Gianluca Rosso
email: gianluca.rosso@sis-statistica.org

( Gradstat

Identification of Risk Extreme Values in a Time Series and Analysis with an Autoregressive Method Application for Climate Risk Events

Keywords: Time series analysis, trend analysis, peaks identification, outliers, peak over threshold POT, linear regression, autoregression, OLS methods, Excel, risk management, operational risk, risk assessment, extreme events, extreme values, climate events, psychology of loss, loss aversion, future discounting, Giddens's paradox

In this article there is no intention to repeat basic concepts about risk management, but we will try to define why often is usefull the time series analysis during the assessment of risks, and how is possible to compute a significative analysis using regression and autoregression. After some basic concepts about trend analysis, will be introduced some methods to identify peaks. This is often usefull when there is no need to use the full time series, because sometimes is more practical to focus only on the extremes. With a correct time series without "not-anomalous" data, the extremes time series are treated with a simply autoregression model. This drives to know if the time series has a correlation between periods, and how many periods could be considered lagged among them. We think that climate events frequently are lagged because the climate show a clear increasing tendency, and that climate risks are potentially increasing during the time. There will be no specific conclusion related with risk management, because the proposed solution with autoregression can be adapted to any time series analysis.

balance is a psychology approach, well known as "psychology of loss". The mile stone of this approach is the loss aversion. Loss aversion is a human characteristic that describes how people are intrinsically afraid of losses. Loss aversion unavoidably leads to risk aversion and a number of predictable behaviours in certain situations. Loss aversion and risk are intrinsically linked. Research into the psychological value drives to the weight that people give to probabilities. And here is again the statistical element described more over. Where the odds of an event are very small, people become almost completely indifferent to variations in levels of risk. In particular people are more heavily influenced (in terms of weighting of probabilities) if an event is described by using frequencies than by using standard indicators of probability of risk. Loss aversion and related biases are an important drivers of human decision making in many situations. On the one hand it can encourage extreme risk averse behaviour, and yet in a different situation it can lead to excessive risk taking. These are heavy evaluation elements in order to assess extreme events, as in climate. Traditionally time series are the basis of any climate analysis. There is only mathematical and statistical reasons? Yes, if you refers to letterature. But probably is possible to add an element in order to make stronger this approch. The element is the loss aversion and the tendency of people to have a bias in favour of the "status quo" because they are more concerned about losses than about future gains. A tendency that is related to "future discounting", a basic concept related with the "Giddens's paradox". It states that in the course of day-to-day life a great number of risks, and climate risk are in that order, many will sit on their hands and do nothing of a concrete nature about them. Yet waiting until they become visible and acute before being stirred to serious action will, by definition, be too late.

The key of this situation is that if people have troubles to see forward, researchers must operate in order to let the people see backward. And from the backward analysis, the time series analysis, we can see then forward. This because analysis of time series can drives automatically to the analysis of trend. This is a natural approach to the study, and there is no other way forward. When we look to natural extreme events we approach to an argument drammatically important for humanity. Social, economic and moral implications are so big that its study will be one of the most important challenge for the next decades. For example, global annual temperatures in last ten years seems to be in average quite stable. But if we wide our view and we take a look to the last 30 years, we could have a point of view very different. Because we can see clearly the tendance: temperature is growing up. It's true that we can observe at least two periods in which temperature are stable, but data mustn't be treated as isolated elements. They must be analized as a time series. It's the natural variability that could show every single period as a record, in positive or in negative. Every portion of our time series could be anomalous, and therefore could contain an extraordinary event.

As wrote in many papers, as said in many news, the problem is not only the amount of the event but also the frequency of occurrence.

There are many ways to analyze these data; in this article we propose a path to have a solution. After some basic concepts about trend analysis, will be introduced some methods to identify peaks. This is often usefull when there is no need to use the full time series, because sometimes is more practical to focus only on the extremes. With a correct time series without "not-anomalous" data, the extremes time series are treated with a simply autoregression model. This brings to know if the time series has a correletion between periods, and how many periods could be considered lagged. We think that single climate events are lagged because the climate show a clear tendency and that climate risks are potentially increasing during the time. At the basis of any analysis of extreme indicators lies the estimation of trends. There are two definitions for "trend". In much of the statistical literature a trend is conceived of as that part of a series which changes relatively slowly (smoothly) over time. Viewed in terms of prediction, a trend is that part of the series which, when extrapolated, gives the clearest indication of the future long-term movements in the series. In many situations these definitions will overlap. But not in all situations. In case of data following a random walk, the latter trend definition does not lead to a smooth curve. Typical examples of the first definition are splines, LOWESS smoothers and Binomial filters. Typical example of the second definition is the IRWtrend model in combination with the Kalman filter. If we scan the climate literature on trend methods, an enormous amount of models arises. We found the following trend models or groups of models (without being complete): low pass filters (various binomial weights; with or without end point estimates), ARIMA models and variations (SARIMA, GARMA, ARFIMA), linear trend with OLS (ordinary least square), kernel smoothers, splines, the resistant (RES) method, Restricted Maximum Likelihood AR, based linear trends, trends in rare events by logistic regression, Bayesian trend models, simple Moving Averages, neural networks, Structural Time-series Models (STMs), smooth transition models, Multiple Regression models with higher order polynomials, exponential smoothing, Mann-Kendall tests for monotonic trends (with or without correction for serial correlations), trend tests against longmemory time series, robust regression trend lines (MM or LTS regression), Seidel-Lanzante trends incorporating abrupt changes, wavelets, Singular Spectrum Analysis (SSA), LOESS and LOWESS smoothing, Shiyatov corridor methods, Holmes double-detrending methods, piecewise linear fitting, Students t-test on sub-periods in time, extreme value theory with a time-varying location parameter and, last not but least, some form of expert judgment (drawing a trend "by hand"). However, the number of trend models applied to extreme indicators, appears to be much more limited. The trend model almost exclusively applied, is the OLS straight line. This model has the advantage of being simple and generating uncertainty information for any trend difference [µt -µs ] (indices "t" and "s" are arbitrary time points within the sample period). The OLS regression model reads as yt = µt + "t = a + b*t + "t (2.1) with "a" the intercept, "b" the slope of the regression line and "t a noise process.

Other examples of OLS linear trend fits can be found in linear trend estimation in combination with ARIMA models for the residuals. In the field of disaster studies OLS trends are the dominant method, albeit that the original data are log-transformed in most cases. Hu et al. (2012) apply Mann-Kendall tests with correction for serial correlation (no actual trend estimated in this approach). Finally, some authors acknowledge that the use of a specific trend model, along with uncertainty analysis, may lead to deviating inferences on (significant) trend changes. Therefore, they chose to evaluate trends using more than one trend model. For example, Moberg and Jones apply two different trend models to the same data: the OLS trend model and the resistant (RES) model. Subsequently, they evaluate all their results with respect to these two trend models. Even more methods are evaluated by Young et al. (2011). They estimate five different trend models to 23-yr wind speed and wave height data and evaluate uncertainty information for each model (their supporting material). We note that the application of more than one trend model to the same data has been published more often (not specifically for the evaluation of extremes).

IDENTIFING PEAKS.

There are two methods in identifying movements of the extreme value are taking the maximum value in a given period (called Block Maxima method) and retrieve the values that pass through a given threshold (called Peaks Over Threshold method).

A dataset of events is given, and it is possible to draw it with a scatterplot (Fig. 3.1). Generalized Extreme Value (GEV) distributionis a family of continuous distribution is built into the EVT to combine the Gumbel, Frechet, and Weibulldistribution, known as the extreme value distribution of type I, II, and III. On the GEV distribution, X is the random variable which has Probability Density Function (PDF) is as follows µ is a parameter of location; σ is a parameter of scale; ξ is a parameter of shape.

Shape parameter ξ determines the behavior of the tail distribution. Distribution type defined with ξ = 0 , ξ > 0 , and ξ < 0 and can be likened to the Gumbel, Frechet, and Weibull distribution.

One of the methods in identifying movements of the extreme value is Block Maxima, which identifies the extreme value based on the value of maximum observation data are grouped according to certain periods. In this method, observation data is divided into blocks in a certain period, example monthly, quarterly, semester, or year. Then for each block is determined the magnitude of observation data and the value is the maximum value of the extremes for each block and were used as the sample (Fig. 3.2).

The POT (Peak Over Threshold) method involves choosing some threshold value, collecting the extreme values above that threshold into a sample, and then drawing conclusions based on that sample. The extreme values are taken as the peak values between two distinct upcrossings. To account for variation in the particular time series chosen, it is beneficial to generate or record multiple data for the same length of time and then form the extreme value sample from the peaks over threshold of all the time series (fig. 3.3).

In our example is shown in Fig. 3.4. Let introduce an important relation between this method of analysis and the regression. This relation will be important for later consideration.

In the scatterplot above we have a distribution of items that has no particulary tendencys. Is a partial indefinite cloud of dots. But if we have a more definite trend of data, we should consider two approach. With a constant threshold the graph is not much different from the other above (Fig. 3.5), but in a quantile regression the aspect of our analysis assume a different perspective (Fig. 3.6). About the exceeding of the value over a threshold, is possible to calculate the probability of the loss during the time. We refer to a Keith Porter work1 . This function is called risk curve.

(3.2)

where -X = uncertain degree of loss to an asset, such as the uncertain damage factor -x = a particular value of X -s = a particular value of the environmental excitation -R(x) = annual frequency with which loss of degree x is exceeded -G(s) = the mean annual frequency of excitation exceeding intensity s

-P[X ≤ x | S = s] = cumulative distribution function of X evaluated at x, given s. If X is lognormally distributed at S = s then ܲሾܺ ≤ ܵ|ݔ = ݏሿ = Φ ቆ ୪୬ ( ೣ ഇ(ೞሻ ሻ ఉ(௦ሻ ቇ (3.3)
where -θ(s) = median vulnerability function, i.e., the value of the damage factor with 50% exceedance probability when the asset is exposed to excitation s -v(s) = coefficient of variation of vulnerability, i.e., the coefficient of variation of the damage factor of the asset exposed to excitation s -β(s) = logarithmic standard deviation of the vulnerability function, i.e., the standard deviation of the natural logarithm of the damage factor when the asset is exposed to excitation s.

About the environmental excitation, a correct definition we can give is that it it is usually the force, deformation, or other degree of loading to which the asset is subjected. In climate events, every measurable force is potentially an environmental excitation. About floods, flood depth or momentum flux or a vector combination of the two at the perimeter of an asset such as a building qualify as excitation by the environment on the asset. The US Army Corps of Engineers uses these measures as inputs to their flood vulnerability functions.

If one has the mean vulnerability function y(s) and coefficient of variation of loss as a function of excitation v(s), to evaluate ߠ(s) and ߚ(s) using the following most general formulas

ߚ = ඥln (1 + ݒ ଶ ሻ (3.4) ߠ = ఓ √ ଵା௩ మ (3.5)
Equation (3.2) can be numerically integrated by

ܴ(ݔሻ = ቆ ିଵ ܩሻݔ( ିଵ ൫1 -݉(ݔ݁ Δݏ ሻ൯ - (௫ሻ ௦ ܩ ିଵ ݉(ݔ݁ቀ Δݏ ሻ ቀΔݏ - ଵ ቁ + ଵ ቁቇ = ∑ ( ିଵ (ݔሻ ⋅ ܽ -∆ (ݔሻ ⋅ ܾ ሻ ୀଵ (3.6)
 (ݔሻ = ܲሾܺ ≤ ܵ|ݔ = ݏ ݅ ሿ = Φ ቆ ln ( ݔ ߠ൫ݏ ݅ ൯ ሻ ݏ(ߚ ݅ ሻ ቇ (3.7) then ∆ (ݔሻ =  (ݔሻ - ିଵ (ݔሻ (3.8)
and

ݏ∆ = ݏ -ݏ ିଵ (3.9) ݉ = ݈݊ ቀ ீ ீ షభ ቁ ݏ∆/ (3.10) for i=1,2,…n 4. REGRESSION.
Just a few words about regression, to introduce the autoregression model. Linear regression rappresents an expected value estimation method, conditioned by a dependent variable Y, given values of others independent variables

ܺ ଵ , … , ܺ : ॱሾܻ|ܺ ଵ , … , ܺ ሿ (4.1)
These values are defined as regressors.

The linear regression model is

ܻ = ߚ + ߚ ଵ ܺ + ݑ (4.2)
where:

-i change between observations, i = 1, ...., n; -ܻ is the dependent variable; -ܺ is the independent variable or regressor; -ߚ + ߚ ଵ ܺ is the regression line or population regression function; -ߚ is the intercept value of the regression line; -ߚ ଵ is the slope of the regression line; -ݑ is the random error. Extimated regression model is

ܻ = ߚ መ + ߚ መ ଵ ܺ (4.3)

With

Identification of Risk Extreme Values in a Time Series and Analysis with an Autoregressive Method, Application

for Climate Risk Events -Gianluca Rosso ( 2016)

ₐє 10 ߚ መ ଵ = ∑( ି ሻ( ି ሻ ∑( ି ሻ మ (4.4) ߚ መ = ܻ -ߚ መ ଵ ܺ (4.5)
In some cases the regression can be considered already sufficient enought to determine the trend. But sometimes the Mann-Kendall2 test must be properly executed after the regression, that is the basis for the analysis. Shortly, we could say that linear regression is usefull to estimate tendency of the parameters, and Mann-Kendall test to estimate significance of the tendence. The choice about only regression or regression plus Mann-Kendall test depends mainly on the modeling of the investigation to be carried out, and then some conditions such as the availability of data, timing, costs.

dependent upon: The magnitude of data, Assumptions of distribution (does not have to have a normal / bell shape distribution), Missing data or Irregularly spaced monitoring periods. Mann-Kendall assesses whether a time-ordered data set exhibits an increasing or decreasing trend, within a predetermined level of significance.

Using one of the test, or using a combination of both, five types of trend tests are possible (as presented in the following table )   The table uses the following notation: -Y = the random response variable of interest in the trend test, -X = an exogenous variable expected to affect the value of Y, -R = the residuals from a regression or LOWESS of Y versus X, and -T = time (often expressed in years). Mann-Kendall is frequently used for analysis on climate time series, which is the argument of this article. So if we refers to a trends in flood flows: Y would be streamflow, X would be the precipitation amount, and R would be called the precipitation-adjusted flow (the duration of precipitation used must be appropriate to the flow variable under consideration. For example, if Y is the annual flood peak from a 25 km 2 basin then X might be the 1-hour maximum rainfall, whereas if Y is the annual flood peak for a 25000 km 2 basin then X might be the 24-hour maximum rainfall). Mann-Kendall test is related with a statistical hypothesis testing. The null hypotesis H0 is that there is no trend in the time series. The three alternative hypotesis are that there is a negative trend, a non-null trend, or a positive trend.

The hypothesis testing is performed according to a predetermined level of significance, in a manner that the test assumes meaning or not. A p-value less than the predetermined significance indicates that the null hypothesis is rejected, i.e. that there is a trend in our time series.

Identification of Risk Extreme Values in a Time Series and Analysis with an Autoregressive Method, Application for Climate Risk Events -Gianluca Rosso (2016) 5. AUTOREGRESSION.

BASIS.

We state that data at particular point of time are (probably highly) correlated with the value that precede and/or succeed. The creation of an autoregressive model generates a new predictor variable by using the Y variable lagged one or more periods. (5.4)

A second-order autoregressive model considers the effect of relationship between consecutive values in a series as well as the correlation between values two periods apart. (5.5)

The selection of an appropriate autoregressive model is not an easy task.

Identification of Risk Extreme Values in a Time Series and Analysis with an Autoregressive Method, Application for Climate Risk Events -Gianluca Rosso ( 2016) 娐Ή Once a model is selected and OLS method is used to obtain estimates of the parameters, the next step would be to eliminate those parameters which do not contribute significantly.

= Z

If the null hypothesis is NOT rejected we may conclude that the selected model contains too many estimated parameters. The highest-order term then be deleted an a new autoregressive model would be obtained through least-squares regression. A test of the hypothesis that the "new" highest-order term is 0 would then be repeated. This testing and modeling procedure continues until we reject H 0 . When this occurs, we know that our highest-order parameter is significant and we are ready to use this model. Refering to the paper wrote about extreme events in Italy (Climate events and insurance demandthe effect of potentially catastrophic events on insurance demand in Italy, G. Rosso, A. Chieppa, A. Ricca, with C. D. Pronzato and I. Pecetto, 2014). We used a complete database with daily climate observations. The database was processed and modified in order to have: first, a smaller monthly database, second, a set with the only extreme events. This database is shown in tab. 5.2.1.1. It represent extrem preciptation events in last eleven years in Italy. The first event assume the "month" number 1, so that di other events have a month numbered with a properly "jump", usefull to draw correctly in a chart, and to calculate a frequency and the "distance" between each single event.

The tab. 5.2.1.1 show also the amount of precipitation in millimeters. You can note that there are no events with precipitations less than 100 mm. This was the threshold assumed in the past research. Substantially, in that research the threashold was linear, as shown in fig. 3.4. Because the study of the regression, seems to be incompatible with charts like 3.5 or 3.6. This is the minimum dataset required for an autoregression analysis. said, you can note that the column "month" jumps. In Our case this is not a problem because the autoregression model correltes the observations among them, properly shiftet for the number of periods we want to analize. But there could be cases in which also the"empty" observation plays an important role. This could be for reinsurance, where the threshold represent the excess amount and the zero items could have significance if the Company payed nothing). We could explain this approach with a very simple example. Let the series x(t) be observed, with t = (1, … , 12) Using the POT peaks over threshold method on order to save the only observations exceeding 100 of value, the result is shown in the following Tab. 5.2.1.3. As said, this dataset can be considerend in two different ways: What we need is to consider these events consequential among them. The Tab A is good for our scopes, and easy treatable with the Autoregression method. With this dataset we can draw a scatterplot and we can conduct a prior analysis in order to evidence if a trend is present in our time series. The scatterplot of data shows an increasing trend (Fig. 5.2.1.3).

month
The regression was calculated as said in par. 4, and the equation of the regression line is y = 14,78x + 189,14 (5.2.1.1)

One of the condition in order to make a good autoregression analysis is that we need to detrend our series; one of the reason is to be able to obtain meaningful sample statistics such as means, variances, and correlations with other variables. Such statistics are useful as descriptors of future behavior only if the series is stationary. For example, if the series is consistently increasing over time, the sample mean and variance will grow with the size of the sample, and they will always underestimate the mean and variance in future periods. And if the mean and variance of a series are not well-defined, then neither are its correlations with other variables. For this reason you should be cautious about trying to extrapolate regression models fitted to nonstationary data. But many analysis uses the original time series not detrended. The reason could be found in the values contained in the database, which propriety and behaviour drives to prefer this kind of analysis. We will use the autocorrelation for both, trended and detrended time series; and we will compare the results.

Many alternative methods are available for detrending. Simple linear trend in mean can be removed by subtracting the least-squares-fit straight line calculated with regression. Therefore the equation (5.2.1.1) is usefull to do detrending procedure. Many statistical software do this operation automatically, but even in Excel a simply formula can resolve this problem. For time series trends, Excel treats the equation as a function of observation number, so that the first observation is 1, the second is 2; and so on. It is necessary to add a column with these observation numbers to support the trend calculations. Figure 5.2.1.3 shows the method of calculation, with the formula used for detrending. It's easy to recognize into the formula the parameters of the regression line. The new detrended series is then shifted for three periods. These data will be used in autoregression analysis. The scatterplot of the detrended series is pictured in figure 5.2.1.5. The first step to use the autoregression model is to replicate all the independent variables, one for each period you think usefull for the analysis. The key is to copy data near the original data, but one-period shifted. As said, this operation can be replicated even if you have di amount of shifted periods you want to analyze. In Fig. 5.2.1.6 you can see the result: the Tab 5.2.1.1 integrated with the series detrended from the Fig. 5.2.1.5, plus the shifting action to generate a numbers of other variables usefull to perform an adeguate autoregressive analysis. The data at the bottom of the table will be never used. The data at the top will be used in correletion to the period of the autoregression you need to perform.

In this model are used detrended data calculated (Fig. 5.2.1.4), but the procedure is the same even with original data not detrended. Analysis for one period with the variable "detr" as dependent and X1 as independent.

Note that because the shifting, the first value of "detr" is not considered. The regression output is typical. All outputs are showed as attachments, both detrended and trended series.

The way to add and examine more periods is easy and is showed in Fig. 5.2.1.8. At the top of the Excel sheet are data that will be used only if needed in relationship with the regressors at the right of the original data. Every single period i of the autoregressive method AR(i) it's none other than a single multiregression, with the exception on the first one that is a single regression with a dependent and an independent variable. the method to read regression output, here are few indications. R 2 is between 0 and 1 and indicates the amount of variation of y i around y bar (its mean) that is explained by the regressors. The ANOVA (analysis of variance) table splits the sum of squares into its components:

Total sums of squares = Residual (or error) sum of squares + Regression (or explained) sum of squares.

Thus

Σ i (y i -ybar) 2 = Σ i (y i -yhat i ) 2 + Σ i (yhat i -ybar) 2 (5.2.1.2) Standard deviation of data is 50,9 (form Descriptive Statistics). The standard error of the autoregression (52,17) and the t Stat (0,79) seems to tell us that the data are not so lagged as we expected. Also autoregressions with more periods drives to the same conclusions. The assumption was that climate extreme events can have a relationship, a lagged effect, because there is an element, or more than one, that influence results, so that the result of one time period tends to spill over into the next period or periods. We know that probably exogenous effects are influencing frequency and intensity of events, and that seems to be real an accumulation situation.

But the analysis with a detrended series is so borderline, even with a conclusion to a ngative direction, that probably the "noise" that define the trend has characteristics so particulary that a detrending action is no needed. In effect the autoregression analysis performed with the original data give other results. Standard deviation of the descriptive statisticis (271,6) is grater than standard error we get with autoregression with one period. So the original data seems to be lagged. We should have a proof with the correlation analysis. In both the correlation is positive, as expected. As in the detrended data value is less than 0.3, the correlation is weak. But in the correlation calculated with original data the correlation is just a little bit better, a moderate correlation that follow the results of autoregression. With two and three periods, also the autoregression with original data tell us that the relationship between the events semms to be non existent. We have demonstrated that a the original dataset has a lightly autocorrelation with one period. Nothing concern other periods and the series. The model building need another important step: the validation. Model validation is a very important step in the model building sequence. Often the validation of a model seems to consist of nothing more than quoting the R 2 statistic from the fit. But we have seen that in our specific case R 2 is particularly low. This is not necessarily bad because R 2 value does not guarantee that the model fits the data well. So, at this point we need to do a graphical residual analysis.

The difference between the observed value of the dependent variable (y) and the predicted value (ŷ) is called the residual (e). Each data point has one residual. Residual = Observed value -Predicted value e = y -ŷ (5.2.2.1)

The residual plot shows the residuals on the vertical axis and the independent variable on the horizontal axis. If the points in a residual plot are randomly dispersed around the horizontal axis, a linear regression model is appropriate for the data; otherwise, a non-linear model is more appropriate.

The residual plot of our model is shown in the figure below (Fig. 5.2.2.1). residual plot shows a fairly random. This random pattern indicates that a linear model provides a decent fit to the data. The residuals should not be either systematically high or low. So, the residuals should be centered on zero throughout the range of fitted values. In other words, the model is correct on average for all fitted values. Nontheless we have a situation particulary good, and the residual plot seems to confirm that the model describe farly well our time series, the plot show in the middle of X axis a point particulary high, that seems to be over the average and is distinctly over the other values.

Displaying the data related to this point we could see that the value is 500 and the difference (residual) is 877.65 (Fig. 5.2.2.2 and Fig. This point is refered to the last observation with 1355 in the 142nd month. After the procedure of identification of peaks, the residual plot help us to find outliers. This event is clearly an outliers. As wrote in a previous paper (G. Rosso, Outilers emphasis on cluster analysis -the use of squared Euclidean distance and fuzzy clustering to detect outliers in a dataset, 2014) outliers can became a precious source of information. In our case we can note that the anomalous event id the last of our time series, and this bring us to the need to persist in carefully observation of events because probably in a close future we could have other anomalous events.

CONCLUSIONS.

Autocorrelation, trend, autoregression. These items are strictly lagged among them. When we try to understand events, frequently time series that describe them have caracteristics of autocorrelation. This means that every single events could be influenced by the previous event, and often not with the only one previous, but with many. As a natural continuation of the research "Climate events and insurance demand -The effect of potentially catastrophic events on insurance demand in Italy" we tried to answer to these questions: these kind of time series have caracteristic of autocorrelation? How can we manage this events? There is the opportunity to assess them correctly and to do predictions? The answer to the first question is: yes, frequently risk time series, and climate risk time series in particulary, have events correlated among them. The answer to the second question was explained in this article. We suggested to manage first the amount of the events, because probably not the entire time series could be usefull for analysis. So if only events exceding a threshold are usefull for the analysis, there are methods dedicated to this work of selection. After that, we suggest to consider that events could be correlated, and that a kind of "noise" can influence the trend of the time series. We can image this "noise" as a persistence of any kind. What supposed often is difficult to demonstrate. The time series object of study must be treated carefully and every result must be interpreted. In our example, autoregression was calculated with the data of the citated research. The scatterplot shows evident fluctuations especially in the last period, however the trend is clear. We know that climate time series are affected by a persistence noise. The world climate is changing, and related risks too. The change seems to be very fast, progressive and exponential increasing. The effect of our anthropogenic activities are subjected to accumulation. This is the "noise" we are speaking. And the increasing trend seems to be in that direction. But if we see the results of the autoregression calculated with our climate data, this assumption is not evident. The time series was detrended before the autoregression calculus, but results tell us that the original time series behave than the detrended series, and only for one period. Results of the autoregression calculated (both on the detrended time series and the original one) seem to suggest that the only effect that evoke the presence of this noise is really the trend. Besides the strong fluctuation of the data values produces outputs with feeble significance of the R squared. But stronger evidence are in the residual analysis. This kind of analysis help to verify the goodness of the model, but also help to identify outliers present in time series. Definitively we can assert that in a climate time series is present a noise that drive to a correlation between periods. The process proposed help to define if the series must be detrended, how many periods are lagged, if model is good enough, if there are outliers in the series and what is their magnitude. The process is long, but supply a lot of informations usefull to provide directions about the analysis.

Identification of Risk Extreme Values in a Time Series and Analysis with an Autoregressive Method, Application for Climate Risk Events -Gianluca Rosso (2016) ࡐі

Fig

  Fig. 1.1

Fig. 3

 3 Fig. 3.1. Source: E. Masiello, Univ. Lyon 1

Fig. 3

 3 Fig. 3.2. Source: E. Masiello, Univ. Lyon 1

Fig. 3

 3 Fig. 3.4. Source: E. Masiello, Univ. Lyon 1

Fig

  Fig. 3.5.

  Dependent variable is a function of itself at the previous moment of period or time. The most often seen form of the equation is a linear form: dependent variable values at the moment t, y t-i (i = 1, 2, ..., p) the dependent variable values at the moment t-i, b o , b i (i=1,..., p) regression coefficient, p autoregression rank, e t disturbance term. order autoregressive model is concerned with only the correlation between consecutive values in a series.

(

  The highest-order parameter does not contribute to the prediction of Yt) using an alpha level of significance, the decision rule is to reject H 0 if α

Fig. 5

 5 Fig. 5.2.1.1 Source: S. Coles, Bristol University

Fig. 5

 5 Fig. 5.2.1.2: S. Coles, Bristol University

Fig

  Fig. 5.2.1.3

Fig. 5

 5 Fig. 5.2.1.6

Fig

  Fig. 5.2.1.7 5.2.1.8

Fig

  Fig. 5.2.1.9

Fig

  Fig. 5.2.1.10

Fig. 5

 5 Fig. 5.2. 1.12

Fig

  Fig. 5.2.1.13

Fig

  Fig. 5.2.1.14

Fig

  Fig. 5.2.2.1

FigFig

  Fig. 5.2.2.3

  Identification of Risk Extreme Values in a Time Series and Analysis with an Autoregressive Method, Application for Climate Risk Events -GianlucaRosso (2016)

	1	200
	2	30
	3	40
	4	120
	5	80
	6	110
	7	180
	8	55
	9	190
	10	110
	11	20
	12	110
	Tab. 5.2.1.2
	ჰѢ
		14

Identification of Risk Extreme Values in a Time Series and Analysis with an Autoregressive Method, Application for Climate Risk Events -GianlucaRosso (2016)

K. Porter, A Beginner's Guide to Fragility, Vulnerability, and Risk, University of Colorado, Denver CO USA.Identification of Risk Extreme Values in a Time Series and Analysis with an Autoregressive Method, Application for Climate Risk Events -GianlucaRosso (2016)

The Mann-Kendall (M-K) Test is a simple test for trend. Mann-Kendall is a non-parametric test and as such, it is not

Identification of Risk Extreme Values in a Time Series and Analysis with an Autoregressive Method, Application for Climate Risk Events -Gianluca Rosso (2016) 䢀ѣ

ACKNOWLEDGEMENTS. Matteo Barigozzi, LSE London School of Economics and Political Science; Esterina Masiello, UCBL Claude Bernard University Lyon 1 -ISFA Institut de science financière et d'assurances; Keith Porter, University of Colorado Boulder and SPA Risk LLC, Denver CO USA.

where yhat i is the value of y i predicted from the regression line and ybar is the sample mean of y.

R 2 = 1 -Residual SS / Total SS (general formula for R 2 )

(5.2.1.3)

F gives the overall F-test of H0: β j = 0 versus Ha: at least one of β j does not equal zero. Excel computes F this as F = [Regression SS/(k-1)] / [Residual SS/(n-k)] (5.2.1.4)

Significance F has the associated P-value. If value > 0.05, we do not reject H0 at signficance level 0.05. Significance F in general has a related Excel formula that is FINV (F, k-1, n-k) where k is the number of regressors including hte intercept.

-Column "Coefficient" gives the least squares estimates of β j .

-Column "Standard error" gives the standard errors (i.e.the estimated standard deviation) of the least squares estimates b j of β j . -Column "t Stat" gives the computed t-statistic for H0: β j = 0 against Ha: β j ≠ 0.

-This is the coefficient divided by the standard error. It is compared to a t with (n-k) degrees of. N is the number of observations and k tha number of regressors. -Column "P-value" gives the p-value for test of H0: β j = 0 against Ha: β j ≠ 0.

-This equals the Pr{|t| > t-Stat}where t is a t-distributed random variable with n-k degrees of freedom and t-Stat is the computed value of the t-statistic given in the previous column. Note that this p-value is for a two-sided test. For a one-sided test divide this pvalue by 2 (also checking the sign of the t-Stat). -Columns "Lower 95%" and "Upper 95%" values define a 95% confidence interval for β j .

We used Excel to perform the analysis, because Excel has a very large diffusion, a lot of tools designed for it are available, frequently for free, and costs for the program are very light if rapported with others statistical programs, expecially for students. But excel own particulary specs that is important to know:

-Excel restricts the number of regressors (only up to 16 regressors); -Excel requires that all the regressor variables be in adjoining columns. You may need to move columns to ensure this; -Excel standard errors and t-statistics and p-values are based on the assumption that the error is independent with constant variance (homoskedastic); -Excel does not provide alternaties, such asheteroskedastic-robust or autocorrelationrobust standard errors and t-statistics and p-values; -more specialized software are STATA, EVIEWS, SAS, LIMDEP, PC-TSP, and many others.

Now, let's see the output of detrended AR(1). for Climate Risk Events -Gianluca Rosso (2016)


E Hadamcik
email: edith.hadamcik@latmos.ipsl.fr

A C Levasseur-Regourd

Imaging polarimetry of comet 73P/Schwassmann-Wachmann 3 main fragments during its 2006 apparition

Keywords: comets, individual, comet 73P/Schwassmann-Wachmann 3, dust, fragmentation, linear polarization

We have observed the dust ejected by parts of the nucleus (so-called fragments or components) of comet 73P/Schwassmann-Wachmann 3 during seven consecutive nights from 2006, April 27 to May 3 by imaging polarimetry using the 0.8 m telescope at OHP (Observatoire de Haute-Provence, France). Three fragments were observed, B and C main fragments on all nights and G fragment on two nights at 24 h interval. Fragment C, which almost behaves as a normal comet, presents some night-to-night evolution on polarization maps together with some sunward-jets morphology.

Fragment B, as noticed by numerous observers, continues to fragment, with clues to the presence of large secondary fragments, tailward on the intensity images; an increase of activity is noticed on May 2. Jets and fans are observed sunward, with a larger extension in fragment C than in B. Fragment G is fainter and, as fragment B, it continues to fragment. A short sunward jet is detected on the rotational gradient image together with an important tailward structure. The integrated polarization for the two main fragments is typical of polarization of high-P max comets. An important evolution is observed from night-to-night on the polarization maps. Fragment C presents, in two nights at 48 h interval, a lower polarization in the inner coma, neither observed in the intermediate night nor later. A high polarization is also observed on the two sides of the lower polarization regions. In fragment B, the regions around the secondary fragments have a higher polarization than the surrounding coma, They are easily detected in the treated intensity images. As usually, the polarization increases when the phase angle increases. Numerous observers have found similar chemical compositions for the two main fragments together with differences in their optical properties, suggesting heterogeneities in the physical properties during the aggregation of the original nucleus and/or changes after the ejection of dust particles.

1-Introduction

Comet 73P/Schwassmann-Wachmann 3 (hereafter SW3) passed close to Earth (perigee on May 12 at 0.079 au) in April-May 2006, allowing a high spatial resolution imaging and numerous remote observations. It is a very active Jupiter-Family Comet (JFC) with an orbital period of 5.36 years and an orbit inclination of 11.39°. Its increasing activity in 1995 was detected by radio observations [START_REF] Crovisier | Comet 73P/Schwassmann-Wachmann 3[END_REF]. The existence of at least four fragments (named A, B, C, D) was revealed in December 1995 by photometry in the red wavelength domain and thermal observations [START_REF] Boehnhardt | Comet 73P/Schwassmann-Wachmann 3[END_REF]. In 2001, the recovered fragments B and C and a new fragment E appeared well separated [START_REF] Boehnhardt | 73P/Schwassmann-Wachmann 3-One orbit after break-up: Search for fragments[END_REF]. Altogether, more than sixty fragments were detected in 2006, although evidence for none of them was found around the main fragment C, which was the primary component [START_REF] Toth | Hubble Space telescope observations of fragment C of the split comet 73P/Schwassmann-Wachmann 3 in 2001 and[END_REF]. Dello [START_REF] Russo | Compositional homogeneity in the fragmented comet 73P/Schwassmann-Wachmann 3[END_REF] studying the volatile composition of fragments B and C found similar compositions, qualified as carbon-chain depleted [START_REF] Disanti | Depleted carbon monoxide in fragment C of the Jupiter-Family comet 73P/Schwassmann-Wachmann 3[END_REF].

The linear polarization of the solar light scattered by dust around main fragments (B and C) was also studied in the visible and the near Infrared [START_REF] Bonev | Broadband imaging and narrowband polarimetry of comet 73P/Schassmann-Wachmann 3, components B and C[END_REF][START_REF] Jones | Evidence of fragmenting dust particles from near-simultaneous optical and near-infrared photometry and polarimetry of comet 73P/Schwassmann-Wachmann 3[END_REF][START_REF] Kiselev | The anomalous spectral dependence of polarization in comets[END_REF]. Linear polarization studies, which are independent of the number of particles in the field of view (for optically thin media), provide complementary clues on the optical and physical properties of dust inside a coma (see e.g. [START_REF] Levasseur-Regourd | Polarization of light scattered by cometary dust particles: observations and tentative interpretations[END_REF]. Polarization values depend not only on the dust properties, but also on the phase angle and the wavelength of observations. Using all the available data in the literature, in wavelength ranges as free as possible of gaseous emission contaminations, a classification of comets by their polarization at large phase angles and large apertures was built with two classes called high-P max comets and low-P max comets [START_REF] Levasseur-Regourd | Evidence of two classes of comets from their polarimetric properties at large phase angles[END_REF].

Both laboratory simulations (see e.g. Hadamcik et al., 2007a[START_REF] Hadamcik | Laboratory measurements of light scattered by clouds and layers of solid particles using imaging technique[END_REF] and numerical simulations (e.g. [START_REF] Kimura | Light scattering by cometary dust numerically simulated with aggregate particles consisting of identical spheres[END_REF][START_REF] Lasue | Cometary dust properties retrieved from polarization observations, Application to C/1995 O1 Hale-Bopp and 1P/Halley[END_REF][START_REF] Kolokolova | Comet dust as a mixture of aggregates and solid particles: model consistent with ground-based and space-mission results[END_REF] provide realistic interpretations of the observations. They are confirmed by correlations with results obtained by other observational techniques such spectroscopy (with silicate infrared emission features). The high polarization values for high-P max comets are usually interpreted by the presence of sub-micron sized grains, which can be parts of fluffy aggregates (Hadamcik and Levasseur-Regourd, 2003a). Low-P max comets may present no jet activity at all or just a few jets in a limited region of the inner coma, with an important decrease in polarization as the aperture increases, as for 9P/Tempel 1 before Deep Impact (Hadamcik et al, 2007b). This high polarization in the inner part can be interpreted by large dark particles, which move slowly. Some comets with an important seasonal effect, as comet 67P/Churyumov-Gerasimenko, may behave like low-P max comets before perihelion and present more jets (with smaller grains) at perihelion or after it [START_REF] Hadamcik | Polarimetric observations of comet 67P/Churyumov-Gerasimenko during 2008-2009 apparition[END_REF]. The presence of large particles (100 m or more) before 2015 perihelion has been confirmed by the first results from COSIMA and GIADA experiments onboard Rosetta spacecraft (e.g. [START_REF] Schulz | Comet 67P/Churyumov-Gerasimenko sheds dust coat accumulated over the past four years[END_REF][START_REF] Rotundi | Dust measurements in the coma of comet 67P/Chruryumov-Gerasimenko inbound to the Sun[END_REF].

In the present paper, after a description of our observations and data reduction method, the results are presented with emphasis on particular structures or events. The intensity images allow to follow the morphologies. Differences in optical properties are underlined in the linear polarization maps.

Comparisons with other comets establish that the two main fragments belong to the high-P max class comets and have similar optical properties than other splitting comets.

2-Observations and data reduction method

The observations have been performed with an imaging polarimetric technique with the Cassegrain 0.8 m telescope at Observatoire de Haute-Provence (OHP) in France. Four polaroid filters are mounted on a rotating wheel, with their fast axis oriented at 45° from one another. During each night, polarized and unpolarized standard stars are observed to control the non-polarization of the instrument and to determine the origin of the instrumental reference system ( 0 corresponding to the position angle (PA) of the polarization of one polarized filter). The intensity, the measured polarization degree P (which has not sign), the measured position angle , the position angle of the polarization plane and finally the values of the polarization Pr (which can be positive or negative), (2)    0.5arctan

I 45  I 135 I 0  I 90 (3)   r    0  (  90) (4)
The polarized intensities are measured in the instrumental reference system.  r =( - 0 ) is the position angle of the polarization plane in the equatorial reference system. The position angle of the scattering plane () is known at each date. For symmetry reason it is defined between 0° and 180°

and can be deduced from the value of the Sun-comet radius vector position angle (Sun-C PA in Table 1). The sign between the parentheses, in Equation 4, is chosen to ensure the condition 0° < ( ± 90°) < 180°.  r is generally of about 90° for comets observed at phase angles larger than 25° [START_REF] Levasseur-Regourd | Evidence of two classes of comets from their polarimetric properties at large phase angles[END_REF]. A more detailed description of the instrument and method can be found in Hadamcik et al. (2007b).

The exposure time for each polarized image was short (10 s to 60 s) to avoid any tracking problem.

Numerous images were recorded to build the polarized components for the different fragments and finally the intensity and polarization maps. The log of the observations is presented in Table 1. The weather and sky conditions were remarkably clear, except during some hours with faint clouds and/or wind. These periods are not used in the data. The region blurred by seeing remained smaller than 200 km radius (and often than 120 km) on the comet. To reduce the contaminations by the gaseous emissions, a Thuan-Gunn 'r' filter has been used (655 nm,  = 90 nm; [START_REF] Thuan | A new four-color intermediate-band photometric system[END_REF].

Results obtained with this filter were validated during previous studies of different comets e.g. comet 103P/Hartley 2 which is a low-P max comet; comparisons with results obtained through ESA cometary continuum narrow band filters (684 nm,  = 9 nm and 443 nm,  = 4 nm) have shown the absence of any major contamination when using the broad Thuan-Gunn filter [START_REF] Hadamcik | Dust in comet Hartley 2 coma, during EPOXI mission[END_REF][START_REF] Hadamcik | Dust coma of comet C/2009 P1 (Garradd) by imaging polarimetry[END_REF].

Nevertheless, the continuum may be contaminated by the O1D band at 630 nm and in a lower ratio at 633.4 nm, NH 2 bands (0,8,0) at 633.5 nm, (0,7,0) at 665 nm and (0,5,0) at 695 nm. In the spectrum (unfortunately limited to 650 nm), published by [START_REF] Yu | Optical spectroscopic observations of comet 73P/Schwassmann-Wachmann 3[END_REF] for observations on 2016/05/02), gas emission bands may be present, but with a relatively small contribution as compared to the continuum.

This will be discussed with the polarization results and comparison with other observations.

Table 1

A center gravity algorithm is used to find the position of the optocenter of the comet on each polarized image. To avoid artifacts in the polarization maps, the polarized components are centered with a precision of 0.1 pixel before any addition or calculation. The sky background estimated in a region outside the coma (in the solar direction at more than 2 arcmin) is subtracted from each image and controlled. The fluxes through apertures of 12 and 24 pixels diameters are measured for each polarized component, and the stability of intensity values is controlled for the whole series. If a difference greater than 2 % is detected, the image is rejected. For some of the observations a median sum is applied to series of six images in case of numerous star tracks to suppress them from the images. This method is efficient but the flux level is reduced and as a consequence the S/N is smaller as compared to just the sum of all the images.

3-Results

In a purpose of comparison, intensity images and polarization maps are presented for the three fragments in Fig 1 . The intensity variations and coma morphologies, the linear polarization maps for SW3-C and SW3-B are presented with their evolution during the period of observations in Fig. 1a and 1b. Observations were made at different times during the night, to try to highlight short-term evolutions, which could be observed night after night. Fig. 1c presents the results obtained for fragment SW3-G on April 29, taking into account the fact that the sky was partly cloudy on April 30.

Figures 1a and 1b and 1c

Intensity images

Figures 1,a,b,c, intensity images show, from left to right, the overall shape of the coma, the isophotes (in log scale for intensity) and some structures emphasized through rotational gradient technique [START_REF] Larson | Coma morphology and dust-emission pattern of periodic comet Halley. I. High resolution images taken at Mount Wilson in[END_REF], with orientation controlled on isophotes images. For each image an azimuthal averaged radial profile in intensity is built and compared to the sunward and tailward profiles. Large variations in slopes are noticed on all profiles. While a value of -1 is expected on the averaged radial decreases for steady-state dust outflow, it is never obtained here at relatively large optocentric distance. Changing slope on the profiles suggest changes in the local optical properties of the particles (e.g. size, size distribution, albedo) and/or in their number density.

Fragment C

It may be noticed in Fig. 1a that jets are detected sunward, at least one is always present, Its position angle changes day after day during the observations, and remains within ± 6° of the solar direction. Outside the inner coma, jets may get curved under the solar radiation pressure; and be separated by 20° to 30°, with position angles between 35° clockwise and 62° anticlockwise from the solar direction. An important tailward structure is present on all the images at a position angle of 188°± 2° anticlockwise from the solar direction, i.e. North from the projected antisolar direction.

Except in the inner coma, on the three dates presented in Fig. 2a, the average profiles are steeper than for a steady state outflow (slope > -1), suggesting a change in size distribution or more absorbing dust particles. In the inner coma (less than 400 km from the optocenter and out of the region blurred by seeing (130 km in average), all the profiles present smaller slopes than -0.7 most likely because of the presence of multi-jets mainly on April 28 and May 3, with breaking of the particles and dynamical effects. These later interpretations were also suggested by [START_REF] Jones | Evidence of fragmenting dust particles from near-simultaneous optical and near-infrared photometry and polarimetry of comet 73P/Schwassmann-Wachmann 3[END_REF], who observed also a flattened slope in the inner coma. The three sunward profiles become steeper for optocentric distances larger than 3000 km with a very important increase of the slope on May 3 suggesting that particles leave the coma pushed back by the solar radiation pressure and can be found tailward. As expected, the slope of decrease on the tailward profiles is smaller than on the other ones and slightly increases on May 3 at optocentric distances larger than 1000 km. To tentatively interpret the polarization results for April 27 and 28, two profiles (E and W), perpendicular to the solar-antisolar direction, are respectively presented. The four profiles are relatively close to the sunward profiles, mainly on April 27 on the East direction. Dust particles are present but without important gradient around the perpendicular directions i.e. they does not originate from radial jets; they may be eventually directly released by the nucleus and/or they may come from the sunward direction when the particles are pushed back by the radiation pressure. On April 28, important dust ejection is detected in the sunward profile; further away the slope is steeper, dust is probably pushed tailward. On May 3, out of the sunward jets, the slope becomes very important (-3) at distances larger than 2000 km showing the end of the coma in that direction. On the perpendicular profiles, except in the inner coma (below 300 km, the intensity decrease is steeper than on April 27. On April 29, the profiles (not presented here) are comparable to those found on April 27.

Figure 2a

Fragment B

In Fig. 1b, short sunward jets may be noticed. A conspicuous tailward jet is situated at a position angle of 220° ± 2° in projection on the sky. Smaller secondary fragments are clearly detected on the treated images, North from the antisolar direction, with position angle within 220°-240°, there is a discontinuity between the dust coming from the nucleus and the dust coming from secondary fragments. More specifically, on May 2, the intensity increases and the overall coma shape changes (Fig 1b); the sunward jets are all around the optocenter (position angles equal 29° 'solar direction', 45°, 90°, 150°, 273°, 305°, 342°); a spike, on the intensity image, extends in the tailward direction.

One fragment is visible on the whole set of images at a position angle of 234° and at an optocentric distance of about 3000 km (noted sf on Fig. 1b isophotes and on Fig. 2b averaged profile). The difference in magnitude in a 170 km diameter circular aperture between the main fragment and the above-mentioned secondary fragment is about 3.8. A large region extending in the tail direction, with probably other fainter fragments that are more difficult to precisely localize, surrounds it.

In Fig. 2b, the profiles for April 27 and May 3 are similar; the tailward profiles present smaller slopes than the other ones. Some faint bumps are observed, which can indicate some secondary

fragments. An important increase of the slope between 500 km and 1500 km is noticed sunward on April 27 and between 400 and 800 km on May 3; further away the slopes are less steep. On May 2, the slope of the average profile is small in the inner coma due to jets and small fragments. It increases between 200 and 1200 km and progressively decreases further away due to the presence of secondary fragment(s) and of their ejected dust. A spike at about 3000 km indicates its position (noted sf). The slope is very small sunward due to presence of important short jets. It is steep between 300 km and 700 km and decreases progressively for larger distances. On the tailward profile the slope decreases in the region where dust is emitted by the secondary fragment. The dust particles ejected by the secondary fragments, which are not all directly detected may be responsible of the changes in slopes.

Figure 2b

Fragment G

This fragment being fainter, its extension on the images, as seen in Fig. 1c, quite small (just the central coma is visible). One fan between position angles 62° and 93° and a faint jet at about a position angle of 30° are detected sunward (Fig 1c). The quite well developed tailward structure at an average position angle of 213° has a curved shape; the irregularities on this structure may originate in noise and/or in smaller secondary fragments.

The slope of the averaged profile is -0.9 between 160 and 500 km, it is -2.3 between 500 and 3000 km, indicating the end of the coma. In the inner coma where jets are observed, the slopes are smaller than -0.6. At distances larger than 500 km from the optocenter, the slope is -1.1 between 500 and 1100 km; the tailward slope is -0.9 between 500 and 3000 km with a bump at about 1800 km probably resulting of some secondary fragment.

Figure 1c

3.2 Linear polarization

Polarization angle

From the standard stars observation, in average  0 is found to be equal to -5° ± 2°. The instrumental polarization is found to be smaller than 0.1 %. The position angle of the polarization plane ( r ) in the equatorial reference system (related to the solar position angle and calculated by the equations 3 and 4) is in average 90° ± 10° for all the observations without any variations close to the optocenter; in consequence P r is about equal to P

Aperture polarization

The polarization values, P, are calculated through increasing apertures (Table 2). In the inner coma important variations are observed mainly for fragment C. The variation of polarization for a specific aperture is studied as a function of the phase angle.

Table 2

Polarization vs aperture

Except in the inner coma, the polarization slightly decreases with increasing apertures up to about 6000 km where it gets rather constant, then takes a value, usually called whole coma polarization (Table 2, Fig. 3).

Figure 3 (a and b)

In Fig. 3a variations in polarization values as a function of aperture are presented for fragment C at = 45° (April 27) with a low polarization in the inner coma (4.5 % at optocenter) and at = 46.4° (April 28) with a high polarization in the inner coma (13 % at optocenter) and finally at = 55.5° (May 3) with a higher polarization in the inner coma (18 % at optocenter). Since the optocententers are in the regions blurred by seeing, the error bars on the corresponding polarization values are large. The phase angle is about the same on the two first dates, the polarization taking about the same values through large apertures (more than 6000 km); then P is about constant as the aperture increases and as expected increases with phase angle.

To better characterize the variation in polarization P when the optocenter distance increases, Fig 3b shows P variation in consecutive concentric annuli as a function of the diameter of the annuli. The decrease in polarization is about 2% (3% for May 3), when the annuli diameter increases from 1500

km to 9000 km. Note that at optocentric distances larger than 500 km radius, out of the inner coma, on April 27 and 28, the polarization values decrease are similar. These decreases may be due to a change in properties of the particles (size, albedo) with e.g. decrease of the albedo induced by a decrease of amorphous C abundance as compared to silicates with increasing nucleus distance [START_REF] Sitko | Infrared spectroscopy of comet 73P/Schwassmann-Wachmann 3 using the Spitzer space telescope[END_REF] or eventually small gas contaminations as suggested by [START_REF] Bonev | Broadband imaging and narrowband polarimetry of comet 73P/Schassmann-Wachmann 3, components B and C[END_REF].

Polarization vs phase angle

The variation of the polarization in the inner coma is compared to the variation of P in the whole coma in Fig. 4. In this phase angle range [40°-60°], the whole coma polarization increases about linearly with the phase angle, as expected from other polarimetric observations (see section 4 and Fig. 5).

Fragment C

As seen for three consecutive nights, the inner coma polarization oscillates of about ± 3 % around the whole coma polarization, which changes only by less than 0.5 %. For higher phase angles, the inner coma polarization is higher by 3-6 % than the whole coma polarization and increases with a comparable rate to that of the whole coma when the phase angle increases. Within the documented phase angle range, whole coma polarization data are almost on the synthetic phase curve corresponding to high-P max comets (Fig. 4a).

Figure 4a

Fragment B

The inner coma polarization is higher than the whole coma polarization by in average 2.5 % and it increases with a rate similar to that of the whole coma when the phase angle increases. The whole coma polarization is slightly below the synthetic phase curve for high-P max comets by 0.7 % in average (Fig. 4b).

Figure 4b

Fragment G On April 29, the inner coma polarization is higher by about 2 % than the whole coma polarization.

However, taking into account the resolution, it is difficult to characterize any structure except a higher polarization for the sunward jet and in the tailward structure. The whole coma polarization (6.4 ± 2.0)

% is close to the low-P max class of comets (7.2 %), the value for high-P max comets being 9.5 % at the same phase angle. On April 30, the inner coma polarization increases slightly and the whole coma polarization seems to have about the same value but with large error bars due to faint stars close to the comet.

Polarization maps

The polarization maps allow characterizing structures, which are not always apparent on the intensity images. A good correlation is usually observed between high polarization structures and jets and fans (Fig. 1a andb). On SW3-C (Fig. 1a), April 27 and 29 polarization maps, two structures present a high polarization (up to 13.5 % at about 250 km from the optocenter in directions perpendicular to the Sun-tail directions) around the lower polarization central regions (down to 5 %).

From the polarization maps and the intensity slopes, it can be inferred that these high polarization regions (also not pointed out by a rotational gradient analysis) contain dust ejected on the sunward side probably along the jets curved under the radiation pressure (Fig. 1a). The dust particles from these regions are different from those present elsewhere in the coma, possibly fragmenting producing smaller aggregates and sublimating changing the albedo. On April 27, the lower polarization region is elongated in a position angle of about 45°, the optocenter being at the center of this region. The region extends on 130 km width and 280 km length. On April 29, the lower polarization region has about the same position angle; it is slightly longer and larger than on April 27.

On SW3-B polarization maps, regions with higher polarization than the surrounding coma are also found around the secondary fragments, while polarization values are not so high than in the jets.

Some fragments in the projected antisolar direction may be present in the tailward structure close to the nucleus and not detected in the images. An increase of activity as on May 2 for SW3-B may be the result of some fragmentation, with an increase of polarization close to the nucleus (Fig. 1b).

Comparison with other polarimetric observations of SW3 fragments and other comets will be discussed in the following section.

4-Discussion

Morphology and changes within a coma

In a coma, the dust particles are pushed tailward by the solar radiation pressure. Its effect is observable for SW3-C curved jets in the sunward direction (Fig. 1a, third column). In the images, relatively day-to-day variations are observed.

On May 2, some secondary fragments are detected tailward in comet SW3-B coma, while they are progressively pushed further away from the nucleus. Fifty-four fragments were observed on May 3

with the 8 m telscope by [START_REF] Fuse | Observations of fragments split from Nucleus B of comet Schwassmann-Wachmann 3 with Subaru telescope[END_REF]; some of these fragments are likely to be within our images. This behavior was previously demonstrated by [START_REF] Desvoivres | Modelling the dynamics of cometary fragments: Application to comet C/1996 B2 Hyakutake[END_REF], using their dynamical model to represent the fragmentation of the nucleus of comet C/1996 B2 (Hyakutake).

Secondary fragments are easy to notice in the coma of SW3-B, North from the tail where the intensity is low and the contrast is quite high. The North-East position of secondary fragments at more than 1000 km from the optocenter is probably the result of rocket effect (speed vector in South-West direction (see e.g. [START_REF] Ishiguro | 2006 fragmentation of comet 73P/Schasswann-Wachmann 3B observed with Subaru/Suprime-Cam[END_REF]. The changes in the morphology of the coma on May 2 with an antisunward spike is reminiscent of Hyakutake observations, possibly because some fragments are too small to be detected. [START_REF] Ishiguro | 2006 fragmentation of comet 73P/Schasswann-Wachmann 3B observed with Subaru/Suprime-Cam[END_REF] and [START_REF] Bertini | Activity evolution, outbursts and splitting events of comet 73P/Schwassmann-Wachmann 3[END_REF] observed also a small outburst in SW3-B on May 2. In the image taken about 12 hours before the present observations by [START_REF] Bertini | Activity evolution, outbursts and splitting events of comet 73P/Schwassmann-Wachmann 3[END_REF], a fragment was observed in the antisolar direction at about 340 km from the main nucleus.

Comparison between polarimetric observations of SW3 and those of other comets

As shown in Fig. 5 similar polarization results were found in the red wavelength domain at about 60° phase angles but not exactly on the same dates, by [START_REF] Bonev | Broadband imaging and narrowband polarimetry of comet 73P/Schassmann-Wachmann 3, components B and C[END_REF] with the narrow band ESA continuum filter and [START_REF] Kiselev | The anomalous spectral dependence of polarization in comets[END_REF] with a red filter, confirming the small, if any, contamination of our Thuan-Gunn broad-band filter (the differences between the three series of measurements being in the error bars). However, at larger phase angles, where phase curves are possibly less constrained, the polarization values seem to be below the high-P max phase curve. In near infrared domain (H and I bands), at about 36°, the data points by [START_REF] Jones | Evidence of fragmenting dust particles from near-simultaneous optical and near-infrared photometry and polarimetry of comet 73P/Schwassmann-Wachmann 3[END_REF] are above the high-P max phase curve in red wavelength as usually observed for comets (Hadamcik and Levasseur-Regourd, 2003b;[START_REF] Kelley | Near-infrared polarimetry and photometry of recent comets[END_REF]) and called 'positive polarimetric spectral gradient', but at larger phase angles than 40° (80°-90°, [START_REF] Bonev | Broadband imaging and narrowband polarimetry of comet 73P/Schassmann-Wachmann 3, components B and C[END_REF][START_REF] Jones | Evidence of fragmenting dust particles from near-simultaneous optical and near-infrared photometry and polarimetry of comet 73P/Schwassmann-Wachmann 3[END_REF][START_REF] Kiselev | The anomalous spectral dependence of polarization in comets[END_REF], they are below the synthetic red phase curve (negative polarimetric spectral gradient). The more neutral or slightly negative spectral gradient in the near infrared domains as observed for the two SW3 fragments at large phase angles, was previously observed for different comets e.g. comet Hale-Bopp (Hadamcik et al., 2003b;Kelley et al., 2005). The infrared data in the I band obtained by [START_REF] Kiselev | The anomalous spectral dependence of polarization in comets[END_REF] are always below the red ones on the whole phase angle ranges 38°-73° for both fragments, and are very low as compared to values obtained by [START_REF] Jones | Evidence of fragmenting dust particles from near-simultaneous optical and near-infrared photometry and polarimetry of comet 73P/Schwassmann-Wachmann 3[END_REF] These values seem to contradict the other results (Fig. 5). The 'I' filter pass-band is not specified, meaning that the results are difficult to compare with other results. It can be noticed that the near infrared polarimetric values at large phase angles are similar to those obtained in red for comet comet C/1995 V1 (West), C/1996 B2 (Hyakutake) and comet C/1999 S4 (LINEAR) after its complete disruption. The very important increase in polarization, observed during disruption of the latter in the red domain (data points not taken into account in the synthetic fit) progressively disappeared. A negative (or neutral) polarimetric spectral gradient was also observed during disruption (Hadamcik and Levasseur-Regourd, 2003c). [START_REF] Kiselev | The anomalous spectral dependence of polarization in comets[END_REF] have studied the spectral gradient on a longer period and found a very important variability before and after complete disruption. Some other comets presented the inverse spectral gradient, or permanently as for comet 21P/Giacobini-Zinner [START_REF] Kiselev | Anomalous wavelength dependence of polarization of comet 21P/Giacobibi-Zinner[END_REF] or just on one night as for comet C/1989 X1 (Austin) [START_REF] Eaton | Polarization studies of comet Austin[END_REF]. The inverse spectral gradient seems to often characterize particles freshly ejected from inside the nucleus. A negative spectral gradient was observed in the inner coma of comet 1P/Halley by HOPE instrument on board the Giotto spacecraft [START_REF] Lasue | Cometary dust properties retrieved from polarization observations, Application to C/1995 O1 Hale-Bopp and 1P/Halley[END_REF]. A similar effect was noticed just after Deep Impact in the plume ejected from comet 9P/Tempel 1 [START_REF] Harrington | Spectropolarimetry of the Deep Impact target Comets 9P/Tempel 1 with HiVIS[END_REF].

Figure 5

As on the polarimetric observations on April 27 and 29 of fragment C, a slightly lower polarization in a region of about 300 km diameter, around the optocenter, was observed in the H band by [START_REF] Jones | Evidence of fragmenting dust particles from near-simultaneous optical and near-infrared photometry and polarimetry of comet 73P/Schwassmann-Wachmann 3[END_REF] on May 12. Lower polarization in the inner coma, at phase angles larger than 30°, was observed in different comets such as comet 1P/Halley [START_REF] Levasseur-Regourd | Similarities between in situ measurements of local dust scattering and dust flux impact data within the coma of 1P/Halley[END_REF], comet 81P/Wild 2 and comet C/1995 O1 Hale-Bopp (Hadamcik and Levasseur-Regourd, 2003a). For comet 2P/Encke an increase in polarization through increasing concentric annuli around the nucleus was noticed by [START_REF] Jewitt | Looking through the HIPPO: Nucleus and dust in comet 2P/Encke[END_REF]. For the same comet, a decreasing P was later observed by [START_REF] Jockers | CCD imaging and aperture polarimetry of comet 2P/Encke: Are there two polarimetric classes of comets?[END_REF] meaning that P in the inner region varies a lot, depending e.g. on the nucleus rotation or on seasonal effects. [START_REF] Jockers | CCD imaging and aperture polarimetry of comet 2P/Encke: Are there two polarimetric classes of comets?[END_REF] nevertheless showed that a decrease in polarization when gas contamination is significant, not only through broadband filters but also through narrow band filters, whenever the central wavelength and the band-pass are not properly chosen. Besides, central region (often called polarization halo) can be partially hidden by jets or angularly small to be detected [START_REF] Levasseur-Regourd | Polarimetric imaging of comet Levy (1990c)[END_REF]Hadamcik and Levasseur-Regourd, 2003a). Different interpretations have been suggested, difference in the size distribution in the inner coma or higher number density of the particles inferring multiple scattering, presence of relatively transparent particles (silicates, low absorption organics, ices; see e.g. [START_REF] Hadamcik | Laboratory measurements of light scattered by clouds and layers of solid particles using imaging technique[END_REF][START_REF] Zubko | Evaluating the carbon depletion found by the Stardust mission in Comet 81P/Wild 2[END_REF][START_REF] Zubko | Characteristics of cometary dust in the innermost coma derived from polarimetry by Giotto[END_REF]. [START_REF] Jewitt | Looking through the HIPPO: Nucleus and dust in comet 2P/Encke[END_REF] interpreted the increasing polarization with increasing nucleus distance by a fading of the particles (fragmentation and sublimation of ices). [START_REF] Jones | Evidence of fragmenting dust particles from near-simultaneous optical and near-infrared photometry and polarimetry of comet 73P/Schwassmann-Wachmann 3[END_REF] suggested a similar interpretation for the lower polarization region in SW3-C on May 12. Short-term day-to-day variability in the dust production rate was observed in SW3-C [START_REF] Bertini | Activity evolution, outbursts and splitting events of comet 73P/Schwassmann-Wachmann 3[END_REF][START_REF] Sitko | Infrared spectroscopy of comet 73P/Schwassmann-Wachmann 3 using the Spitzer space telescope[END_REF]. This high variability mainly in the inner coma polarization maps is also displayed in the intensity images (Fig. 1a). At the opposite the whole coma polarization remains stable, depending only on the phase angle for the two fragments.

Jets are often observed in the inner coma. They can be made of very small sub-micrometer sized grains in eventually large high-porous aggregates with high-speed ejection, allowing them to be found far from the nucleus (Hadamcik and Levasseur-Regourd, 2003a). For some comets, a high polarization is observed in a restricted area due to the low speed of large relatively dense particles.

These large particles are often found in low-P max comets with an important decrease when the aperture size increases [START_REF] Kiselev | Polarimetry of comets[END_REF]. An important decrease of P is observed for SW3-G but it is difficult to definitely conclude, the S/N ratio being low. The particles are ejected from SW3-C nucleus mainly by jets activity; they may undergo sublimation and/or fragmentation in the coma.

Some particles in SW3-B are ejected within jets, while others result from successive fragmentations, inducing a large size distribution (e.g. from submicron-sized to large millimeter-sized particles). A comparison with observations with other techniques is necessary to go further in the interpretation of physical properties.

Color of the dust

Changes in the dust properties in the coma, as derived from the polarimetric observations, are also suggested by the study of the color of the dust [START_REF] Bertini | Activity evolution, outbursts and splitting events of comet 73P/Schwassmann-Wachmann 3[END_REF]. The color depends on the size of the particles and in aggregates on their constituent grains. The surface composition of the grains may be also responsible of some change in color if the complex refractive index dependence on the wavelength is significant as for many organics and for some iron-bearing materials. [START_REF] Bertini | Activity evolution, outbursts and splitting events of comet 73P/Schwassmann-Wachmann 3[END_REF] observed spectroscopically the three fragments through a slit oriented North-South as a function of the optocentric distance in the visible domain (between red and blue continuum). They found important variations in dust color in the comae, depending on the fragment and the date of observations. Fragment B was observed on May 2 (about 12 hours before our observations). The normalized reddening is higher in the South direction than in North direction mainly at 750 km and 6000 km from the optocenter. These differences of reddening could be related to large particles resulting of fragmentation in the South direction and to small particles ejected along jets in the North direction. Nevertheless, for other dates (out of our period of observations), [START_REF] Bertini | Activity evolution, outbursts and splitting events of comet 73P/Schwassmann-Wachmann 3[END_REF] found an inverse color meaning that a more detailed analysis could be done, For fragment G, on April 29, an average reddening was measured, with a higher normalized color in North than in South, but the difference is in the error bars. In conclusion, changes in color are correlated to changes in structure and polarization but color maps would be necessary at the same dates.

Correlation between high polarization and mid-infrared emission features

A correlation was found between dust-rich comets (i.e., comets with a high positive branch of their polarization phase curve) and the strength of 10 and 18-22 m silicates emission features [START_REF] Chernova | Polarimetric characteristics of dust particles as observed in 13 comets: Comparisons with asteroids[END_REF]). An excellent correlation has been found between the very high polarization and high polarization regions as in the jets and arcs of comet C/1995 O1 Hale-Bopp [START_REF] Hadamcik | CCD imaging polarimetry of comet Hale-Bopp[END_REF]Hadamcik and Levasseur-Regourd, 2003a), with well structured 10 m emission features partially due to crystalline silicates [START_REF] Kolokolova | Physical properties of cometary dust from light scattering and thermal emission[END_REF]Harker et al., 2008). Similar results were found in the 'plume' ejected by Deep Impact at comet 9P/Tempel 1 (Hadamcik et al., 2007;[START_REF] Harker | The dust grains from 9P/Tempel 1 before and after encounter with Deep Impact[END_REF]. Jupiter Family Comets are usually low-Pmax comets with weaker continuum thermal emission and weak or inexistent silicate emission features [START_REF] Kiselev | Polarimetry of comets[END_REF]. The usual explanation is the presence of large dark compact particles (> 5 m). From the thermal emission in mid-infrared, physical properties of the particles are inferred. They are used to interpret the light scattering and polarization results for SW3. [START_REF] Harker | Mid-infrared spectrophotometric observations of fragments B and C of comet 73P/Schwassmann-Wachmann 3[END_REF] used observations at 10 m with Gemini, and [START_REF] Sitko | Infrared spectroscopy of comet 73P/Schwassmann-Wachmann 3 using the Spitzer space telescope[END_REF] used the Spitzer Space telescope at 16 m and 22 m. They both, through dust models, confirm similarities in mineralogy and grain properties between SW3-C and SW3-B. Amorphous carbon and amorphous silicates dominate the composition of the two fragments, with an excess of amorphous carbon as compared to other comets. In the anti-solar direction, [START_REF] Harker | Mid-infrared spectrophotometric observations of fragments B and C of comet 73P/Schwassmann-Wachmann 3[END_REF] found a carbon abundance between 45 and 23 % in weight for SW3-B and 60 to 42 % for SW3-C, with a carbon abundance of 21 % for Hale-Bopp and 28 % for the plume of 9P/Tempel 1. They found a slightly higher, model-dependent, silicate-to-carbon ratio in SW3-B than in SW3-C, increasing with the nucleus distance for both fragments. [START_REF] Sitko | Infrared spectroscopy of comet 73P/Schwassmann-Wachmann 3 using the Spitzer space telescope[END_REF] noted that while the emissivity spectra of amorphous carbon and nano-phase Fe are both similar and featureless in thermal infrared, the abundance of amorphous carbon in SW3 may be slightly smaller than previously estimated. For the polarization results, this difference has no influence. The crystalline mass fraction of silicates with a modest population of crystalline olivine (25-30 %) is about the same in SW3-B and SW3-C, for submicrometer and micrometer sizes [START_REF] Harker | Mid-infrared spectrophotometric observations of fragments B and C of comet 73P/Schwassmann-Wachmann 3[END_REF]. [START_REF] Sitko | Infrared spectroscopy of comet 73P/Schwassmann-Wachmann 3 using the Spitzer space telescope[END_REF] found a slight difference in the grain size distribution with a slope of -4.0 for SW3-C coma similar to that found for active comets with strong jets, and -3.5 for SW3-B, (close to that expected for dust in collisional equilibrium). The peak size distribution was 0.5 m in radius for SW3-B and 0.3 m for SW3-C, slightly larger than the peak for Hale-Bopp (0.15-0.20 m) and for the plume of 9P/Tempel 1 (0.18 m) [START_REF] Williams | Measurements of submicron grains in the coma of comet Hale-Bopp C/1995 O1 during 1997 February 15-20 UT[END_REF][START_REF] Harker | Grain properties of comet C/1995 O1 (Hale-Bopp)[END_REF]2005).

The small grains in fluffy aggregates allow the emission of high continuum for the silicate IR thermal emission and important emission features when small crystalline silicates are present e.g. in Hale-Bopp or Tempel 1 after Deep impact. In these two comets the crystallinity is higher than in SW3.

The high thermal continuum is consistent with the small size of grains in relatively porous (fluffy)

particles and with the high abundance of amorphous carbon [START_REF] Harker | Mid-infrared spectrophotometric observations of fragments B and C of comet 73P/Schwassmann-Wachmann 3[END_REF]). The polarization is consistent with mid-infrared results for high P max comets, but lower than for Hale-Bopp and for Tempel 1 after Deep Impact. SW3 is thus an exception among Jupiter Family comets with a high continuum in mid-infrared and a high polarization in large apertures. The small difference in polarization between SW3-C and SW3-B could be related to the differences in the size distribution of the grains and/or the slightly higher carbon-to-silicate ratio in SW3-C (or an eventual small gaseous contamination through the Thuan-Gunn filter for SW3-B). The small decrease in polarization, out of the inner coma, with increasing nucleus distance could result from the decrease in the carbon-to-silicate ratio. In a similar way, an alternative interpretation of the low polarization values in the innermost coma could be related to an anomalous lower carbon-to-silicate ratio on April 27 and April 29 and eventually on May 12 (Jones et al., 2008 observations) and/or to large particles, which progressively fragment.

5-Summary

Three Dates, intensity images, isophotes in log scale for the intensity, rotational gradient treated images and polarization maps. All fields of view equal to 9000 km x 9000 km. North is up and East on the left side, solar direction is shown on the isophotes by a segment. For each display, the level is chosen to emphasize the different structures (thus, levels of intensity and polarization not comparable from one night to another, red color is for high values, blue for small ones, in each image and polarization map). On SW3-B isophotes: sf = secondary fragment also visible on all images.

  and the position angle  r in the coordinate system referring to the scattering plane are calculated by the following expressions:  I  I 0  I 90  I 45  I 135 (1)  P  200 (I 0  I 90 ) 2  (I 45  I 135 ) 2 I 0  I 90  I 45  I 135

  Fig. 2a and Fig. 2b present profiles on April 27, April 28 and May 3 for fragment C, on April 27, May 2 and May 3 for fragment B. They confirm the presence of structures in the coma.

  [START_REF] Sitko | Infrared spectroscopy of comet 73P/Schwassmann-Wachmann 3 using the Spitzer space telescope[END_REF] relate ground-based observations showing for SW3-C an important variability in mid-infrared and tentatively interpret it by two kinds of dust particles, one from 'old' surface alterning with dust with different properties from the 'new' surface. This variability observed also in the intensity images and polarization maps is more important for SW3-C than for SW3-B.

  main fragments of comet 73P/Schwassmann-Wachmann 3 were extensively observed at OHP in France, in April-May 2006. Observations of main fragments C and B during seven consecutive nights allowed us to observe the evolution of the different structures in intensity images and linear polarization maps, while the smaller fragment G was observed during two nights. Fragment SW3-C, in spite of behaving like a comet with a high maximum in polarization, showed important variations, mainly in the inner coma. Fragment SW3-B continued to fragment and variations were noticed mainly in the tailward direction.The rotational gradient treated intensity images allowed us to underline different structures in the jets and the presence of secondary fragments. A clear deviation from -1 in the slope of the azimuthally averaged radial profile for the two main fragments (C and B) proves that the dust particles behavior is far from a free expanding outflow. Significant changes in the slopes suggest changes in the spatial distribution of the particles and in their properties.The whole coma integrated polarization seems to indicate that the two main fragments belong to the high-P max class of comets. Contaminations by gas emissions seem to be very small accounting for less than 1 % on the whole coma polarization. The polarization results are perfectly consistent with the high thermal emission in the mid-infrared. The color of the dust, in SW3-B suggests that the particles are larger than within the jets and/or are fragmenting on short time scales. The high polarization regions correspond to jets and to fragments regions. Besides, for SW3-C, dust along the jets, pushed back by solar radiation pressure presents an even higher polarization due to fragmentation, and a lower polarization is observed in the inner coma. Numerous secondary fragments and fragment regions are observed tailward in the coma of the fragment SW3-B in the treated intensity images but also in the polarization maps. Correlated results for the same observing dates by different techniques help to interpret the observations. Comet SW3 fragments provide unique results. The variation of the light scattering properties through the coma and their evolution may give an insight on the properties of inner nuclei; differences in their optical properties suggests heterogeneities in the physical properties during the aggregation of the original nucleus and/or changes after the ejection of dust particles.

  Figure captionsFigures 1 1a. Fragment C; 1b. Fragment B; 1c. Fragment G. From April 27 to May 3, 2006, for fragments C and B, and on April 29, 2006, for fragment G.

  Figures 2 2a.Fragment C: April 27 and 28, May 3; 2b. Fragment B: April 27, May 2 and 3.

  Fragment C: April 27 and 28, May 3; 2b. Fragment B: April 27, May 2 and 3. Radial profiles: azimuthally integrated intensity, sunward, tailward, and for fragment C, perpendicular to the solar direction West and East for the 2 first dates. SW3-B: sf = spike corresponding to the secondary fragment observed on the images. The HWHM of the point-spread function averaged on the whole observing time, for each date and fragment, as measured using stars in each field is indicated by a vertical line.

Figure 3 3a.Figures 4 4a.

 34 Figure 3 3a. Fragment C variations of polarization vs aperture diameter. The error bars (smaller than 0.5 % except at the optocenter in the region blurred by seeing) are smaller than the symbols. 3b. Fragment C variations of polarization vs annulus diameter. Note that the decreases with optocenter distance are remarkably similar except in the inner regions where important variations are noticed.

Figure 5

 5 Figure 5Fragments C, B, G polarimetric phase curves. Comparison with other comets and classification.Red domain and near-infrared.The synthetic phase curves are fits of all available observations for whole coma in the red wavelength domain excluding comet C/1999 S4 LINEAR during its disruption(Hadamcik and Levasseur-Regourd, 2003). Observations by[START_REF] Bonev | Broadband imaging and narrowband polarimetry of comet 73P/Schassmann-Wachmann 3, components B and C[END_REF] are noted Cb and Bb; observations by[START_REF] Jones | Evidence of fragmenting dust particles from near-simultaneous optical and near-infrared photometry and polarimetry of comet 73P/Schwassmann-Wachmann 3[END_REF] are noted Cj, Bj; observations by[START_REF] Kiselev | The anomalous spectral dependence of polarization in comets[END_REF] are noted Ck and Bk. Fragmented comets C/1975 West, C/1996 B2 Hyakutake and C/1999 S4 LINEAR data are also shown for comparison.

  Figure 1aClick here to download high resolution image

Table 1 .

 1 Log of the observations Dates correspond to the beginning of each night e.g. April 27 = 27-28 night. R h = heliocentric distance, = geocentric distance, m vis = visual magnitude, (°) = phase angle, Sun-C PA = extended Sun-comet radius vector position angle (JPL Horizons ephemeris); Exposure time= total exposure time for each polarized component.

	Dates, 2006	R h (au)	au)	m vis	(°)	Sun-C PA (°) Exposure time (s)
	Fragment C
	April 27	1.110	0.155	9.6	45.0	215.4	1020
	April 28	1.103	0.147	9.4	46.4	216.1	840 + 180
	April 29	1.095	0.140	9.3	47.9	217.1	720
	April 30	1.088	0.133	9.1	49.5	218.1	1200
	May 1	1.081	0.126	9.0	51.3	219.4	1200
	May 2	1.074	0.119	8.9	53.3	211.1	620
	May 3	1.068	0.113	8.7	55.5	215.2	900
	Fragment B
	April 27	1.110	0.155	11.8	43.3	205.6	1260
	April 28	1.103	0.147	11.7	44.6	205.8	840 + 180
	April 29	1.095	0.140	11.5	45.9	206.2	1320
	April 30	1.088	0.133	11.4	47.4	206.9	960
	May 1	1.081	0.126	11.3	49.0	207.7	360
	May 2	1.074	0.119	11.0	50.7	208.9	1260
	May 3	1.068	0.113	10.8	52.6	210.3	900 + 600
	Fragment G
	April 29	1.095	0.140	13.2	45.7	204. 3	540
	April 30	1.088	0.133	13.1	47.1	204.8	240 partly cloudy

Table 2 :

 2 Polarization vs aperture diameter. phase angle, scattering plane PA,  obs = measured polarization angle. Error bars correspond to a confidence level of 95 % (2 At optocenter, but in the region blurred by seeing. NA = not available, G too faint and diffuse.

	DATES			obs	Center	500 km	1000 km	2000 km	4000 km	6000 km
		(°)	(°)	(°)	(%)	(%)	(%)	(%)	(%)	(%)

Acknowledgements

The authors thank D. Bardin for his help during the observations. We thank OHP for time allocation, PNP (Programme National of Planetologie) and CNES for partial funding, and CNES and ESA for funding the PROGRA2 microgravity flights.