Franc ¸ois Petitjean

Camille Kurtz
email: ckurtz@unistra.fr

Pierre Ganc ¸arski
email: gancarski@unistra.fr

Spatio-Temporal Reasoning for the Classification of Satellite Image Time Series

Keywords: Multi-temporal Analysis, Satellite Image Time Series, Data Mining, Segmentation, Information Extraction

Satellite Image Time Series (SITS) analysis is an important domain with various applications in land study. In the coming years, both high temporal and high spatial resolution SITS will be available. In the classical case, these data are studied by analysing the radiometric evolution of the pixels with time. When dealing with high spatial resolution images, object-based approaches are generally used in order to exploit the spatial relationships of the data. However, these approaches require a segmentation step to provide contextual information about the pixels. Even if the segmentation of single images is widely studied, its generalisation to a series of images remains an open-issue. This article aims at providing both temporal and spatial analysis of SITS. We propose first segmenting each image of the series, and then using these segmentations in order to characterise each pixel of the data with a spatial dimension (i.e. with contextual information). Providing spatially characterised pixels, pixel-based temporal analysis can be performed. Experiments carried out with this methodology show the relevance of this approach and the significance of the resulting extracted patterns in the context of the analysis of SITS.

Introduction

Satellite Image Time Series (SITS) is a major resource for Earth monitoring. Since the last decades, these image series are either sensed with a high temporal resolution (daily coverage at a kilo-metric spatial resolution) or with a high spatial resolution (weekly coverage at a metric spatial resolution). However, since a few years, satellites such as the Taiwanese Formosat-2 are providing both high temporal and high spatial resolution SITS (HSR SITS), but with a limited coverage of the Earth's surface and with only four spectral bands. In the coming years, these kinds of data will become widely available thanks to the ESA's Sentinel program. The growing availability of such images, periodically acquired by satellite sensors on the same geographical area, will make it possible to produce and regularly update accurate land-cover maps of a given investigated site.

In order to efficiently use the huge amounts of data that will be produced by these new sensors, adapted methods for SITS analysis have to be developed. Such methods should allow the end-user to obtain satisfactory results (e.g. relevant and accurate temporal evolution behaviour) with minimal time (by automating the tasks which do not require human expertise), and minimal effort (by reducing the parameters).

In the standard methods, these data are studied by analysing the radiometric evolution of the pixels through the time series. Due to the high spatial resolution of these future images, the geometrical information of the scene could also be considered in the classification process by using region-based approaches. In these methods, a segmentation process is required to extract segments based on radiometric homogeneous hypotheses. Once these segments are extracted, it is possible to characterise them using spatial/geometrical properties, to enhance the classification process.

However, the problem of using a segmentation step in a temporal classification remains an open issue, since neither the mapping between mono-temporal segmentations, nor the temporal segmentation are resolved. A review of the available literature on SITS analysis shows a lack of existing methods responding to this issue. This article aims at addressing this issue by characterising a pixel with spatial properties in order to improve the analysis of SITS.

This article is organised as follows. Section 2 gives an overview of existing methods for SITS analysis. Section 3 introduces our method for spatio-temporal analysis of SITS. Section 4 describes the experiments carried out with this methodology. Section 5 presents the results obtained using the proposed methodology. Conclusions and perspectives will be found in Section 6.

State of the art

SITS allows the analysis, through observations of land phenomena, of a broad range of applications such as the study of land-cover or even the mapping of damage following a natural disaster. These changes may be of different types, origins and durations. For a detailed survey of these methods, the reader should refer to [START_REF] Coppin | Digital change detection methods in ecosystem monitoring: a review[END_REF][START_REF] Lu | Change detection techniques[END_REF].

In the literature, we find three main families of methods. Bi-temporal analysis, i.e., the study of transitions, can locate and study abrupt changes occurring between two observations. Bi-temporal methods include image differencing [START_REF] Bruzzone | Automatic analysis of the difference image for unsupervised change detection[END_REF], image ratioing [START_REF] Jensen | Urban change detection mapping using landsat digital data[END_REF], image composition [START_REF] Ouma | Multiscale remote sensing data segmentation and post-segmentation change detection based on logical modeling: Theoretical exposition and experimental results for forestland cover change analysis[END_REF] or change vector analysis (CVA) [START_REF] Johnson | Change vector analysis: a technique for the multispectral monitoring of land cover and condition[END_REF]. A second family of mixed methods, mainly statistical methods, applies to two or more images. They include methods such as post-classification comparison [7], linear data transformation (PCA and MAF) [START_REF] Howarth | Time-Series Analysis of Medium-Resolution, Multisensor Satellite Data for Identifying Landscape Change[END_REF], image regression or interpolation [START_REF] Kennedy | Trajectory-based change detection for automated characterization of forest disturbance dynamics[END_REF] and frequency analysis (e.g., Fourier, wavelets) [START_REF] Andres | Fourier analysis of multi-temporal AVHRR data applied to a land cover classification[END_REF]. Then, we find methods designed more for image time series and based on radiometric trajectory analysis [START_REF] Jönsson | Timesat-a program for analyzing time-series of satellite sensor data[END_REF][START_REF] Verbesselt | Detecting trend and seasonal changes in satellite image time series[END_REF][START_REF] Petitjean | Analysing satellite image time series by means of pattern mining[END_REF][START_REF] Kennedy | Detecting trends in forest disturbance and recovery using yearly Landsat time series: 1. LandTrendr -Temporal segmentation algorithms[END_REF].

Whatever the type of method used in order to analyse satellite image time series, there is a gap between the amount of data representing these time series, and the ability of algorithms to analyse them. Firstly, these algorithms are often dedicated to the study of a change in a scene from bi-temporal representation. Secondly, even if they can map areas of change, they are not able to characterise them. Thirdly, and this point is even more problematic, the geometrical/spatial properties of the data are never taken into account, except for the use of the pixel coordinates. High Spatial Resolution SITS have given rise to the need for spatially and temporally dedicated methods.

To improve the analysing process by using the spatial relationships of the data, object based methods have been more recently proposed [START_REF] Blaschke | Object based image analysis for remote sensing[END_REF]. In a first step, the images are segmented/partitioned into sets of connected regions. Then for each region, spatial and/or geometric features (e.g. area, elongation, smoothness) are computed in order to characterise the regions [START_REF] Herold | Spatial metrics and image texture for mapping urban land use[END_REF][START_REF] Carleer | Urban land cover multilevel region-based classification of VHR data by selecting relevant features[END_REF]. Finally, these regions are classified using these features [START_REF] Kurtz | Multiresolution region-based clustering for urban analysis[END_REF].

Object based methods have shown promising results in the context of mono-image analysis. However, it seems currently difficult to extend these methods to deal with the spatial properties of SITS. Indeed, although several methods have been proposed in order to map segments from one image to another [START_REF] Gueguen | Analysis of satellite image time series based on information bottleneck[END_REF], or to directly build spatio-temporal segments [START_REF] Moscheni | Spatio-temporal segmentation based on region merging[END_REF], their scalability to wide sensed areas and their robustness to local disturbance (temporally and spatially) remains problematic.

This article aims at solving the above-mentioned issues. We therefore propose to classify SITS as the radiometric evolution of atomic sensed areas (i.e., pixels) with time. Then, in order to take into account the spatial properties of the data, we propose to characterise each pixel with spatial and geometrical attributes obtained using a pre-segmentation step. This formulation allows the study of spatial characteristics over time while abstracting from the correspondence between segments since the data remains the pixel. Moreover, this formulation is aimed at obtaining accurate and reliable evolution behaviour maps both by preserving the geometrical details in the images and by properly considering the spatial context information. This paradigm, using spatially characterised pixels, was previously introduced and studied for multi-level segmentation of a single image in [START_REF] Bruzzone | A multilevel context-based system for classification of very high spatial resolution images[END_REF] and has shown promising results. We propose, in this article, to extend it to the analysis of multi-temporal satellite images.

Spatio-temporal analysis methodology

In this section, we present the proposed approach, which is composed of five main steps that are applied sequentially: These steps are fully described in the remainder of this section. The reader may also refer to Figure 1 for a visual outline of the workflow of the proposed approach.

Input / Output

Let us briefly define the input (resp. the output) of the proposed method.

Input. The method takes as input a series S image = I1 , . . . , I N of N ortho-rectified 1 images of width W and height H. Let B be the number of bands of the images composing the series. Each multivalued (i.e. with multiple bands) image

I n (n ∈ [1, N]
) can be seen as a function: Output. The method gives as output a classification/clustering of the major behaviours of evolution. Such classification can be modelled by a label image

I n : [[1, W]] × [[1, H]] → Z B (x, y) → B b=1 I n b (x, y) (1
I C : [[1, W]] × [[1, H]] → [[1, C]],
which associates to each sensed area (x, y) a class value C(x, y) among the C possible ones. Addendum: Each class from the classification is also modelled by a centroid sequence, which provides another source of information of the main evolution behaviours.

Segmentation of images

A segmentation of a multivalued image

I n is a partition R n = {R n i } R n i=1 of [[1, W]] × [[1, H]]; broadly speaking, the scene visualised in I n is "decomposed" into R n distinct parts R n
i , which are supposed to present specific radiometrical properties. We will denote R n i as a region of the image I n . To any segmented image I n , we then associate a region image

I n R : [[1, W]] × [[1, H]] → [[1, R n ]] (x, y) → I n R (x, y) (2)
Such region image is a function that associates to each sensed area (x, y) a region label I n R (x, y) among the R n possible ones.

Once the N images have been segmented (producing N region images

I n R , (n ∈ [[1, N]]
)), it is then possible to characterise each region of each segmentation by following the next step.

Regions characterisation

Numerous features (spectral, geometrical, topological, etc.) can be computed for the regions of a segmentation in order to characterise them. Each feature can be seen as a function F associating to each region

R n i (i ∈ [[1, R n ]]) of a segmentation S n a corresponding feature value F(R n i ) ∈ R. F : [[1, R]] → R R n i → F(R n i ) (3)
Once a region is characterised by a feature value, it is then possible to affect this value to all the pixels composing the region. Let A be the number of features chosen to describe every sensed area (x, y) of every image.

Construction of vector images

At this step, each pixel of a multivalued image I n can be characterised by two types of information:

-directly sensed values (i.e., B values, denoted

P n b with b ∈ [[1, B]] where P n b (x, y) = I n b (x, y)); -region-associated values (i.e., A values, denoted F a with a ∈ [[1, A]]).
All of these values are normalised over the image time series. It is then possible to combine these features to build "enriched" pixels in order to better characterise them. To process, a vector of features is created and associated to each one of the pixels contained in the image I n . Finally, by applying this step to each image of the series, we build N vector images defined as:

V n : [[1, W]] × [[1, H]] → [0, 1] B+A (x, y) → B b=1 P n b (x, y) × A a=1 F a (I n R (x, y)) (4)

Construction of time series

Let S be the dataset built from the image time series. S is the set of sequences defined as:

S = V 1 (x, y), • • • , V N (x, y) | x ∈ [[1, W]] , y ∈ [[1, H]] (5)
In these sequences, each element is (B+A)-dimensional. Since high-dimensional spaces do not often provide the best solutions, we will study different projections of this (B+A)-dimensional space in the experiment part (e.g., time series where each pixel is characterised by a 5-tuples composed of three directly sensed values and two region-associated values).

Classification of the time series

The extraction of relevant temporal behaviours from satellite image time series can be realised using a classification algorithm. Once these time series have been built, it becomes possible to classify them into different clusters/classes of interest. To this end, the proposed methodology makes it possible to use either supervised or unsupervised classification algorithms.

A classification of a set of sequences S is a partition

C = {C i } C i=1 of [[1, W]] × [[1, H
]]; broadly speaking, as each temporal sequence is associated to a sensed area (x, y), the whole scene can be "decomposed" into C distinct parts C i , which are supposed to represent similar temporal evolution behaviours. We will denote C i as a cluster/class. The classification can be modelled by a label image

I C : [[1, W]] × [[1, H]] → [[1, C]],
which associates to each sensed area (x, y) a class value I C (x, y) among the C possible ones

I C : [[1, W]] × [[1, H]] → [[1, C]] (x, y) → I C (x, y) (6)
4. Experiments

Applicative context: Crop monitoring

The analysis of agronomical areas is important for the monitoring of physical variables, in order to give information to the experts about pollution, vegetation health, crop rotation, etc. This monitoring is usually achieved through remote sensing. Indeed, by using classification processes, satellite image time series actually provide an efficient way to monitor the evolution of the Earth's surface. Moreover, when the classes of interest are temporal (e.g., wheat crop, maize crop), the time dimension of the data has to be taken into account by the classification algorithms. For instance, the reflectance levels of the maize crop and of the wheat crop are very similar while their temporal behaviours are quite different (i.e., the wheat crop is ahead of the maize crop).

Thus, the usual strategy for land-cover mapping consists of classifying the temporal radiometric profiles of atomic sensed areas (x, y). With the arrival of SITS with high spatial resolution (HSR), it becomes necessary to use the spatial information held in these series, in order to either study the evolution of spatial features, or to help characterising the different land-cover classes. Our experiments focus on the second point. The underlying idea is that several spatiallybuilt features can be used in the classification process. For example, some crops are usually cultivated in smaller parcels than others, while having the same radiometric behaviour (e.g., sunflower crop vs. wheat crop). Another (non restrictive) example could be the use of the smoothness of the regions, which could help to distinguish between tree-crop and forest.

Material: The Satellite Image Time Series used

We detail here the main information concerning the images used for this work. The area of study of this work is located near the town of Toulouse in the South West of France. We have at our disposal 15 cloud-free Formosat-2 images sensed over the 2007 cultural year. These images cover an area of 64 km 2 . An image from the series is given in Figure 2 while the temporal repartition of the sensed images is given in Figure 3.

From these images, we use the multi-spectral product at a spatial resolution of 8 m with the four bands Near-Infrared, Red, Green and Blue. Before being used in this work, the Formosat-2 products have been ortho-rectified (guaranteeing that a pixel (x, y) covers the same geographic area throughout the image series). All images also undergo processes in order to make the radiometric pixel values comparable from one image to another. These processes consist of converting the digital counts provided by the sensor into a physical magnitude and in restoring their own contribution to the surface by correcting for atmospheric effects.

From the instrument radiometric model, digital numbers are first converted into reflectances (normalised physical quantities of solar irradiance). The absolute calibration coefficients used in this step come from the monitoring of Formosat-2 sensors conducted by the French Space Agency (CNES). The inversion of the surface reflectance is then made by comparing the measured reflectance in simulations at the top of the atmosphere, carried out for atmospheric and geometric conditions of measurement. The elevation is taken into account by carrying out simulations for various altitudes, including a weighting of the atmospheric pressure and the amounts of aerosols and water vapour. The state of the atmosphere at the time of the sensing is in turn characterised using meteorological sources (NCEP for the pressure and the humidity), using ozone data sources (TOMS or TOAST) and using aerosol data (SeaWiFS, Aeronet). Otherwise, climatological values are used.

Settings

The five steps described in Section 3 have been performed as follows.

Segmentation of images. In the literature, the most efficient segmentation algorithms are obtained by combining several basic/standard ones. Furthermore, the parameters of these basic segmentation algorithms are tuned according to the characteristics of the image modality (used as input) and the features of the objects to be segmented. To avoid this parametrisation problem, we have chosen to use the Mean-Shift segmentation algorithm [START_REF] Comaniciu | Mean shift: a robust approach toward feature space analysis[END_REF] to segment each image. Indeed, this algorithm is simple to configure and has shown promising results in the context of the segmentation of remote sensing images [START_REF] Huang | An adaptive mean-shift analysis approach for object extraction and classification from urban hyperspectral imagery[END_REF].

For a given pixel, this algorithm builds a set of neighbouring pixels within a given spatial radius and colour range. The spatial and colour centre of this set is then computed and the algorithm iterates with this new spatial and colour centre. There are three main parameters: the spatial radius (denoted by h s ) used for defining the neighbourhood, the range radius (denoted by h r ) used for defining the interval in the colour space and the minimum size M for the regions to be kept after segmentation.

Since the level of geometrical information extracted by the segmentation algorithm depends on its parametrisation, we have run the algorithm using different parameter configurations. Experiments have shown that the best parameter configuration for the extraction of agricultural areas from these data is (h s = 5, h r = 25, M = 25). Figure 4 illustrates different segmentation results obtained on an extract of the image presented in Figure 2. One can note that the parameter configuration (h s = 5, h r = 5, M = 25) provides over-segmented results while the parameter configuration (h s = 5, h r = 25, M = 25) provides satisfactory results to deal with for the extraction of agricultural areas. We have used the OTB implementation2 of the Mean-Shift algorithm.

Region characterisation. Several characteristics can be useful for the classification of agronomical scene. For instance, the size of the regions could be used to discriminate small/large fields, while the smoothness could be used to dissociate forest regions from fields. In this way and without loss of generality, the following region-associated features have been computed: -the mean of the infra-red band of the region (F NIR ); -the mean of the red band of the region (F R ); -the mean of the green band of the region (F G ); -the mean of the blue band of the region (F B ); -the area of the region (F Area ); -the elongation of the region (F Elong. ); -the smoothness of the region (F S mooth. ).

The elongation is computed as the highest ratio between the width and the length of several bounding boxes (computed for different directions, i.e., each π/8). The smoothness is computed as the ratio between the perimeter of the morphologically opened region and the original region. To this end, we use an opening structuring element invariant to the scale (i.e., with a size depending on the area of the original region). The size of the structuring element was set to √ F Area .

Construction of vector images. As it was explained previously, each pixel can be characterised by two types of information: directly sensed values (denoted as P NIR , P R , P G , P B ), and region-associated values (denoted as F NIR , . . . , F S mooth. ). All the values are normalised in [0, 1], attribute by attribute over the series. This allows each attribute to be of comparable weight for the classification step.

Construction of time series. In order to find the best separation of thematic classes, we tested several combinations of the nine attributes over the time series. Twelve combinations were selected:

Note that the Kappa index is an agreement measure between two partitions and thus does not require to "align" the clusters with the reference classes.

Results

This section presents the results of the proposed contextual approach for multi-temporal analysis. These experiments aim at showing a non-restrictive example of the use of our approach in the classification of an agronomical HSR SITS.

Table 1 summarises the F-measure and the Kappa values obtained for the experiments with several attributes projections (see Section 4). Experiment ⋆ gives the reference score obtained by a pixel-based classification of the SITS. From this table, one can first note that these baseline scores are quite high, demonstrating the relevance of the temporal dimension for land-cover classification. In addition, note that due to the significant number of classes in the reference map, the Kappa scores are quite high and a small increase of the score can correspond to a major difference in the classification.

These experiments show that the mean value of the regions, as well as the smoothness of the regions, are relevant features for the classification of the specific studied area. However, the area feature seems actually not relevant for this dataset, since each crop class contains different size of fields. A similar observation can be made for the elongation feature. Nevertheless, these observations are not questioning the genericity of the approach, since these spatial characteristics (and others) could be used in other applications. In our case, the best result has been obtained with the use of the radiometric mean of the regions combined with their smoothness.

In order to visually assess the results provided by the proposed method, Figure 5 shows the best clustering result obtained on sequences from 2007, as well as the corresponding reference land cover map. Finally, Figure 6 focuses on a restricted area in order to visualise the differences between the pixel-based approach and the proposed pixel-enriched approach. One can see that, in the details, the land-cover map obtained with the proposed pixel-enriched approach is spatially more consistent and regular than the result obtained with the pixel-based approach. Furthermore, one can note that the orange and yellow classes, corresponding respectively to corn and wheat crop fields, as well as the dark green class corresponding to hardwoods, are well separated. More generally, these results demonstrate statistically and visually the relevance of the proposed pixel-enriched approach compared to the pixel-based analysis.

Conclusion

This article has introduced a novel approach for the analysis of satellite image time series. The originality of this approach lies in its consideration of spatial relationships between pixels in each remotely sensed image. We have seen that characterising pixels with contextual features computed on segments, allows us to enhance the classification process. This methodology has been carried out on a SITS; the first results obtained have shown the relevance of this approach.

We believe this work opens up a number of research directions. Firstly, the choice of spatial features and of their weighting in the classification process has to be deeply studied. For instance, textural and topological features could be used. Secondly, we also plan to validate the proposed methodology by using another segmentation strategies. For instance, it has been proposed in [START_REF] Kurtz | Extraction of complex patterns from multiresolution remote sensing images: A hierarchical top-down methodology[END_REF] a new segmentation approach enabling to decompose the scene at different semantic levels. Such an approach could be extended to SITS to analyse the scene in a multi-temporal/multi-level fashion. Finally, the higher the spatial and temporal resolution, the more relevant our approach will be. In this way, the next step of this study could consists of applying this paradigm to a series of images couples Multi-Spectral / Panchromatic. The spatial accuracy of Panchromatic images will help to preserve the fine details and structures. This hybrid paradigm combines the possibilities offered by the (per-pixel) multi-temporal analysis and the relevance of the (mono-image) region-based frameworks for spatio-temporal analysis. The coming Sentinel-2 satellite will provide at the same time images with different spatial and radiometric resolutions at a high temporal frequency. In this context, the methodology proposed in this article provides a first trend to deal with such data.

Figure 1 :

 1 Figure 1: Workflow of the proposed approach.

  ) with be the Cartesian product and [[a, b]] denote the interval on Z, bounded by a, b ∈ Z.

Figure 2 :

 2 Figure 2: One image from the series (August, 4 th 2007). day of year Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Figure 3 :

 3 Figure 3: Sensing distribution of images sensed over 2007. Each spot represents a sensed image.

Figure 4 :

 4 Figure 4: Results of the Mean-Shift segmentation algorithm carried out on an extract of the satellite image presented in Figure 2. (a) Extract of one image of the series. (b) Segmentation result (h s = 5, h r = 5, M = 25). (c) Segmentation result (h s = 5, h r = 25, M = 25). Note that the boundaries of the regions are depicted in white.

Figure 5 :Figure 6 :

 56 Figure 5: Results of the proposed method carried out on the satellite image time series. (a) Land cover reference map. (b) Clustering map obtained (Experiment 7). Note that this clustering map has been recoloured according to the corresponding land cover reference map.

Table 1 :

 1 Results of the experiments. NIR , P R , P G , P B , F S mooth. , F Area , F Elong. ) 48.5 87.1 7 (F NIR , F R , F G , F B , F S mooth. ) NIR , F R , F G , F B , F Elong. ) 45.7 87.3 10 (F NIR , F R , F G , F B , F S mooth. , F Area , F Elong. ) 47.6 86.9 + s d ) • (s s + d s ) + (s d + d d ) • (d s + d d ) (s s + s d + d s + d d ) 2

			Experiment	F	K
	⋆	(P NIR , P R , P G , P B )	51.9 87.7
	1	(F NIR , F R , F G , F B )	52.3 87.8
	2	(P NIR , P R , P G , P B , F NIR , F R , F G , F B )	51.8 87.8
	3	(P NIR , P R , P G , P B , F S mooth. )	52.7 88.0
	4	(P NIR , P R , P G , P B , F Area )	51.7 86.6
	5	(P NIR , P R , P G , P B , F Elong. )	46.0 87.3
	6	(P 53.2 88.0
	8	(F NIR , F R , F G , F B , F Area )	51.8 86.6
	9 (F 11 All attributes	51.2 87.2
		The best scores are shown in boldface.
	Thus, the Kappa index is computed with:
			Pr(a) =	s s + d d s s + s d + d s + d d	(10)
	and	Pr(e) =	(s s

Note that, the radiometric levels of the images do not have to be comparable from one image to another.

ORFEO Toolbox (OTB) is an open source library of image processing algorithms developed by the French Space Agency (CNES). http://www.orfeo-toolbox.org

Note that other distances (and more relevant temporal ones[START_REF] Petitjean | Clustering of satellite image time series under time warping[END_REF]) could also be used. However, we remind that this article is focused on the data representation and not on the relevance of the classification algorithm.

Acknowledgements

The authors would like to thank the French Space Agency (CNES) and Thales Alenia Space for supporting this work under research contract n°1520011594 and the researchers from CESBIO (Danielle Ducrot, Claire Marais-Sicre, Olivier Hagolle and Mireille Huc) for providing the land-cover maps and the geometrically and radiometrically corrected Formosat-2 images.

10. (F NIR , F R , F G , F B , F S mooth. , F Area , F Elong. ); 11. All attributes.

Classification of time series. In the machine learning literature, classification problems are usually addressed using supervised or unsupervised algorithms. Supervised classification algorithms require training examples to learn the classification model. In our case, as we want to demonstrate the relevance of the proposed data representation, the choice and the suitability of the examples would create a bias, which would make difficult to identify the benefits provided by the spatial features. For these reasons, we have chosen to use an unsupervised classification algorithm. We have then applied the classical K-means clustering algorithm [START_REF] Macqueen | Some methods for classification and analysis of multivariate observations[END_REF] to classify the time series previously constructed. The distance used to compare the time series of S is the Euclidean distance 3 .

In agreement with the expert, the K-means algorithm has been used with as many classes (see Appendix A.) as in the reference map (i.e., 25 seeds), and with 15 iterations. Note that any clustering algorithm dealing with numerical data could also be used.

Validation

The obtained results have been compared to a land cover map produced by the method described in [START_REF] Idbraim | An unsupervised classification using a novel ICM method with constraints for land cover mapping from remote sensing imagery[END_REF] using a comprehensive ground reference data set. Several evaluation indexes have been computed. To assess the accuracy of the extracted thematic classes, we have computed the mean, for each class, of the F-measures obtained F . To this end, for each thematic class, the best corresponding clusters (in terms of partitions) were extracted. Then, we have computed: the percentage of false positives (denoted by f (p) ), the percentage of false negatives (denoted by f (n) ) and the percentage of true positives (denoted by t (p) ). These measures are used to estimate the precision P and the recall R of the results obtained by using the proposed method:

For each experiment, we have then computed the geometrical mean P of the precisions obtained and the geometrical mean R of the recalls obtained. Finally, we have computed the mean F-measure F which is the harmonic mean of the mean precision and the mean recall:

To assess the relevance of the results, we have also computed the Kappa index [START_REF] Congalton | A review of assessing the accuracy of classifications of remotely sensed data[END_REF]. The Kappa index K, which is a measure of global classification accuracy, is defined as:

where Pr(a) is the relative agreement among the observers, and Pr(e) is the hypothetical probability of chance agreement. The Kappa index takes value in [0, 1] and decreases as the classification is in disagreement with the ground-truth map. We have computed this index as follows. The approach consists of considering all point couples (x 1 , x 2 ) = ((x 1 , y 1 ), (x 2 , y 2 )) and see the configuration of these two points in each partition (the clustering result and the ground-truth). There are four possible configurations; for each one, a counter is associated and incremented each time a configuration appears:

1. x 1 and x 2 belong to the same partition both in the clustering and in the reference map (counter s s ); 2. x 1 and x 2 belong to the same partition in the clustering but not in the reference map (counter s d );


Laure Cabantous
email: laure.cabantous@nottingham.ac.uk.

Olivier Chanel
email: chanel@univmed.fr.

Jean-Christophe Vergnaud
email: christophe.vergnaud@univ-paris1.fr

Transport, health and climate change: Deciding on the optimal policy*

Keywords: climate change, model of decision-making under uncertainty, irreversibilities, transport policy. JEL code: C6, D62, D81, R48

Transport generates many externalities, some related to atmospheric pollution. In this paper, we focus on two: greenhouse gases, and local pollution. In the search for optimal transport policies, these two externalities have usually been analysed separately. Here, we study them jointly, in a sequential decision-making model. Our model allows for the irreversibility of the policies undertaken, as well as the possibility of a progressive reduction of uncertainties with the arrival of information. We find that when both sources of externalities are analysed jointly, structural measures enabling private transport requirements to be reduced are identified as being more advantageous economically than technological measures to reduce emissions of pollutants. We illustrate the usefulness of a joint analysis of externalities with two examples: tax measures on cars and housing policy.

Introduction

Beyond their primary vocation, which is to carry goods and people, our means of transport produce numerous harmful effects and generate considerable externalities (CGP, 2001;Infras-IWW, 2004;[START_REF] Mailbach | Handbook on estimation of external costs in the transport sector[END_REF]. They are responsible for road congestion, landscape disturbance and road accidents. Moreover, they contribute to a deteriorating environment in terms of noise levels, aesthetics and atmospheric pollution (greenhouse gas emissions and local pollutants). 4The impact of transport with respect both to climate change (greenhouse gas emissions, or GGEs) and to local air pollution (particles, NO X ) is a particularly topical issue. In both cases, progress on fixed-source emissions has actually led to transport becoming the prime source of GGEs: it produced an average of 21% of emissions by the European Union-15 (EU-15), in 2005 (European Environment Agency, 2007). 5 Transport is also the prime emitter of local, health-damaging pollutants, producing 36% of total particle emissions in 2007 (European Environment Agency, 2009a). Finally, transport is the major source of emissions of NO X , generating some 50% of total emissions in 2007 (European Environment Agency, 2009b). Road transport accounts for roughly two-thirds of particle and NO X emissions, as well as 90% of GGEs of all transport emissions.

Although local pollution and climate change share the same source, they have often been analysed separately when pricing policies for transport are determined. In France, for example, the government-organised public forum on the environment (Grenelle de l'Environnement) recommended introducing a tax system based on both CO 2 and particle emissions ("Eco-pastille"). This system, however, was adopted only for CO 2 . Similarly, in the United Kingdom, where the Department of Transport (1998Transport ( , 2008) ) has been pleading for over ten years for joint management of transport and environmental policies, local pollutants and GGEs are analysed separately [START_REF] Begg | Transport Policy and vehicle emission objectives in the UK: is the marriage between transport and environment Policy over?[END_REF].

Moreover, to date most debates on policies for reducing atmospheric pollutant emissions from transport have not satisfactorily taken into account the prevailing major uncertainties, be they scientific (the extent of environmental change), economic (the method of evaluating the costs of externalities) or technical (the effectiveness of pollution cleanup policies). Thus, in 2000, when the French government was updating its recommended reference values for the costs of effects ("valeurs tutélaires"), in order to take into account the external costs of transport (CGP, 2001), the two Ministries responsible for the environment and for transport disagreed over the value of a tonne of CO 2 and the health costs of exposure to air pollution. The Ministry for the Environment pleaded for much higher values than the Ministry for Transport. The latter cited the great uncertainty involved in the evaluation of externalities to defend lower valuations. Since these reference values are intended to be applied when assessing infrastructure projects, higher values would have a particularly favourable impact on the development of public transport systems. More recently, similar differences emerged in the debate on the updating of the value of a tonne of CO 2 (Centre d'analyse stratégique, 2009) and on the amount of the "climate and energy contribution" tax (Conférence des experts sur la contribution Climat et Energie, 2009).

A fuller grasp of the uncertainties, influenced less by special interests, thus seems to be needed. Given this context, it appears relevant to place uncertainty in a dynamic framework. This requires a sensitivity analysis, clarifying to what extent the values adopted impact on the effectiveness of policies. It also means that policy choices will be conditioned by the effects of the arrival of information. For instance, different policies for reducing atmospheric pollution could be compared, while taking into account the arrival of information showing that the negative effects of one of these externalities are more important than expected.

In this article, we present a model to support public decision-making in the field of transport policy. Our model has the advantage of jointly considering both types of atmospheric pollution generated by transport (GGEs and particle pollution which we will use as an indicator of local pollution). It also includes the principal characteristics of the context with respect to uncertainty and irreversibilities, thus providing scope for taking into consideration the effect of the arrival of information to influence the choice of policies. The model shows that when both sources of externalities are analysed jointly, structural measures enabling private transport requirements to be reduced are more advantageous economically than technological measures aimed at reducing emissions of pollutants.

The article begins with a description of the decision-making context. We then present the general hypotheses of the model in Part 3. Part 4 gives the principal findings. We discuss and illustrate the concrete implications of this model in Part 5. Part 6 concludes.

The decision-making context: Uncertainty and irreversibilities in transport

The debate over policies for reducing emissions of atmospheric pollutants by transport is complicated by the existence of numerous uncertainties. Despite major advances in the body of knowledge on climate change and local pollution, there are persistent scientific, economic and technological uncertainties about the effects of GGEs and particle pollution. Moreover, both these types of pollution have the effect of producing economic and/or ecological irreversibilities: i.e. they lead to alterations in the ecological and/or economic systems such that it becomes impossible subsequently to return to the initial state. Since no future action will enable a return to starting point, the decision-maker has to take into account, from the outset, the potential effects of the policies that s/he plans to conduct. Irreversibilities therefore complicate the decision-making process, suggesting that the decision-maker should be placed in a dynamic framework with the arrival of information enabling him/her to allow for the value of expected information (called quasi-option value or information option value).

Scientific uncertainties about GGEs

Both the nature and extent of the greenhouse effect and the consequences of GGEs are highly controversial (see for instance [START_REF] Michaels | Shattered Consensus: The True State of Global Warming[END_REF]or Singer, 2008).

Uncertainties about the extent of climate change

With the object of estimating the extent of climate change, numerous scenarios using complex models of evaluation based on various hypotheses (climate sensitivity, thermal inertia in oceans, net balance of carbon sinks, etc.) have been proposed. Whatever the standpoint, it is estimated that the rate of climate change will be faster than any other change experienced over the past 10,000 years [START_REF] Godard | Les instruments des politiques internationales de l'environnement : la prévention du risque climatique et les mécanisme de permis négociables[END_REF].

Beyond this certainty, it remains very difficult to estimate precisely the extent of climate change, so that scientific studies usually give a range of values. For example, the Fourth Report of the Intergovernmental Panel on Climate Change (IPCC) estimates that if no action is taken, there will be an increase in global temperatures of between 1.1 and 6.4° C by the end of the 21 st century, compared to the 1990-99 period. It further predicts an increase in the sea level ranging from 18 cm to 59 cm (IPCC, 2007) and a decrease in rainfall of roughly 15% between latitudes 30°N-55°N and latitudes 0°-20°S.

Uncertainties about the consequences of climate change

Estimating the consequences of an increase in the mean temperature of the planet is also difficult, since this brings into play a number of factors such as relief, proximity to a body of water or latitude. While there is a relative consensus on the potential consequences of climate change (changes to temperature distribution, increase in the frequency and intensity of natural disasters and contagious diseases), scientific studies do not offer a precise estimation of these consequences. The public decision-maker thus has to decide on the basis of scientific reports which, while rigorously compiled, contain loose estimates of the consequences of climate change. For example, the scientific studies concerning France conclude that climate change will bring about changes to rainfall varying from -20% to +50%, depending on location and season. Moreover, while the scientific studies all conclude that these alterations will have major economic impacts on the fields of insurance, tourism, agriculture, health, the extent and the timing of these effects have not been precisely determined. The IPCC report (2007) estimates the socio-economic cost of a 4° C rise in temperature at between 1 and 5% of Gross National Product. It evaluates the actualised net economic cost of the damage caused by global change at $12/tonne of CO 2 in 2005, with a range of $3 to $95/tonne of CO 2 (IPCC, 2007).

Scientific uncertainties about particles

The scientific uncertainties surrounding particles are more limited than those surrounding GGEs. The processes by which particle pollution affects health and buildings, for example, have already been established. Scientific studies have revealed two major health risks: respiratory and cardiovascular morbidity and short-term mortality [START_REF] Who | Health effects of transport-related air pollution[END_REF]. However, this does not mean that particle pollution carries no scientific uncertainties. Its consequences in terms of morbidity and long-term mortality, as well as on the determination of number of life years lost, for example, are not yet fully understood. There are also uncertainties concerning the harm to health caused by particles as a function of their source, their size and their chemical composition [START_REF] Hüglin | Contribution du trafic routier aux immissions de PM10 et PM2.5, summary of "Anteil des Strassenverkehrs an den PM10-und PM2.5-Immissionen NFP41/C4[END_REF][START_REF] Samoli | Short-term effects of nitrogen dioxide on mortality: an analysis within the APHEA project[END_REF]. Moreover, these uncertainties are added to those arising from the transfer of Exposure-Response functions (established for certain geographic zones) to other geographic zones for which specific data are not available.

Finally, there are uncertainties over obtaining the full, expected benefit from a particular technological measure (such as particle filters). The composition of atmospheric pollution is constantly changing (under the impetus of technological evolution and changes in air pollution standards) and the indicators measured are just one piece of the picture which, it is hoped, are as representative as possible. 6 Since they come from the same sources and are subject to the same conditions of dispersion by weather, measurements of the concentrations of numerous pollutants are strongly correlated. This leads to the risk of attributing excessive environmental and health damage to a particular pollutant, because it is measured more efficiently, and, as a corollary, underestimating the contributions of the other, associated pollutants. This type of uncertainty, however, disappears when there is a simultaneous reduction in all emissions.

Economic and technological uncertainties

In addition to scientific uncertainties about physical phenomena, the public decision-maker faces technological uncertainties linked to estimating the effectiveness of pollution cleanup measures. For example, it is clear that developing electric vehicles will eliminate local pollution, but the results in terms of GGEs are more mixed, and will depend on whether or not fossil fuels are used to produce the electricity.

Finally, the public decision-maker faces uncertainties of an economic nature when seeking to evaluate the costs of the externalities. These economic uncertainties arise essentially from the choice of methodology. The decision-maker can choose from various methods of evaluating harm, from various monetary values (cost of a morbidity episode, inability to work, the value of human life, damage to buildings, impacts on agriculture) and various discount rates. Taken together, these choices have major consequences on decisions, so that methodology is often the subject of lively political debate, as illustrated by the discussions on the CGP (2001) report on reference values in France. Certain approaches, however, are leading to a relative homogenisation and stabilisation in methodology: for example, the New Elements for the Assessment of External Costs of Energy Technologies [START_REF] Newext | New Elements for the Assessment of External Costs from Energy Technologies[END_REF] or Clean Air for Europe [START_REF] Cafe | Clean Air for Europe Cost Benefit Analyses: Baseline Analysis 2000 to 2020[END_REF].

The irreversibilities

It is common practice to distinguish between irreversibilities of an ecological nature and those of an economic nature. In this paper, ecological irreversibilities only concern GGEs. Current emissions are adding to an atmospheric stock which is very slow to dissipate. For instance, [START_REF] Devolder | Le facteur 4 : origine et pertinence, miméo[END_REF] estimates that 50% of CO 2 emitted disappear in 30 years, a further 30% in a few centuries and the remaining 20% in a few millennia. If we want to limit global warming to 2°C, we will have to decrease total world-wide greenhouse gas emissions by between 50% to 85% by 2050 [START_REF] Devolder | Le facteur 4 : origine et pertinence, miméo[END_REF]. In terms of the life span of a human being, therefore, the phenomenon can effectively be considered irreversible, and the whole range of potential strategies needs to be considered from a 20-to 50-year perspective. Conversely, particle pollution is not actually irreversible:7 mean particle concentrations in the air can decrease rapidly (by 90% in a few days); thus natural regeneration is fairly rapid and there is no problem of stock build ups.

The economic irreversibilities connected with GGEs and with particle pollution concern the costs entailed in putting policies into practice. Transport use is closely linked to lifestyle, so that implementing a fundamental policy takes a relatively long time and probably involves sunk costs.

Thus, climate change appears to be a unique phenomenon affecting the whole planet (with the problems of coordination that this implies) and involving a long time horizon (with the attendant problem of discounting). In addition, it brings into play both irreversibilities and the possibility of uncertainties being progressively reduced through the arrival of information. All these features argue for it to be treated within the framework of economic theory relating to decision-making, under uncertainty with an irreversibility effect [START_REF] Arrow | Environmental preservation, uncertainty and irreversibility[END_REF][START_REF] Freixas | On the Irreversibility Effect[END_REF][START_REF] Henry | Investment decisions under uncertainty: the irreversibility effect[END_REF]. Particle pollution, on the other hand, seems adequately characterised. Thus, the relative stabilisation of the economic evaluations of health costs means that it can be considered as relatively well-known, and hence certain.

The Model: A Reduced Representation of Transport Economics

We present a sequential model to support decision-making. This model is a reduced representation of transport economics, in which the public decision-maker can control different decision variables in order to regulate emissions of pollutants from transport. 8 The model has two periods, which makes it possible to take into consideration the effect of an arrival of information on the choice of policies. It focuses on two pollutants (GGEs and particle pollution) and considers the two major types of public sector policies currently under consideration in most countries: technological measures and structural measures. In essence, this stylised model, designed along the lines of the models introduced by [START_REF] Arrow | Environmental preservation, uncertainty and irreversibility[END_REF] and [START_REF] Henry | Investment decisions under uncertainty: the irreversibility effect[END_REF], highlights in a didactic manner the logical arguments in favour of one or the other of the policies.

General structure of the model

In order to take into account the arrival of information and the irreversibilities involved in climate change and particle pollution, the model covers two periods, t = 1,2. During each of the two periods, the public decision-maker can take action using the three decision variables at his/her disposal:

• the proportion of journeys made in private vehicles V t (the proportion of journeys made using public transport naturally being complementary

t V - 1
),

• the rate of reduction of GGEs t G ,

• the rate of reduction of particle pollution t P .

Taking the proportion of journeys made in private vehicles t V as a decision variable, we assume that the decision-maker can choose the equilibrium level in the transport market. In addition, as we are dealing with the proportion of journeys made either in private or public transport, we therefore also assume that there is a fixed volume of kilometres covered by an individual. This strong assumption is intended to simplify analysis. We would otherwise have to introduce a collective utility function dependent on total volume. Adding this feature to the model however, would make no difference to the qualitative results on the optimal internalisation of pollution.

These three decision variables concern both the structural and technological types of policy available to the public decision-maker attempting to regulate atmospheric pollution.

The first variable is the product of structural measures: the number of kilometres covered in private road transport is altered, thus affecting overall externalities. Such policies aim at, for example, increased car-sharing, reduced city centre access and parking, as well as more efficient public transport. The public decision-maker is absolutely free to choose how policies are implemented (urban tolls, enlarged public transport networks, special traffic lanes reserved for vehicles carrying at least three people, increased taxes on all fuels). Because these policies affect total emissions, they are less sensitive to uncertainty than technological measures.

The other two variables for the reduction of emissions of pollutants concern technological measures targeting the emission rates of pollutants per kilometres travelled. These are, for example, policies aimed at improving technologies like particle filters or NO X filters; at renewing the car fleet with less polluting vehicles (low-emission or zero-emission levels) or at traffic management (speed limits). These technological policies can be implemented in various ways, by imposing standards on fuels, through green tax incentives, through the "Ecopastille" system. However, these measures have limitations, generally targeting emissions from a single pollutant (CO 2 , SO 2 , particles, NO X ). Moreover, the gains obtained from technological improvements can very easily be partially lost following an increase in the volume of individual journeys, for instance.9

The objective function of the public decision-maker

In order to take both types of policy into account, the objective function of the public decision-maker needs to distinguish between costs associated with structural policies and costs associated with technological policies. Formally, the public decision-maker is looking to minimise the total social costs of transport, which can be broken down as follows:

Total costs = Variable costs of transport + Pollution cleanup costs + Pollution costs

• The variable costs of transport are given by a function ) 1 ( .

t t V C V - + α
where (.) C is differentiable and convex. α is a unit cost per km travelled in a private vehicle. • The costs of the pollution vary with the proportion of individual journeys:

( ) ( ) ( ) t t t V P G . 1 . 1 . - + - φ θ
where θ (φ ) represents the extra cost per GGE (particle) unit of pollution emitted from a private vehicle as compared to public transport. We consider the rates of emissions per passenger-kilometre from public transport to be lower than those from private vehicles. We also assume that whatever the values taken by θ and φ in the model, we still have β θ ≥ and γ φ ≥ and that therefore any attempt at pollution cleanup is always effective.

In the end, the function of the total cost that the public decision-maker has to minimise at t = 1,2 is the following:

( ) ( ) ( ) ( ) . 1 . 1 . ) ( ) ( . . . ) 1 ( . ) ; ; ; ; ( t t t t t t t t t t t t t V P G P E G D V P G V C V P G V CT - + - + + + + + - + = φ θ γ β α φ θ
This cost function has the advantage of being relatively simple, while realistically synthesising the genuine social costs generated by transport. Operational applications would no doubt require an adjusted version.

Uncertainty and information

We take into account the fact that the public decision-maker has to decide on a policy to reduce emissions of pollutants from transport, even though s/he is aware that there are uncertainties about the consequences of the emission of these pollutants. We have already mentioned that the main uncertainties concern the values of the costs connected with GGEs. Hence, our analysis focuses on these values.

Formally, we use θ and φ to denote the expected supplementary costs per unit of GGE and of particle pollution respectively. In the case of GGEs, we assume that this cost can be higher or lower (with equal probability) by an amount ε with < ε θ (this assumption excludes the possibility of discovering that climate change is economically favourable).

Moreover, we assume that the public decision-maker is able to learn the exact costs of GGEs between period 1 and period 2 and we call g = 0,1 the indicator of the arrival of such information. This assumption makes it possible to study the impact that anticipating the arrival of information has on the optimal decision.

Irreversibilities and intertemporal externalities

In order to obtain a suitable representation of the situation, our model includes two types of decisional irreversibilities (or intertemporal externalities).

The first irreversibility concerns the decision about developing public transport. This kind of decision usually needs to be taken well in advance. Accordingly, a relevant assumption is that it will not be possible in the second period to adjust the proportion of public transport upwards:

1 2 1 1 V V - ≤ - , that is 1 2 V V ≥ .
This extreme assumption is consistent with certain structural decisions, such as development of heavy infrastructures for public transport (tramways, undergrounds), urban development policy involving a more or less denselypopulated habitat. We will call i = 0,1 the indicator of the presence of such an irreversibility constraint:

[ ] 1 ; . 1 2 V i V ∈
. This first intertemporal externality is induced from the fact that the costs of investments in public transport depend on the level of public transport in the preceding period. The indicator shows the presence (i = 1) or the absence (i = 0) of intertemporal investment costs.

The second irreversibility concerns the choices of rates of reduction of emissions 2 G and 2 P :

1 2 G G ≥ and 1 2 P P ≥ . Indeed, these levels of emission reduction are usually fixed by environmental standards. Experience shows that there is no going back on environmental regulations (for example, the enforcement of the desulphurisation of diesel or the use of unleaded petrol), and that any changes always tend towards reinforcing standards, thus diminishing rates of emission. It can therefore justifiably be assumed that these standards will not be relaxed in the future. We introduce c = 0,1 as an indicator of the presence of irreversibility constraints:

[ ] [ ] 2 , 1 , 1 ; . , 1 ; . 1 1 = ∈ ∈ - - t P c P G c G t t t t

The reference solution

To examine the effect of the arrival of information, we consider the optimal stationary solution in the absence of information as the reference situation. It is therefore not necessary to consider the sequential nature of the model, and the total cost function can simply be minimised over one period. We call this the optimal solution ( ) * * * P G V , ;

; it represents a situation where (i) marginal costs of private and public transport are the same, and (ii) marginal costs of pollution cleanup are the same as marginal profits. (

) * * * P G V , ;
is therefore the solution to the following system:

( ) ( )      = ′ + = ′ + - ′ = - + - + + + * * * * * * * * * * * V P E V V G D V V C P G P G . ) ( . . ) ( . ) 1 ( 1 . 1 . . . φ γ θ β φ θ γ β α
This optimal stationary solution is not dependent on the presence of irreversibility constraints.

The results

We look at sequential choices over the two periods, with the arrival of information between Period 1 and Period 2.

To simplify the presentation of the results, we choose to eliminate two extreme cases which are unrealistic. The first entails the public decision-maker taking no pollution cleanup measures, so that we have a zero cost of pollution cleanup, and a maximum cost of extra pollution. The second case that we eliminate is where the public decision-maker reduces the proportion of private transport to zero, so that we get maximum cost of pollution cleanup and a zero cost of extra pollution.

Having eliminated these two unrealistic extreme cases, we seek an optimal intermediate solution. By definition, this optimal solution minimises overall total costs over the two periods. It should be noted that the second period costs may depend on decisions taken during the first period under irreversibility constraints 1 = i and/or 1 = c :

( ) [ ] [ ] [ ] ) ; ; ; ; ( ; ; ; ; ; ; 2 2 2 1 ; . , 1 ; . , 1 ; . 1 1 1 1 2 1 2 1 2 φ θ φ θ P G V CT Min c i P G V J P c P G c G V i V ∈ ∈ ∈ =
We use ( ) ;; ;

g c i P G V J ; ; ; ; ; ; ; • No information (g = 0): ( ) ( ) c i P G V J c i P G V J ; ; ; ; ; ; 0 ; ; ; ; ; ; ; 1 1 1 1 1 1 φ θ φ θ = • Information on climate change (g = 1): ( ) ( ) ( ) [ ] c i P G V J c i P G V J c i P G V
1 1 1 1 1 1 1 1 1 φ ε θ φ ε θ φ θ - + + =
The optimal choices of the first period are denoted ( )

g c i V ; ; 1 * , (
) and( )

g c i G ; ; 1 *
g c i P ; ; 1 *
. These optimal choices are the solution to the problem of minimising overall costs over the two periods:

[ ] [ ] [ ] (
)

g c i P G V J P G V CT Min P G V ; ; ; ; ; ; ; ) ; ; ; ; ( 1 1 1 1 1 1 1 ; 0 , 1 ; 0 , 1 ; 0 1 1 1 φ θ φ θ + ∈ ∈ ∈
In the absence of information, even though two periods were considered, the optimal stationary solution is also the optimal solution to the problem of minimisation, whatever the nature of constraints i and c:

( ) ( ) ( ) . 0 ; ; , 0 ; ; , 0 ; ; , 1 1 1 * * * * * * = = = ∀ P c i P G c i G V c i V c i
Similarly, anticipating information has no effect in the first period without irreversibility constraints ) 0 ( = = c i . The value function for the second period, ( )

g P G V J ; 0 ; 0 ; ; ; ; ; 1 1 1 φ θ
, is thus independent of the decisions taken in the first period, and we obtain:

( ) ( ) ( ) . ; 0 ; 0 , ; 0 ; 0 , ; 0 ; 0 , 1 1 1 * * * * * * = = = ∀ P g P G g G V g V g
Solving this model reveals two types of effects: effects related to the expectation of information and those related to the presence of irreversibility. We present the results concerning these two types of effects in the following subsection.

The effects of expecting the arrival of information

In models that take into account decisional irreversibilities and where there exists a quasioption value, expecting the arrival of information generally encourages the decision-maker to make more flexible choices (due to the irreversibility effect). Indeed, more flexible choices enable the decision-maker to use the information better.

Owing to the structure of the model detailed herein, an irreversibility effect is expected for the rates of emission reduction. In other words, greater flexibility is expected to result in lower rates of emission reduction as compared to the reference situation. Expecting the arrival of information on climate change leads to setting a reduction rate for GGEs in Period 1 which is lower than that of the stationary reference solution ( ( ) ( )

0 ; ; 1 ; ; 1 1 c i G c i G * * ≤ ).
Expecting an arrival of information leads not only to direct irreversibility effects but also to supplementary effects, as described in Proposition 1 below.

Proposition 1: When an arrival of information is expected, the optimal policy is to reduce the proportion of private transport and the reduction levels for emission rates of both pollutants:

c i, ∀ (1) ( ) ( ), 0 ; ; 1 ; ; 1 1 c i V c i V * * ≤ (2) ( ) ( ) 0 ; ; 1 ; ; 1 1 c i G c i G * * ≤ , (3) ( ) ( ) 0 ; ; 1 ; ; 1 1 c i P c i P * * ≤
Result (2) corresponds to the direct irreversibility effects usually encountered. The other results correspond to the supplementary effects and can easily be understood. Because there is a higher rate of emission for the pollutant concerned by the arrival of information, the marginal cost of private transport is increased compared to the reference situation and the optimal policy is to increase the proportion of public transport (Result (1)). In the process, the economic advantages from the pollution cleanup of the other pollutant (particles) are reduced, and the optimal policy is to reduce the level of reduction in emissions of this pollutant as well (Result (3)). The formal proof is contained in the Appendix.

The effects of irreversibilities and intertemporal externalities

Irreversibilities and investment costs have an effect comparable to that of information. If information arrives, 1 = g , irreversibility constraints lead to reducing simultaneously the proportion of private transport and the level of reduction of emissions.

Proposition 2: The introduction of irreversibility constraints leads to a lower proportion of private transport and lower levels of reduction in emission rates:

a) when these constraints concern the proportion of public transport:

, c ∀ (1) ( ) ( ), 1 ; ; 0 1 ; ; 1 1 1 c V c V * * ≤ (2) ( ) ( ) 1 ; ; 0 1 ; ; 1 1 1 c G c G * * ≤ , (3) ( ) ( ) 1 ; ; 0 1 ; ; 1 1 1 c P c P * * ≤ .
(b) when these constraints concern the rates of emission reduction: i

∀ (1) ( ) ( ), 1 ; 0 ; 1 ; 1 ; 1 1 i V i V * * ≤ (2) ( ) ( ) 1 ; 0 ; 1 ; 1 ; 1 1 i G i G * * ≤ , (3) ( ) ( ) 1 ; 0 ; 1 ; 1 ; 1 1 i P i P * * ≤
Irreversibility constraints introduce a problem of flexibility.

In the second period, if the information received indicates that environmental impacts are in the upper range, then the optimal policy will be to develop public transport. The irreversibility constraint concerning public transport will thus force the public decision-maker to take a more flexible decision in the first period, that is, to increase the proportion of public transport starting from the first period (Result (a-1)). Since the proportion of private transport is reduced, the economic advantages of pollution cleanup are also reduced, which explains Results (a-2) and (a-3).

Those irreversibility constraints which concern rates of emission reduction pose a problem when the information received indicates that the impacts of climate change are in the lower range. To increase flexibility and allow the GGEs reduction rate to be adjusted downwards, the public decision-maker has to limit this emission reduction rate, starting from the first period (Result (b-2)). The marginal cost of private transport in the first period is thus higher, which explains Result (b-1), indicating that the proportion of private transport should be reduced. As a final consequence, the economic advantages from pollution cleanup are also reduced, explaining Result (b-3). The formal proof is contained in the Appendix.

Discussion and Two Illustrative Examples

These results show that the presence of uncertainties has two effects. First, uncertainty strengthens the economic attractiveness of structural measures which reduce the need for private transport. Second (and conversely), it weakens the attractiveness of technological measures which reduce pollutant emissions. This runs contrary to the arguments of economic and industrial lobbyists, who generally prefer technological measures to structural measures which can pose a threat to their economic importance or competitiveness, and who use uncertainty to support the scenarios most favourable to themselves.

The presence of irreversibilities also adds to the attractiveness of structural measures. However, these irreversibilities are not necessarily taken into account in the right way during political decision-making. Countries currently planning, for example, to invest in the extensive development of electric vehicles, like France or the Netherlands, are not making fully explicit the irreversibilities that such a plan will entail in terms of electricity supply and generation.

In the model, we presumed that the public decision-maker was simultaneously setting the rates for emission of the pollutants, conferring on him/her a high level of rationality and an advanced level of sophistication in the treatment of the externalities. In practice, decisions are often taken in a less-integrated manner; the public decision-maker only considers one single dimension of the problem at a time. This way of working means that the optimisation problem is dealt with by acting only on the proportion of private transport ( t V ), and on one of the two rates of emission ( t G or t P ) -the other rate being considered as fixed. It can be conjectured that in many cases this truncated analysis may lead to reducing private transport less than optimal policy would suggest.

In practice, truncated analyses of this nature produce even worse consequences because the respective choices of rates of emission are not so independent. For example, in the case of the 'Eco-pastille', inciting people to purchase diesel vehicles only moderately reduces GGE rates, but considerably increases local pollutant emission rates. Box 1 quantifies the environmental consequences of the Eco-pastille in terms of external costs in the EU-15. The "urban sprawl" phenomenon described in Box 2 illustrates the disastrous economic and environmental consequences of government policy that encourages single-family housing without taking the associated externalities into consideration.

Box 1: The Eco-pastille for Cars in the EU-15

Tax policies on vehicles and fuel have proved effective in reducing the CO 2 emissions of a fleet (see for example [START_REF] Ryan | The impact of fiscal and other measures on new passenger car sales and CO2 emissions intensity: Evidence from Europe[END_REF]. Thus, the Eco-pastille or green incentive scheme operating in several European countries (Austria, Denmark, Spain, Italy, France, Portugal, the Netherlands, Sweden) bases an "environmental" tax policy solely on emissions of CO 2 . In doing so, it favours the vehicles with the lowest CO 2 emissions (especially diesel engines) but which produce highly damaging NO X or ultrafine particles that are particularly harmful to health. [START_REF] Mayeres | Should diesel cars in Europe be discouraged?[END_REF] pointed out that the cost in environmental terms of diesel vehicles was far higher than that of petrol vehicles, because of the high social cost associated with their particle emissions. The gains expected from lowering average emissions of CO 2 can easily be quantified and compared with the external costs expected from extra emissions of particles (and of NO X ), ignoring the other external costs (accidents, traffic congestion, noise, landscape disturbance due to roads) which obviously remain identical.

The weighted average emission rates for urban/non-urban road journeys in EU-15 are 0.045 g/veh-km for petrol vehicles and 0.2185 g/veh-km for diesel vehicles (Infras-IWW, 2004). For CO 2 emissions, the average emission rates of the current fleet are estimated at 165 g/km for petrol vehicles and 150 g/km for diesel vehicles.

The external cost of 1 kg of PM 10 emitted is estimated at €212 (Infras IWW, 2004), while a tonne of CO 2 is subjected to a sensitivity analysis: between €20 (a low estimate falling within the Kyoto objectives) and €140 (a high estimate yielding a reduction of 50% in 2030).

Thus, the external costs associated with CO 2 and with particles for 100 km travelled are €1.28 (€3.26 under the high estimate) for a petrol vehicle, compared to €4.93 (respectively €6.73) for a diesel vehicle, in other words roughly 4 (respectively 2) times less. Projecting this on the average number of kilometres travelled by a vehicle over its lifetime (200,000 km according to European Union, 2007), then this represents a difference of some €7,300, rising to €10,200 if NO X emissions are included at the values given by European Union (2007). These orders of magnitude are confirmed by [START_REF] Mailbach | Handbook on estimation of external costs in the transport sector[END_REF], who perform a similar analysis simultaneously taking into account NO X , non-methane volatile organic compounds (NMVOC), SO 2 and PM 10 . Using data on emissions from the German car fleet, they obtain an external cost 4 times higher for diesel vehicles than for Euro 5-standard petrol vehicles, whatever the engine size.

So the Eco-pastille system actually provides support for the boom -and technological progress -in vehicles that create an environmental footprint, through their emissions of local pollutants (particles), which is two to four times more harmful, in exchange for a 10% gain on CO 2. A paradoxical consequence of this dieselisation of the fleet is that GGEs from petrol vehicles decreased by 17% between 1990 and 2005 for EU-15, while those from diesel vehicles increased by 83% (source EEA 2007).

The European Union recently addressed the problem by drawing up a proposal for a directive obliging the authorities in Member States, when they buy transport vehicles, to take into account not only the exploitation costs linked to energy consumption and CO 2 emissions, but also those linked to NO X , NMVOC and particle pollution (European Union, 2007). This obligation is intended to take effect from 1 st January 2012 -if indeed it is adopted. Meanwhile, certain countries are applying more judicious Eco-pastille policies. The Netherlands, for example, requires all new diesel vehicles to have a particle filter, and since 2008 some German cities (like Berlin, Cologne and Hanover) have been regulating traffic in certain zones, according to emission standards.

Box 2: Urban Sprawl and the "House for €15 per Day"

Since the beginning of the 1970s, households have been moving out of most European city centres and inner suburbs into peri-urban zones. Yet planners and environmental researchers alike point to the negative impact of this urban sprawl, which increases the length of journeys between home and workplace (see for example [START_REF] Bartholomew | Land use-transportation scenario planning: promise and reality[END_REF]. Certain policies encouraging single-unit housing promote this phenomenon. For example, the so-called "house for €15 a day" offered in France aptly illustrates the negative consequences of a lack of overall analysis of externalities. In this box, we estimate the extent of the external effects, in terms of emissions of CO 2 and particles, that this policy could generate.

The private vehicle of a household resident in a peri-urban zone travels on average 25,300 km/yr, as compared to 13,700 km/yr for a household living in an urban or suburban area (average weighted by zones of residence, Insee, 2009). In view of the findings from Box 1 and the current composition of the French car fleet (50% are diesel vehicles and the proportion is increasing constantly), the external cost relative to atmospheric emissions can be estimated at €3,670 over 10 years (respectively €5,870, using the upper range for a tonne of CO 2 ). When the external costs linked to accidents, noise and traffic congestion (€0.0614 per vehicle per km, Infras IWW, 2004) are added in, the total estimate is as high as €10,800 (respectively €13,000) over 10 years.

Over and above the external effects, the supplementary private expenditure of someone purchasing this kind of house represents on average €4,570 over 10 years in fuel costs [START_REF] Lemaître | Dépenses de carburant automobile des ménages : relations avec la zone de résidence et impacts redistributifs potentiels d'une fiscalité incitative[END_REF], added to which are the effects of lower energy efficiency in the home. Thus, expenditure on household energy weighted by type of heating is twice as high when the dwelling is single-unit (6.7% of household budget) than when it is multi-unit (3.34%) [START_REF] Marcus | Effet sur les prix et le pouvoir d'achat des ménages[END_REF], i.e. some €6,000 difference over 10 years according to [START_REF] Grosmesnil | La consommation d'énergie à usage domestique depuis quarante ans[END_REF].

In total, the extra cost associated with a "house for €15 a day" in a peri-urban zone is €20,000 over 10 years, and this is bound to rise with the application of the "climate and energy contribution" tax (CCE), which has not yet been implemented, but which is set to be €100 in 2030 (Conférence des experts sur la contribution Climat et Energie, 2009). It should be noted that under its affordable housing legislation, the French government intends to extend this measure to apartments, which would mitigate the analysis in this Box.

Thus, externalities linked to the effects of emissions from road transport are the object of varying uncertainties, depending of what is being evaluated: uncertainties are lessened when the reduction envisaged is global rather than concentrated on one pollutant. During the phase of internalisation, therefore, it is useful to think about the effects of the policy chosen: does it act on overall externalities (structural policy) or does it target only one of them (technological policy)?

Conclusion

We have analysed the choice of a policy to reduce GGEs and local pollutant emissions linked to transport in a model of sequential decision-making under uncertainty. We have looked at two types of policy, structural policies acting on overall emissions and technological policies targeting certain pollutants. We have examined the effects of the arrival of information and of the irreversibility of the policies adopted on the optimal decision. One of our findings is that jointly taking both pollutants into account weakens the attractiveness of a policy aimed at reducing local (particle) emissions alone and, instead, strengthens the attractiveness of a structural policy aimed at limiting private transport in general. Some promising avenues for future research would be to take into account explicitly the uncertainties concerning the consequences of GGEs and the costs associated with particle pollution, as well as to investigate the question of the irreversibility linked to stocking these gases.

( ) ( ) ( ) ( ) . if 0 . ) ( . or . if 0 . ) ( . . if 0 . ) ( . or . if 0 . ) ( . . if 0 ) 1 ( 1 . 1 . . . or . if 0 ) 1 ( 1 . 1 . . . 1 2 2 2 2 1 2 2 2 2 1 2 2 2 2 1 2 2 2 2 1 2 2 2 2 2 2 1 2 2 2 2 2 2 P c P V P E V P c P V P E V G c G V G D V G c G V G D V V i G V C P G P G V i V V C P G P G = ≥ - ′ + > = - ′ + = ≥ - ′ + > = - ′ + = = - ′ - - + - + + + > = - ′ - - + - + + + φ γ φ γ θ β θ β φ θ γ β α φ θ γ β α
We show first that the 2 nd period optimal choices increase with the 1 st period choices 1

V , 1 G and 1 P .

(a) Assume

1 1 V V > . Let
; ; 2 1 1 1 2 2 1 1 1 2 2 1 1 1 2 2 1 1 1 2 2 1 1 1 2 2 1 1 1 2 * * * * * * * * * * * * = ≥ = = ≥ = = ≥ = P P G V P P P G V P G P G V G G P G V G V P G V V V P G V V • Suppose ad absurdum that we have * * < 2 2 V V , * * ≤ 2 2 G G
and * * ≤ 2 2 P P

. Then, we show that:

( ) ( )

0 ) 1 ( 1 . 1 . . . 2 2 2 2 2 < - ′ - - + - + + + * * * * * V C P G P G φ θ γ β α Since 1 1 2 2 V V V V > ≥ > * * , ( ) ( ) 0 ) 1 ( 1 . 1 . . . 2 2 2 2 2 = - ′ - - + - + + + V C P G P G φ θ γ β α Given that θ β ≤ , the assumption that * * ≤ 2 2 G G implies that ( ) ( ). 1 . . 1 . . 2 2 2 2 * * * * - + ≤ - + G G G G θ β θ β Similarly, we have ( ) ( ). 1 . . 1 . . 2 2 2 2 * * * * - + ≤ - + P P P P φ γ φ γ
The assumption that

* * < 2 2 V V implies that ) 1 ( ) 1 ( 2 2 * * - ′ > - ′ V C V C
. Therefore, we have:

( ) ( ) ( ) ( ) 0 ) 1 ( 1 . 1 . . . ) 1 ( 1 . 1 . . . 2 2 2 2 2 2 2 2 2 2 = - ′ - - + - + + + < - ′ - - + - + + + * * * * * * * * * * V C P G P G V C P G P G φ θ γ β α φ θ γ β α
which proves that the optimality conditions are not satisfied.

• Note that * * < 2 2 V V and * * > 2 2 G G
is not feasible, since then we would have .

0 . ) ( . . ) ( . 2 2 2 2 2 2 ≥ - ′ + > - ′ + * * * * * * V G D V V G D V θ β θ β But since 1 2 2 .G c G G ≥ > * *
, the optimality conditions imply that 0 . ) ( .

2 2 2 = - ′ + * * * V G D V θ β
which contradicts the hypothesis. A similar argument can be used to prove that

- ′ + < - ′ + V G D V V G D V θ β θ β Because 1 2 .G c G ≥ * , we have 1 2 .G c G > *
, and the optimality conditions imply that 0 . ) ( .

2 2 2 = - ′ + * * * V G D V θ β . We therefore have 0 . ) ( . 2 2 2 < - ′ + * * * V G D V θ β
, which proves that the optimality conditions cannot be satisfied. We can show that * * ≥ 2

2 V V et * * < 2 2 P P
is not feasible using a similar argument.

On the whole, we obtain that *

* ≥ 2 2 V V , * * ≥ 2 2 G G and * * ≥ 2 2 P P . (b) Assume 1 1 G G > . Note that if 1 2 .G c G ≥ * , then , 2 * V * 2
G and * 2 P satisfy the optimality conditions when the first period choices are ( )

1 1 1 ; ; P G V . Consequently * * = 2 2 V V , * * = 2 2 G G
and * * = 2 2 P P .

Let us consider the case where

1 2 1 . . G c G G c ≥ > * . Since 1 2 .G c G ≥ * , then * * > 2 2 G G . Assume that we do not have * * ≥ 2 2 V V
, and that instead we have * * < ≤

2 2 1 V V V . Since φ γ ≤ , then * * * * * * - ′ + > - ′ + 2 2 2 2 2 2 . ) ( . . ) ( . V P E V V P E V φ γ φ γ
+ + + - + - + - + 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ). ( ) ( ) ( . . . . ) ( ) 1 ( . ). ( ) ( ) ( . . . . ) ( ) 1 ( . V P E G D V P V G V C V V P E G D V P V G V C V φ θ φ γ θ β α φ θ φ γ θ β α whereas ( ) ( ) * * * * * * * * * * * * * * * * * * + + + + - + - + - + < + + + + - + - + - + 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ). ( ) ( ) ( . . . . ) ( ) 1 ( . ). ( ) ( ) ( . . . . ) ( ) 1 ( . V P E G D V P V G V C V V P E G D V P V G V C V φ θ φ γ θ β α φ θ φ γ θ β α
Thus, by reduction, we would have: *

* * * * * * * - + - < - + - 2 2 2 2 2 2 2 2 . . ) ( . . ) ( . . ) ( . . ) ( V G V G V G V G θ β θ β θ β θ β and since 0 ≤ -θ β , we would have * * * * * * * * + ≥ + 2 2 2 2 2 2 2 2 . . . . . V G V G V G V G or ( )( ) 0 . 2 2 2 2 ≥ - - * * * * V V G G which contradicts * * > 2 2 G G and * * < 2 2 V V . Yet, since * * ≥ 2 2 V V
, we also have * * ≥ 2 2 P P which proves that the 2 nd period choices increase with 1 G .

A similar argument can be used to prove that the 2 nd period choices increase with P 1 .

Step 2 : Property of the 2 nd period value function.

We will show that the 1 st order derivatives of the function ( )

c i P G V J ; ; ; ; ; ; 1 1 1 φ θ
with respect to the 1 st period choices are positive. We have:

∂J ∂V 1 = α + β.G 2 * + γ.P 2 * + θ. 1-G 2 * ( ) + φ. 1-P 2 * ( ) -′ C (1-V 2 * ) ( ) ∂V 2 * ∂V 1 + β.V 2 * + ′ D (G 2 * ) -θ.V 2 * ( ) ∂G 2 * ∂V 1 + γ.V 2 * + ′ E (P 2 * ) -φ.V 2 * ( ) ∂P 2 * ∂V 1 If ( ) 0 . ) ( . 2 2 2 > - ′ + * * * V G D V θ β , then 1 2 .G c G = * and therefore 0 1 2 = ∂ ∂ * V G . Consequently, ( ) 0 . ) ( . 1 2 2 2 2 = ∂ ∂ - ′ + * * * V G V G D V θ β and similarly ( ) . 0 . ) ( . 1 2 2 2 2 = ∂ ∂ - ′ + * * * * V P V P E V φ γ If ( ) ( ) 0 ) 1 ( 1 . 1 . . . 2 2 2 2 2 > - ′ - - + - + + + * * * * * V C P G P G φ θ γ β α then 1 2 .V i V = * and therefore i V V = ∂ ∂ * 1 2 . Thus ( ) ( ) ( ) 0 ) 1 ( 1 . 1 . . . . 2 2 2 2 2 1 ≥ - ′ - - + - + + + = ∂ ∂ * * * * * V C P G P G i V J φ θ γ β α Similarly, we have ( ) ( ) 0 . . ) ( . 0 . . ) ( . 2 2 2 1 2 2 2 1 ≥ - ′ + = ∂ ∂ ≥ - ′ + = ∂ ∂ * * * * * c V P E V P J c V G D V G J φ γ θ β
Step 3: Proof of proposition 1. • First, suppose that ( ) . This implies that one of the following three values have to satisfy, is not satisfied.

For P G V x , , = , ( ) ( ) * * * * * * = = x c i P G V x c
, 1 ; ; 1 1 * * * > = V V c i V ( ) * * * ≥ = G G c i G
• Suppose that ( )

, 1 ; ; 1 1 * * * > = V V c i V ( ) * * * < = G G c i G
; ; P G V
is not the optimal solution.

• Similarly, if we assume that ( ) . Note that this assumption cannot hold, since then we would have 0 . ) ( .

, 1 ; ; 1 1 * * * > = V V c i V ( ) * * * < = G G c i G
1 1 1 > - ′ + * * * V G D V θ β and because 0 1 ≥ ∂ ∂ G J , 0 . ) ( . 1 1 1 1 > ∂ ∂ + - ′ + * * * G J V G D V θ β Similarly, ( ) * * * ≤ = V V c i V

  Let us assume that proposition 1 is not true, and see what happens.

  strictly positive. Since we have shown that the 1 st order derivatives of J are positive, then one of the optimality conditions that

  We sketch out the proof for result (a). Result (b) can easily be proven using a similar argument. Let us start with the benchmark case 0 binding for *

  10

•

  Pollution cleanup costs are linked to technological measures and can be broken down

	into two parts. The linear part (	t G . β +	γ	.	t P	) t V .	represents variable costs where β (γ ) is
	the unit cost per km travelled of a total reduction in GGEs (particle emissions). The
	non-linear part	D	( G t	)	+	E	(	t P	)	represents the fixed costs which are taken to be
	positive, increasing and convex.

  us assume that we do not have

	and	( ( (	; ;	; ;	) ) )	( ( (	; ; ;	; ; ;	) ) ) . ,

Greenhouse gases include CO 2 , CH 4 , N 2 O, CF 4 , C 2 F 6 and SF 6 . They are usually expressed in CO 2 equivalent or global warming potential for the purposes of analysis.

GGEs from the transport sector increased by 25.5% between 1990 and 2005, while emissions from all other sources decreased by 7% over the same period.

Up until the introduction of unleaded petrol, "atmospheric lead content" acted as a good measure of automobile-generated pollution. The disappearance of lead did not, however, lead to the disappearance of automobile-generated pollution, nor of the associated health hazards.

However, from the point of view of those people affected by particle pollution, it can appear irreversible insofar as it can cause premature death.

Since the public decision-maker is the only agent explicitly considered in this model, this model reduces to an individual decision-making model.

The results of a study by the European EnvironmentAgency (2000) on the evolution of energy efficiency in transport since the beginning of the 1970s are instructive. Although engines have become more fuel-efficient (using roughly 30% less fuel), the energy used per passenger-kilometre in car transport has remained constant. The fact is that this period has witnessed increased traffic congestion, fewer passengers per car, increased car weight (in the same categories), and shorter average car journeys.

To suppose that private transport costs are linear (αV t ) is probably unrealistic. However, it should be noted that a convex function in (1-V t ) is formally equivalent to a concave function in V t . Hence, the function C(.) can formally take account of concave costs for private transport, which is fairly realistic (amortisation of vehicle purchasing costs by households decreases with the total number of kilometres travelled, for example).

* Authors thank two anonymous referees for helpful comments and suggestions and Marjorie Sweetko for her translation of this article into English. They also tank participants at the University Paris 1 seminar, 2009 for comments on an earlier version of the paper. Financial support of French Research National Agency (ANR-08-RISKNAT-007-01 RISKEMOTION) is gratefully acknowledged.

Appendix

Proof of Proposition 1.

Step 1: Properties of 2 nd period choices

These optimal solutions are functions of ( )