2009). Computing extreme indices requires reliable daily data. Thus the development of suitable techniques to homogenize daily data is necessary.

Homogenization of temperatures at a daily time scale is much more difficult than at monthly or annual scales. This is not due to the detection of shifts, since this information may be provided by the analysis of annual or monthly series. Thus this is mainly an adjustment problem. When considering annual or monthly data, the effect of the changes affecting the series can be assumed to be a bias that may vary according to the season. These biases are very easy to estimate and remove using linear techniques (e.g., Caussinus and Mestre 2004). This is no longer the case when daily temperature data are processed, however, where adjustments should vary according to the meteorological situation of each day. Differences in shelter radiative properties may dramatically influence observations, as shown in shelter intercomparison experiments (Lefèvre 1998). For example, on average, the difference between a standard French BMO 1050 shelter and a CIMEL Electronique shelter, which was provided to nonprofessional observers, is approximately +0.5°C, but for individual days this difference may rise up to 1.8°C. This occurs especially during hot sunny days with little wind, where the natural ventilation of this small shelter fails to compensate for radiative heating. A recent intercomparison study of nine widely used screens also shows increasing absolute temperature differences with decreasing cloud cover and wind speed (Brandsma and Van der Meulen 2008).

For temperature adjustment, multiple regression models—including other parameters such as wind speed and direction, sunshine duration, and parallel measurements—are the best way to proceed, as was achieved for the De Bilt series (Brandsma et al. 2002; Brandsma 2004). The Netherlands Meteorological Institute (KNMI) has kept all original instruments as well as complete metadata and photographic archives of the environment of earlier site positions. Using this unique material, Brandsma et al. (2002) were able to carefully plan parallel measurement experiments, not only for temperature measurements, but also for wind speed and sunshine duration. The conditions in which the De Bilt series was homogenized are unique, however. Wind speed or sunshine duration data are extremely rare when considering older data, for which usually only precipitation and temperature were observed. Furthermore, metadata simply do not exist in many cases. Reproducing the old measurement conditions (Brandsma et al. 2002; Brandsma 2004; Brunet et al. 2004, 2007) is a way to correct the series, but this approach is expensive, is time consuming, and requires waiting a long time to get a sufficient archive after the experiment has started. For these reasons, some authors have limited themselves to assessing homogeneity using graphical analysis of time series of annual indices derived from daily data to suppress inhomogeneous stations from any further analyses (Peterson et al. 2002; Aguilar et al. 2005).

If there is a need for daily data adjustment, the simplest adjustment method relies on interpolation of monthly adjustment coefficients (Vincent et al. 2002; denoted as the Vincent method in the following), a procedure also applied by Moberg et al. (2002) and Brunet et al. (2006) to obtain a better performance in the calculation of extreme indices based on daily temperature. This method provides adjustments only for the mean of an inhomogeneity and not for its higher-order moments. Note that in Brunet et al. (2006) data are “prehomogenized” by means of transfer functions obtained through a shelter intercomparison experiment before applying Vincent’s method.

Other methods characterize the changes of the entire distribution function using overlapping data between observing systems. Trewin and Trevitt (1996) use overlapping observations between temperature observing systems (e.g., when there is a change in shelter type or location) to build a transfer function between the probability density function (PDF) of the old and new measurement systems. Their method was used to homogenize Australian daily temperature measurements (Trewin 2001). Della-Marta and Wanner (2006) use a similar approach that models the changes to PDFs; it does not need overlap observations, however, and instead uses information from nearby reference stations. The main improvement of this homogenization method, called higher-order moments (HOM), in comparison with that of Trewin and Trevitt (1996) is the use of a nonlinear model, making it capable of handling inhomogeneities in higher moments. This method has been applied to summer daily maximum temperature at 26 western European stations (Della-Marta et al. 2007).

In the following, we propose a variation of the HOM method for homogenization of daily measurement temperature series. Although part of the principle involved is similar, relying on the definition of homogeneous subperiods, we propose a very different direct nonlinear spline regression approach rather than an adjustment that is based on quantiles. Our proposed method is referred to as spline daily homogenization (SPLIDHOM).

The SPLIDHOM model and the cubic smoothing spline estimation are described in section 2. In section 3, a simulation study is realized, by means of bivariate autoregressive models. This simulation allows one to compare SPLIDHOM, HOM, and Vincent’s adjustments. Advantages and drawbacks of each method are then discussed. In section 4, the example of Toulouse, France, daily minimum temperature (TN) series demonstrates the usefulness of the SPLIDHOM method.

## 2. Method

Our goal is to provide realistic adjustments of individual temperature measurements of a candidate series *Y* (the series to be adjusted), given the temperature of the series itself, by means of an estimated transfer function. The estimation of this function has to be possible even in the absence of overlapping parallel measurements. Like in Della-Marta and Wanner (2006), we rely on the existence of a close and well-correlated reference series *X*. This reference series does not necessarily need to be totally homogeneous but should be homogeneous on subperiods of at least 2 yr around each break affecting the candidate series, since 1) fitting spline models requires a minimum amount of data and 2) data have to cover a range of situations that is large enough to avoid extrapolation of the functions. Note that the definition of homogeneous subperiods provided in section 2a is exactly the same as that in Della-Marta and Wanner (2006).

### a. Notation

In the following, we denote *Y* as the candidate series and *X* as the reference series. Let *j* = 1, … , *k* be the set of changepoints affecting *Y*. For practical algorithmic reasons, we introduce dummy changepoints *j* = 0 that corresponds to the last observation of *Y* and *k* + 1 that corresponds to the day before the first observation of *Y*. Note that 1 refers to the most recent nondummy changepoint and that *k* indicates the oldest one. Let us denote HSP_{Xj}_{aft} as the homogeneous subperiod of *X* after the *j*th changepoint on *Y* and HSP_{Xj}_{bef} as the homogeneous subperiod of *X* before (see Fig. 1). The homogeneous subperiod on *Y* between changepoints *j* and *j* − 1 is denoted as HSP* _{Yj}*. Because

*X*may be affected by changepoints also, homogeneous subperiods HSP

_{Xj}_{aft}and HSP

_{Xj}_{−1bef}may be shorter than HSP

*. Let*

_{Yj}*m*

_{YXj}_{aft}be the nonlinear regression function of

*Y*on

*X*after the

*j*th changepoint and

*m*

_{YXj}_{bef}be the nonlinear regression function of

*Y*on

*X*before the

*j*th changepoint;

*m*

_{XYj}_{bef}is the nonlinear regression function of

*X*on

*Y*before the

*j*th changepoint.

### b. Model

The changepoint effects are adjusted sequentially, from the most recent (denoted as 1) to the oldest (*k*). The last period HSP_{Y}_{1} remains unchanged. Adjustment is first applied to HSP_{Y}_{2}, then to HSP_{Y}_{3}, up to HSP_{Yk}_{+1}. For adjustment of the whole subperiod HSP_{Yj}_{+1} (corresponding to the effect of the *j*th changepoint), the first step is to estimate *m _{YXj}*

_{bef}(

*m*

_{YXj}_{aft}), that is the regression of

*Y*on

*X*before (after) the break on HSP

_{Xj}_{bef}(HSP

_{Xj}_{aft}) subperiods (Fig. 1).

*m*

_{YXj}_{bef}and

*m*

_{YXj}_{aft}do not coincide, and their difference

*m*

_{YXj}_{aft−YXjbef}=

*m*

_{YXj}_{aft}−

*m*

_{YXj}_{bef}is not null, at least on parts of the data range. We adjust HSP

_{Yj}_{+1}so that the

*m*

_{YXj}_{bef}regression function matches the regression

*m*

_{YXj}_{aft}estimated on HSP

*. Thus, adjustments are given by the estimation of*

_{Yj}*m*

_{YXj}_{aft−YXjbef}(denoted as

*X*, if estimates of

*m*

_{YXj}_{bef}and

*m*

_{YXj}_{aft}are unbiased, then their difference is an unbiased estimator of

*m*

_{YXj}_{aft−YXjbef}=

*m*

_{YXj}_{aft}−

*m*

_{YXj}_{bef}. Any observed value

*Y*may be adjusted using this function and the corresponding

_{t}*X*value, according towhere

_{t}*X*is homogeneous on HSP

_{Yj}_{+1}(i.e., HSP

_{Yj}_{+1}and HSP

_{xj}_{bef}coincide) then adjustments can be directly applied to

*Y*before the

*j*th changepoint using (1). In the general case, however, reference

*X*itself might be inhomogeneous, or missing, on parts of HSP

_{Yj}_{+1}. So an additional step is performed. The

*m*

_{XYj}_{bef}regression function is estimated. This is the regression of

*X*on

*Y*for subperiod HSP

_{Xj}_{bef}. It allows the substitution of

*Y*into “pseudo”

_{t}*X*values:

_{t}*m*

_{XYj}_{bef}on HSP

_{Xj}_{bef}. Last, the SPLIDHOM adjusted observations

Although it is based on the same definition of subperiods as HOM is, the adjustment proposed by SPLIDHOM differs in its principle. SPLIDHOM is based on regression only, whereas HOM is based on distribution fitting. Note that, in the practical implementation of our algorithm, the model may be applied for each month or each season separately.

### c. Fitting

In practice, the various regressions involved are almost linear, whereas a large proportion of the useful information might be hidden in the nonlinear part of the regressions. For estimating the regression function, several techniques have been tested: kernel smoothers (Brockmann et al. 1993; too noisy at the edge for data-scarcity reasons), wavelet thresholding (Nason 2008; too sensitive to small outliers), and locally weighted polynomial regression and scatterplot smoothing (LOESS; Cleveland and Grosse 1991; too computationally demanding when applied for cross-validation techniques). Our final choice relies on a classical cubic smoothing spline that does not have the previously mentioned drawbacks for our application. In the following we recall the basics of smoothing splines. Readers may refer to Hastie and Tibshirani (1990) for a more complete overview of this technique.

*X*,

_{i}*Y*) for

_{i}*i*= 1, … ,

*n*be a sequence of observations, modeled by the relation

*E*(

*Y*|

_{i}*X*) =

_{i}*m*(

*X*). The smoothing-spline estimate is defined as the function

_{i}*m*″ as the second derivative of m and

*λ*as the smoothing parameter) that minimizes the penalized residual sum of squares:Interval [

*a*,

*b*] corresponds to the range of

*X*. This problem has a unique (and explicit) solution that is a natural cubic spline with knots at the values

*X*. This model may seem overparameterized, but spline continuity constraints at knots bring down its dimension dramatically.

_{i}*λ*(

*λ*≥ 0) controls the trade-off between fidelity to the data and roughness of the function estimate. Larger values of

*λ*correspond to smoother solutions. If

*λ*→ ∞,

*m*″(

*t*) → 0 and the minimizer is the least squares line. The smoothing parameter is estimated for each regression by means of a standard cross-validation technique to avoid overfitting. Let

*λ*obtained by leaving out observation

*i*, which mimics training and test-sample procedures. Estimated

*λ*is the value that minimizes the cross-validation sum of squares:This cross-validation technique gave satisfactory results in our application, selecting most of the time solutions having an equivalent degree of freedom from 2 to 4, roughly corresponding to polynomials of degree 1–3. This is a significant difference from HOM, in which the LOESS smoothing parameter is fitted empirically, as stated by the authors themselves.

Because the range of the data within different HSPs can be different, we often face an additional extrapolation problem. Linear extrapolation of *m _{XYj}*

_{bef}is easy to achieve, but extrapolation of

*m*

_{YXj}_{aft−YXjbef}may lead to incorrect results. So, we also choose to bound adjustments at the edges, as in the HOM method. In practice, adjusting values greater (lower) than the largest (lowest) observed value of

*X*on the estimation interval is performed using adjustment computed for the largest (lowest) observed value of

*X*on the estimation interval.

## 3. Results

### a. Simulation study

This experiment has two purposes: first, to establish the correlation necessary to obtain good results with the HOM and SPLIDHOM methods and, second, to show SPLIDHOM improvements relative to results from Vincent’s method and HOM in a variety of situations. We show the influence of HOM, SPLIDHOM, and Vincent’s method on several indices computed on daily maximum temperatures, including root-mean-square error (RMSE), annual mean, summer [June–August (JJA)] mean, 5% (Q05) and 95% (Q95) quantiles, and annual absolute maximum temperature.

^{2}. Pairs of correlated candidate and reference series are then simulated using the following procedure. First, we generate correlated noise terms

*U*

_{1t}and

*U*

_{2t}by means of a bivariate AR(1) process {

*U*

_{1},

*U*

_{2}} (Neumaier and Schneider 2001) described asthat is, the noise term {

*U*

_{1},

*U*

_{2}} of the process follows a centered bivariate normal distribution, with the correlation between

*ε*

_{1}and

*ε*

_{2}being controlled by parameter

*r*. In practice, we set

*φ*

_{1}=

*φ*

_{2}= 0.672 and

*σ*

^{2}= 8.6, which are values estimated on the real Toulouse temperature series. Pairs of series are created by summing the same trend and seasonal cycle (estimated on Toulouse temperatures) to the noise terms

*U*

_{1}(first series) and

*U*

_{2}(second series). Inhomogeneities are added to the first series to create the candidate, with the second series being the reference.

We choose to add three different synthetic inhomogeneities to the candidate series to study a variety of situations: type-I inhomogeneity consists in adding a normal random variable of mean −1.5°C and standard deviation 0.5°C to the daily data (pure noise). This “type I” inhomogeneity roughly reproduces temperature-independent errors. For example, an error related to sun exposure is likely independent of the actual observed temperature, since it may occur on hot days as well on cold late-winter days with snow cover. Type-II inhomogeneity consists in transforming data using transfer function *t* → *t* + (*t* − 18)/10 + *ξ* (*ξ* being random normal noise with standard deviation 0.2°C). Type-II inhomogeneity enlarges the distribution of daily data. The type-III transfer function is given by *t* → *t* + (*e ^{t}*

^{/10})/20 +

*ξ*(

*ξ*is defined as in type II). Type III results in larger skewness. The parameters of those functions are set empirically so that they roughly match the effect of potential shelter changes, as observed in shelter intercomparison experiments (not shown here). We applied type I to the period of 1966–70, type II to the periods of 1951–65 and 1986–95, and type III to the period of 1971–85 to study the adjustment of multiple inhomogeneities of various types in the data. The effect of such transforms on Toulouse TX distribution is shown in Fig. 3.

For *r* taking values of 0.80, 0.85, 0.90, 0.95, 0.96, 0.97, 0.98, and 0.99, 50 pairs of candidate and reference series are generated. A candidate series (“truth”) is then perturbed as described previously to give the “raw” candidate. The raw candidate series is then adjusted using the HOM and SPLIDHOM methods. A pseudo-Vincent method is also used: for each subperiod, 12 monthly adjustment coefficients are computed, computing the monthly mean differences between truth and raw over the whole subperiods. Because those estimates are much more accurate than they would be in reality, noise is added, consisting of a random centered normal variable of standard deviation 0.3, which is roughly the standard error estimate observed on monthly adjustment coefficients computed using the Caussinus and Mestre (2004) analysis-of-variance (ANOVA) model. The annual cycle of adjustments is then interpolated using a spline as described in Vincent’s method. Note that the multivariate ANOVA model takes all available monthly series in a regional neighborhood. In this experiment, we consider that the average regional network density does not vary but that *r* can take a wide range of values within the regional network.

For each correlation and for 50 pairs of simulated series, we compare differences between “true” candidate and raw series, and differences between true candidate series and series adjusted by means of Vincent’s method, HOM, and SPLIDHOM, on a variety of indices: RMSE of the adjusted daily values versus truth, and annual indices, such as annual means (average of the 365 values), annual absolute minimum and maximum temperatures (respectively the lowest and highest temperature that occurred during the year), and annual quantiles Q95 and Q05 of the daily values of the considered year. For each correlation, we compute box plots of the 50 corresponding RMSE as well as box plots of differences (raw minus true or “adjusted” minus true) observed on annual indices (for each simulated series, and each year). Results for *r* = 0.80, 0.90, and 0.98 are provided in Figs. 4–6. Perfect adjustments would result in null differences and RMSE.

From these results, a number of comments can be made. 1) All three methods improve the data: inhomogeneities are reduced when comparing adjusted series to raw series. 2) The Vincent method is able to correct the means (annual and JJA) and outperforms both HOM and SPLIDHOM for lower correlations, in terms of RMSE, but is strongly biased regarding adjustment of annual maxima as well as extreme quantiles. The biases of the annual maxima, Q95, and Q05 are about 1.0°, 0.4°, and −0.4°C, respectively, using the Vincent method (Fig. 6, bottom panels) in our experiment. 3) The HOM and SPLIDHOM improvements relative to the Vincent method are hardly noticeable for *r* < 0.90. For example, when *r* = 0.8, the biases of annual maxima and Q95 are about 0.8° and 0.4°C, respectively, for SPLIDHOM, but higher correlations ensure for both HOM and SPLIDHOM a good adjustment of the means and significant improvements for extreme quantiles. 4) SPLIDHOM clearly performs better than HOM in terms of RMSE.

Because *r* is a really crucial parameter, we plot median and interquartile range of RMSE for the three methods as a function of *r*, for each of the scores (Fig. 7). This confirms that both HOM and SPLIDHOM need well-correlated series (*r* > 0.90) to outperform the Vincent method, in terms of RMSE and bias reduction for extreme quantiles. The performance of the Vincent method is less sensitive to *r* value, at least for the range of correlations we tested. In looking at a comparison of HOM and SPLIDHOM, it is seen that SPLIDHOM clearly exhibits lower RMSE. Adjustment of annual maxima is equivalent for both methods, but SPLIDHOM performs generally better than HOM for means (annual and JJA) and Q05. For Q95, SPLIDHOM is more biased for *r* ≤ 0.90 but gets the best results for *r* > 0.96. If we roughly consider that SPLIDHOM is superior to the Vincent method for a correlation of 0.90 and delivers trustworthy results at a correlation of 0.95, those correlation thresholds are important to take into account. For maximum temperatures, in a region of flat terrain such as the Paris, France, region, a correlation of 0.95 (0.90) is achieved for an approximate station distance of ~75 km (~150 km). In the more-mountainous area around Lyon, France, those distances are, respectively, 18 and 60 km (not shown here).

### b. Application to the Toulouse–Blagnac temperature series

Toulouse–Blagnac (Toulouse civil airport; professional station with index number 31069001) TN and TX series are affected by several abrupt changepoints. Those changes are detected using “PRODIGE” software (Caussinus and Mestre 2004), which relies on multiple pairwise comparisons of annual Toulouse series with regional neighbors. Statistical detection itself is performed by means of a dynamic programming algorithm (Hawkins 2001) to find positions of changes together with an adapted penalized likelihood criterion (Caussinus and Lyazrhi 1997) assessing the significance of changes. Metadata allow validation of those detections and provide causes and precise days for changes: 20 June 1962 (new instrumental park), 15 October 1968 (relocation and new shelter), 1 May 1972 (sensor change), and 17 June 1986 and 8 November 1991 (for both, relocations of instrumental park caused by construction of new runways). The reference data are provided by the Toulouse–Francazal series (French “Armée de l’Air” station; military airport), situated 12 km south of the Toulouse–Blagnac airport. This series is affected by a large changepoint on 14 November 1955 (relocation and shelter change). The Toulouse–Blagnac series starts in 1951. Correlation of the series is high: *r* = 0.98 (at a daily time scale, with seasonal cycle removed), justifying the use of the SPLIDHOM technique. Changepoint effects are adjusted sequentially, for each season, from the period before the most recent changepoint (1991) to the oldest one (1962). In this example, we choose seasonal instead of monthly estimates, because the results appeared to be more stable. Let us analyze in detail the period 1986–91 for the autumn season [September–November (SON)]. The top-left panel in Fig. 8 shows the scatterplot of observed daily Toulouse–Blagnac TN (candidate *Y*) as a function of daily Toulouse–Francazal TN (reference *X*), for homogeneous subperiod 1 September 1986–8 November 1991 for the SON season. The solid gray line corresponds to the smoothing-spline estimation of regression function *m _{YXj}*

_{bef}. In a similar way, the top-right panel of Fig. 8 shows the scatterplot of daily data and estimation of regression function

*m*

_{YXj}_{aft}after the target change on 8 November 1991 for the SON season over subperiod 8 November 1991–30 November 2009. The bottom-left panel of Fig. 8 shows the estimation of this difference of the latter two functions, as a function of

*X*(Toulouse–Francazal). This corresponds to the estimation of the function

*m*

_{YXj}_{aft−YXjbef}in (1). In this example,

*m*

_{YXj}_{aft−YXjbef}can be considered to be linear. An estimation of the

*m*

_{XYj}_{bef}additional transfer function used in (2) is also provided in Fig. 8 (bottom-right panel). Figure 9 shows the estimated SPLIDHOM adjustment function

When analyzing the adjustments of the previous break (1986) for the same SON autumn season, we find a different shape (Figs. 10 and 11). The estimation of *m _{YXj}*

_{aft−YXjbef}is not linear (Fig. 10, bottom-left panel), resulting in a nonlinear adjustment function (Fig. 11) and thus justifying the use of nonlinear models in SPLIDHOM. In this example, the adjustments are less important. When analyzing adjustments of Toulouse series for every breakpoint and every season, roughly one-half of the adjustment functions have a linear or quasi-linear (including constant) shape, with the other one-half exhibiting a nonlinear shape.

The examples given above show that adjustments of our method are sensitive to temperature itself, thus taking into account the meteorological situation of each day in a crude way. This is the main contrast and improvement to the adjustments of the method provided by Vincent et al. (2002). This method depends on the seasonal variations of monthly adjustments and of the position of the day in the year. It is indeed simple to apply, and it keeps coherency between the usual homogenization methods (applied to annual and monthly data) and daily adjusted data, but adjustment of higher quantiles is a bit less realistic, as shown by the experimental study.

In addition, it can be shown that our method reaches also a good agreement with standard homogenization procedures: comparing time series of annual means of TN, homogenized using SPLIDHOM (daily homogenization) and by means of PRODIGE (monthly homogenization; Caussinus and Mestre 2004), we find very good agreement (Fig. 12). This is a remarkable result, since the PRODIGE method relies on a completely different principle in which mean biases are estimated using an ANOVA model, applied on a large set of monthly series in the same climatic area. When considering annual averages, we get very similar results by using two completely different methods applied independently.

## 4. Conclusions

Although parts of the principles involved in HOM and SPLIDHOM are very similar, especially the definition of subperiods, SPLIDHOM adjustments differ: they are based on nonparametric regression (by means of a cubic smoothing spline) whereas HOM involves fitting data to several candidate distributions. The use of a smoothing parameter set by means of cross-validation avoids overfitting during the estimation process. On simulated examples, our SPLIDHOM technique shows an improvement over HOM (especially in terms of RMSE) and Vincent’s method for the correction of extreme quantiles if the correlation is high enough (above 0.90). When the correlation is lower than 0.9, Vincent’s method is often superior and thus should not be neglected. Vincent’s method is not designed for adjusting extreme values, however. Therefore, the adjustment of extreme quantiles is only sensible if a highly correlated reference station exists. A very important result of our study is that correlation of the candidate is the essential parameter that drives performance of both HOM and SPLIDHOM.

On practical examples, SPLIDHOM adjustments are compatible with more-classical homogenization techniques applied to monthly or annual series, which is a highly desirable feature. Also, when the individual errors cannot be considered to be “temperature dependent” (type-I errors in our simulation), SPLIDHOM still removes the main biases.

SPLIDHOM should be compared with new emerging techniques that have been recently developed, such as an improved version of HOM known as higher-order moments for autocorrelated data (HOMAD; Toreti et al. 2010), and a quantile-matching technique (Wang et al. 2010). Performance of those methods will be investigated further using various benchmarks and more types of inhomogeneities, during the last phase of European Cooperation in Science and Technology (COST) Action ES0601 “Advances in homogenization methods of climate series: An integrated approach (HOME).”

SPLIDHOM software is freely available through a simple request to the corresponding author.

This work was partly funded by European Union COST Action ES0601 HOME. The authors thank Paul Della-Marta for fruitful discussions and for providing his HOM algorithm and three anonymous reviewers for very useful suggestions. The authors also credit the contributors of the R project.

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