Jose I Orlicki

Pablo I Fierens

J Ignacio Alvarez-Hamelin

Faceted Ranking of Egos in Collaborative Tagging Systems

Multimedia uploaded content is tagged and recommended by users of collaborative systems, resulting in informal classifications also known as folksonomies. Faceted web ranking has been proved a reasonable alternative to a single ranking which does not take into account a personalized context. In this paper we analyze the online computation of rankings of users associated to facets made up of multiple tags. Possible applications are user reputation evaluation (egoranking) and improvement of content quality in case of retrieval. We propose a solution based on PageRank as centrality measure: (i) a ranking for each tag is computed offline on the basis of the corresponding tag-dependent subgraph; (ii) a faceted order is generated by merging rankings corresponding to all the tags in the facet. The fundamental assumption, validated by empirical observations, is that step (i) is scalable. We also present algorithms for part (ii) having time complexity O(k), where k is the number of tags in the facet, well suited to online computation.

Introduction

In collaborative tagging systems, users assign keywords or tags to their uploaded content, or bookmarks, in order to improve future navigation, filtering or searching (see, e.g., Marlow et al. [START_REF] Marlow | HT06, tagging paper, taxonomy, Flickr, academic article, to read[END_REF]). These systems generate a categorization of content commonly known as a folksonomy.

An example is the collaborative URL tagging system Delicious [Del], which was analyzed in depth by Golger and Huberman [START_REF] Golder | Usage patterns of collaborative tagging systems[END_REF], discovering temporal stability in the relative proportions of tags within a given tagging subject. In this system Internet resources (URLs) are bookmarked and classified with tags by users. Other two well-known collaborative tagging systems for multimedia content are YouTube [You] (videos) and Flickr [Fli] (photos), which are the focus of this paper.

YouTube and Flickr differ from Delicious in that the resources are uploaded by users, so that all contents bookmarked as favorites are inside the system. That is, YouTube and Flickr can be considered closed systems.

Users can be ranked in relation to a tag or set of tags which we call a facet. Some applications of these faceted (i.e., tag-associated) rankings are: (i) searching for content through navigation of the best users inside a tag-facet; (ii) measuring reputation of users by listing their best rankings for different tags or tag sets.

The order or ranking can be determined by a centrality measure, such as PageRank [START_REF] Page | The PageRank Citation Ranking: Bringing Order to the Web[END_REF][START_REF] Nicole | Survey: Deeper Inside PageRank[END_REF], in a recommendation or subscription graph. Given a facet, a straightforward solution is to compute the centrality measure based on an appropriate facet-dependent subgraph of the recommendation network. However, the online computation of the centrality measure is unfeasible because its high time complexity, even for small facets with two or three tags. Moreover, the offline computation of the centrality measure for each facet is also unfeasible because the large number of possible facets. Therefore, alternative solutions must be looked for. A simple solution is to use a general ranking computed offline, which is then filtered online for each facet query. Using a single ranking of web pages or users within folksonomies has the disadvantage that the best ranked ones are those having the highest centrality in a global ranking, which is facet-independent. In the information retrieval case, this implies that the returned results are ordered in a way that does not take into account the focus on the searched topic. This problem is called topic drift [START_REF] Richardson | The Intelligent Surfer: Probabilistic Combination of Link and Content Information in PageRank[END_REF].

In this paper we propose a solution to the problem of topic drift in faceted rankings which is based on PageRank as centrality measure. Our approach follows a two-step procedure: (i) a ranking for each tag is computed offline on the basis of the corresponding tag-dependent subgraph; (ii) a faceted order is generated by merging rankings corresponding to all the tags in the facet.

The fundamental assumption is that step (i) in this procedure can be computed with an acceptable overhead which depends on the size of the dataset. This hypothesis is validated by two empirical observations. On one hand, in the studied recommendation (tagged) graphs most of the tags are associated to very small subgraphs, while only a small number of tags have large associated subgraphs (see Section 3). On the other hand, the mean number of tags per edge is finite and small as explained in Section 4.8.

The problem then becomes to find a good and efficient algorithm to merge several rankings in step (ii). In Section 4, we present several alternatives. We concentrate our effort on facets that correspond to the logical conjunction of tags (match-all-tags-queries) because this is the most used logical combination in information retrieval (Christopher [Chr08], Chapter 1).

The rest of the paper is organized as follows. We discuss prior works and their limitations in Section 2. In Section 3 we explore two real examples of tagged graphs. In particular, we analyze several important characteristics of these graphs, such as the scale-free behavior of the vertex indegree and assortativeness of the embedded recommendation network (see Section 3.3). The proposed algorithms are introduced in Section 4, including an analysis of related scalability issues in Section 4.8. We discuss experimental results in Section 5 and we conclude with some final remarks and possible directions of future work in Section 6.

Related work

Theory and implementation concepts used in this work for PageRank centrality are based on the comprehensive survey of Langville and Meyer [START_REF] Nicole | Survey: Deeper Inside PageRank[END_REF]. This centrality measure for directed graphs is a variation of eigenvector centrality which includes the notion of a random surfer, i.e., an imaginary surfer that, in arriving to a vertex with no out-links, jumps to a randomly chosen vertex. The PageRank algorithm is based on the iterated multiplication of the adjacency matrix of the directed graph (modified to add the random surfer), and a vector representing the probability that a surfer is in a particular vertex. The iteration stops when each vector component does not change more than a given error ǫ. Only a hundred of matrix multiplications are needed for ǫ = 10 -6 and standard parameters (see [START_REF] Nicole | Survey: Deeper Inside PageRank[END_REF] for details).

Basic topic-sensitive PageRank analysis was attempted biasing the general PageRank equation to special subsets of web pages by Al-Saffar and Heileman [START_REF] Al | Experimental Bounds on the Usefulness of Personalized and Topic-Sensitive PageRank[END_REF], and using a predefined set of categories by Haveliwala [START_REF] Taher | Topic-sensitive PageRank[END_REF] extracted from the Open Directory Project [ODp]. Although encouraging results were obtained in both works, they suffer from the limitation of a fixed number of topics biasing the rankings. Another variations of personalized PageRank were augmented with weights based on usage by Eirinaki and Vazirgiannis [START_REF] Eirinaki | Usage-Based PageRank for Web Personalization[END_REF] and on access time-length and frequency by Guo et al. [START_REF] Yong Zhen Guo | Personalized PageRank for Web Page Prediction Based on Access Time-Length and Frequency[END_REF] by previous users, they built a unique PageRank vector adapted to usage but the result is not user dependent nor query dependent as we prefer.

Hotho et al. [START_REF] Hotho | Information Retrieval in Folksonomies: Search and Ranking[END_REF] adapted PageRank to work on a tripartite graph of users, tags and resources corresponding to a folksonomy. They also developed a form of topic-biasing on the modified PageRank, but the generation of a faceted ranking implies a new computation of the adapted PageRank algorithm on the network for each new facet.

There has also been some work done on faceted ranking of web pages. For example, the approach of DeLong, Mane and Srivastava [START_REF] Delong | Concept-Aware Ranking: Teaching an Old Graph New Moves[END_REF] involves the construction of a larger multigraph using the hyperlink graph with each vertex corresponding to a pair webpage-concept and each edge to a hyperlink associated with a concept. Subgraph ideas are suggested by them: "It might be faster to simply run PageRank on sub-graphs pertaining to each individual concept (assuming there are a small number of concepts)." Although DeLong et al. [START_REF] Delong | Concept-Aware Ranking: Teaching an Old Graph New Moves[END_REF] obtain good ranking results for single-keyword facets, they do not support multi-keyword queries.

Query-dependent PageRank calculation was introduced in Richarson and Domingos [START_REF] Richardson | The Intelligent Surfer: Probabilistic Combination of Link and Content Information in PageRank[END_REF] to extract a weighted probability per keyword for each webpage. These probabilities are summed up to generate a query-dependent result. They also show that this faceted ranking has, for thousands of keywords, computation and storage requirements that are only approximately 100-200 times greater than that of a single query-independent PageRank. As we show in Section 4.8, our facet-dependent ranking algorithms have similar time complexity. Scalability issues were also tackled by Jeh and Widom [START_REF] Jeh | Scaling personalized web search[END_REF] criticizing offline computation of multiple PageRank vectors for each possible query and preferring another more efficient dynamic programming algorithm for online calculation of the faceted rankings based on offline computation of basis vectors. They found that their algorithm scales well with the size of set H, the biasing page set, and they criticize previous ideas in [START_REF] Richardson | The Intelligent Surfer: Probabilistic Combination of Link and Content Information in PageRank[END_REF]: "[Richarson and Domingos] suggested that importance scores be precomputed offline for every possible text query, but the enormous number of possibilities makes this approach difficult to scale."

In this paper, we propose a different alternative to the problem of faceted ranking. Instead of computing offline the rankings corresponding to all possible facets, our solution requires only the offline computation of a ranking per tag. A faceted ranking is generated by adequately merging the rankings of the corresponding tags. Section 4 deals with different approaches to the merging step.

Construction of a tagged graph

In this section we introduce the basic definitions related to tagged graphs and we present the network analysis of two real cases.

Basic definitions

Let G = (N, E, T ) be a simple Let M = {(u 1 , m 1 , T 1 ), . . . , (u r , m r , T r )} be a set of tagged contents, where u i is the user, m i is the content and T i is the preferred set of tags included by the user2 , and let V = {(c ′ 1 , m ′ 1 ), . . . , (c ′ p , m ′ p )} the set of favorite recommendations, where c ′ i is a recommender user and m ′ i is the recommended content3 , then a tagged graph G = (N, E, T ) is build, where

N := {u i : ∃i(u i , m i , T i ) ∈ M }∪ {c ′ j : ∃j(c ′ j , m ′ j ) ∈ V }, E := {(c ′ j , u k ) : (c ′ j , m ′ j ) ∈ V ∧ (u k , m ′ j , T k ) ∈ M }, and
T ((c ′ j , u k )) := {T k : (c ′ j , m ′ j ) ∈ V ∧ (u k , m ′ j , T k ) ∈ M ∧ (c ′ j , u k ) ∈ E} .
We show an example of the application of these definitions in Figure 1. Given G = (N, E, T ) and a tag t then G(t

) := (N ′ , E ′ , T ′ ) is a tagged subgraph, where E ′ = {e : e ∈ E ∧ t ∈ T (e)}, N ′ = {a, b : (a, b) ∈ E ′ } and T ′ = {t ∈ T (e ′ ) : e ′ ∈ E ′ }. M0 = {(A, song1, {blues}) V0 = {(A, song2) (B, song2, {blues,jazz}) (B, song4) (C, song3,{blues}) (B, song5) (C, song4,{jazz}) (A, song3) (D, song5,{blues}) (A, song4) (D, song6,{rock}) } (C, song6) } A blues,jazz / /
blues,jazz @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @

V 0 . If G 1 = (N 1 , E 1 , T 1 ) and G 2 = (N 2 , E 2 , T 2 ) are graphs then G 1 ∩ G 2 := (N ′ , E ′ , T ′ ) where E ′ = E 1 ∩ E 2 , N ′ = {a, b : (a, b) ∈ E ′ } and T ′ (e) := T 1 (e) ∩ T 2 (e). Also G 1 ∪ G 2 := (N ′ , E ′ , T ′ ) where E ′ = E 1 ∪ E 2 , N ′ = {a, b : (a, b) ∈ E ′ } and T ′ (e) := T 1 (e) ∪ T 2 (e).
The conjunction and disjunction graphs can be defined as

G(t 1 ∧ . . . ∧ t k-1 ∧ t k ) := G(t 1 ∧ . . . ∧ t k-1 ) ∩ G(t k ) (see Figure 6(c)) and G(t 1 ∨ . . . ∨ t k-1 ∨ t k ) := G(t 1 ∨ . . . ∨ t k-1 ) ∪ G(t k ) (see Figure 6(d)) .
The number of edges of a graph G is denoted |E(G)| and the number of vertices in a graph is denoted by |N (G)|.

Two real cases: YouTube and Flickr

In this section, we present two examples of collaborative tagging systems where content is tagged and recommendations are made. These systems actually rank content according to the number of visits, recommendations or relevance of the text accompanying the content. However, to our knowledge, no use of graph-based faceted ranking is made.

The taxonomy of tagging systems in Marlow et al. [MNBD06] allows us to classify YouTube [You]

and Flickr [Fli] in the following ways:

• regarding the tagging rights, both are selftagging systems;

• regarding the aggregation model, they are set systems;

• regarding the object-type, they are called nontextual systems;

• regarding source of material, they are classified as user-contributed ;

• finally, regarding tagging support, while YouTube can be classified as a suggested tagging system, Flickr must be considered a blind tagging system.

In our first example the content is multimedia in the form of favorite videos recommended by users. The information was collected from the service YouTube [You] using the public API crawling 185852 edges and 51490 vertices in Breadth-First Search (BFS) order starting from the popular user jcl5m that had videos included in the top twenty top rated videos during April 2008. From this information and following the the definitions in Section 3.1, we constructed a complete tagged graph G and several sample subgraphs such as G(music ∨ f unny), G(music), G(f unny) and G(music ∧ f unny) (other subgraphs present a similar behavior). Table 1 presents the number of vertices and edges of each of these networks. We must note that mandatory categorical tags such as Entertainment, Sports or M usic, always capitalized, were removed in order to include only tags inserted by users.

Graph

G(music ∧ f unny) 1,406 1,147
Table 1: Sizes of the video tagged graph and some of its subgraphs.

In our second example the content are photos and the recommendations are in the form of favorite photos4 . The information was collected from the service Flickr [Fli] using the public API crawling 229709 edges and 35210 vertices in BFS order starting from the popular user junku-newcleus. The complete tagged graph G and the sample subgraphs G(blue ∨ f lower), G(blue), G(f lower) and G(blue ∧ f lower) were constructed. The number of vertices and edges of these graphs are shown in Table 2

Network analysis

Graph analysis was made using the tool Network Workbench [N06], except for the calculation of PageRank. Figures 2, 3 and 4 show vertex indegree distribution, vertex outdegree distribution and correlation of indegree of in-neighbors with indegree of vertices for the YouTube and Flickr networks. All graph-analytical parameters, except those for small subgraphs like G(music ∧ f unny) were binned and plotted in log-log curves. This is the reason why some degree points appear below zero and one (x-axis), because there exist vertices with either indegree or outdegree equal to zero. Vertex indegree, in both video and photo networks, is characterized by a power-law distribution: P (k) ≈ k -γ , where 2 < γ < 3 (see Figure 2). Random variables modelled by this type of heavy-tailed distributions have a finite mean, but infinite second and higher non-central moments. Furthermore, there is a non-vanishing probability of finding a vertex with an arbitrary high indegree. Clearly, in any realworld network, the total number of vertices is a natural upper-bound to the greatest possible indegree. However, experience with Internet related networks shows that the power-law distribution of the indegree does not change significantly as the network grows and, hence, the probability of finding a vertex with an arbitrary degree eventually becomes non-zero (for more details see, e.g., Pastor-Satorras and Vespignani [START_REF] Pastor | Evolution and structure of the Internet: A statistical physics approach[END_REF]).

Since recommendation lists are made by individual users, vertex outdegree does not show the same kind of scale-free behavior than vertex indegree. On the contrary, each user recommends only 20 to 30 other users on average (see Figure 3). Moreover, since vertex outdegree is mostly controlled by human users, we do not expect its average to change significantly as the network grows.

The correlation of indegree of in-neighbors with vertex indegree (see Figure 4) indicates the existence of assortative (positive slope) or disassortative behavior (negative slope). Assortativeness is commonly observed in social networks, where people with many connections relates to people which is also well-connected. Disassortativeness is more common in other kinds of networks, such as information, technological and biological networks (see, e.g., Newman [New02]). In the favorite videos network there is no clear correlation (small or no slope), but the photo network there is a slight assortativeness indicating a biased preference of vertices with high indegree for vertices with high indegree (see Figure 4).

We also computed the PageRank of the sample graphs, removing dangling vertices with indegree 1 and out degree 0, because most of them correspond to vertices which have not been expanded by the crawler (BFS), having the lowest PageRank (a similar approach is taken in [START_REF] Page | The PageRank Citation Ranking: Bringing Order to the Web[END_REF]). Figure 5 shows that PageRank distributions are also scale-free, i.e., they can be approximated by power law distributions. Note that the power law exponents are very similar for the complete tagged graph and subgraphs, on each network.

Faceted Ranking on Tagged Graphs

Given a set M of tagged content, a set V of favorite recommendations and a tag set or facet F , the faceted ranking problem consists in finding the ranking of users according to facet F . In this section we present six different approaches to the faceted ranking problem using tagged graphs. The first two algorithms (E-intersection and Eunion/N -intersection in Sections 4.2 and 4.3 respectively) are not scalable for online queries because their computation requires the extraction of a subgraph which might be very large in a large network 5 and the calculation of the corresponding PageRank vector. Moreover, the offline computation of those rankings for each possible facet F ⊆ e∈E T (e) is also unfeasible because the large number of such facets. However, they serve as a basis of comparison for the other four online algorithms because they are a good approximation to the desired result.

We should note that the focus of this paper is on conjunction-based queries in which all words must be matched, as opposed to disjunction-based ones where the match of any word is sufficient. Conjunctionbased queries are the most common type of boolean queries [Chr08].

Before presenting faceted ranking algorithms, we need some preliminary definitions related to vertex centrality which are given in the following section.

Vertex Centrality

Given a graph G = (N, E), C(G) : N → R is a vertex centrality function and R(C(G)) : N → N is a vertex ranking function associates a complete order such that to the highest centrality vertex of C(G) corresponds the number one, the second highest has number two and so on. PageRank C(G) is a vertex centrality function associating probabilities according to a random surfer traversing the graph G [START_REF] Nicole | Survey: Deeper Inside PageRank[END_REF]. The vertex ranking function R(G) := R(C(G)) will 5 We have observed that as the network grows the relative frequency of tags usage converges. Similar behavior was observed for particular resources by [START_REF] Golder | Usage patterns of collaborative tagging systems[END_REF].

be our default ranking for graphs.

A blues,jazz / / blues,jazz @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @

B blues / / D C (a)
A blues,jazz / / blues,jazz @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ B jazz C

A blues,jazz / / blues,jazz @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ B C

A blues,jazz / / blues,jazz @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @

E-intersection

Given a set of tags, a ranking may be calculated by computing the centrality measure of the subgraph corresponding to the recommendation edges which include all the tags. This approach, called E-intersection, cannot be implemented for online queries, as explained above, but serves as a reasonable standard of comparison because we use the exact information available for the PageRank in a conjunctive query. The E-intersection ranking for tagged graph G according to facet

F = {t 1 , . . . , t k } is R(G(t 1 ∧ . . . ∧ t k )).
As an example see Figure 6(c). Assuming a previously built inverted index (Christopher [Chr08], Chapter 1) for the tagged graph mapping tags into sets of edges, the complexity can be decomposed on the retrieval time for each subgraph, which takes proportional to

(k × |E(G(t i ))| max + m log m), where |E(G(t i ))| max is computed for the largest subgraph G(t i ).

E-union/N-intersection

Consider the example given in Figure 1 under the query blues ∧ rock. According to the E-intersection algorithm, there is no node in the network satisfying the query. However, it may seem reasonable to return node D as a response to such search. In order to take into account this case, we devised another algorithm called E-union/N -intersection. In this case, the union of all edge recommendations per tag is used when computing the PageRank, but only those vertices involved in recommendations for all tags are kept. The latter filtering is included because we want vertices recommended for each of the tags in the facet.

The E-union/N -intersection ranking for vertex n in a tagged graph G according to facet

F = {t 1 , . . . , t k } is R(C(G(t 1 ∨ . . . ∨ t k )))(n),
where C is restricted to vertices in vertex intersection N (G(t 1 )) ∩ . . . ∩ N (G(t k )), the other vertices having centrality 0. Note that, in general, there are

more vertices in N (G(t 1 )) ∩ . . . ∩ N (G(t k )) than in N (G(t 1 ) ∩ . . . ∩ G(t k )).
The time complexity of this algorithm is proportional to

k i=1 |E(G(t i ))| + m log m, where m = |E(G(t 1 ∨ . . . ∨ t k ))|. Then, the total time complexity for this algorithm is O (k × |E(G(t i ))| max + m log m),
where |E(G(t i ))| max is computed for the largest subgraph G(t i ).

Single ranking

A simple online faceted ranking consists of a monolithic ranking, without considering the facet, which is then filtered to exclude those vertices that are not related to all tags in the facet. That is, one ranks by the monolithic global rank of the complete tagged graph and the only vertices remaining for facet {t 1 , . . . , t k } are the ones in

N G(t 1 ) ∩ . . . ∩ N G(t k ) .
Assuming a precomputed inverted index, mapping tags into nodes, the complexity of this algorithm is

O(k×|N (G(t i ))| max ),
where k is the number of different tags in the facet, and |N (G(t i ))| max is computed for the biggest subgraph G(t i ). It is also possible to retrieve a (small) constant number of top elements to intersect, yielding a time complexity of O(k).

P R-product

In order to approximate efficiently the edge intersection we can precompute individual rankings for each tag and then combine them by element-wise multiplication. This approximation is inspired on the probability product of independent events.

If G 1 and G 2 are subgraphs of graph G we define PageRank ranking product as

R(G 1 ) • R(G 2 ) := R(C(G 1 ) • C(G 2 )), where (C(G 1 )•C(G 2 ))(n) := C(G 1 )(n)•C(G 2 )(n) (real product). The P R-product ranking for tagged graph G according to facet F = {t 1 , . . . , t k } is k i=0 R(G(t i )).
Assuming the individual rankings for all tags have been computed, the complexity of this algorithm is

O(k × |N (G(t i ))| max ).
Here, it is also possible to reduce the time complexity to O(k) taking a (small) constant number of top elements to make the product.

R-sum

Consider a recommendation graph G larger than that in Figure 1 and the query blues ∧ jazz. Assume that the PageRank of the top three nodes in the rankings corresponding the subgraphs G(blues) and G(jazz) are as given in Table 3. Ignoring other nodes, the ranking given by the P R-product rule is a, b and c. However, it may be argued that node b shows a better equilibrium of PageRank values than node a. Intuitively, one may feel inclined to rank b over a given the values in the table. In order to follow this intuition, we devised the R-sum algorithm which is also intended to avoid topic drift inside the queried facet, that is, any tag prevailing over the others.

The R-sum ranking for a tagged graph G according to facet

F = {t 1 , . . . , t k } is k i=0 R(G(t i )),
where we define PageRank ranking sum as

R(G 1 ) + R(G 2 ) := R(-(R(G 1 ) + R(G 2 ))).
Notice that in this sum we are using as centrality the sum of ranking positions in a reverse order, and according to the R-sum algorithm, the ranking of nodes in the example of Table 3 is b, a and c.

The complexity of this algorithm is similar to that of P R-product.

τ -N-intersection

In this case, edge intersection is computed involving only vertices (and associated edges) that are on the top w positions of the individual rankings. The τ -Nintersection ranking for tagged graph G according to Table 3: Comparison of P R-product and R-sum in an example.

facet

F = {t 1 , . . . , t k } is R k i=0 G(τ (R(G(t i )), w))(t i ) ,
where τ (R(G), w) is the set of vertices including the top w vertices ranked using PageRank and G({a, b, . . .}) is the maximal subgraph of G including vertices {a, b, . . .} and edges connecting them. In other words, this algorithm has the following steps: (i) for each t i , the subgraph G(t i ) is constructed; (ii) a ranking of users is computed on the basis of the PageRank of G(t i ); (iii) the w winners of each t iassociated ranking are extracted; (iv) given a facet

F = {t 1 , • • • , t k },
a new subgraph including only the winners for tag t i is constructed; (v) a facetassociated ranking is constructed based on the new graph. Steps (i)-(iii) are computed offline. In this presentation, we have fixed the number of top items selected at five hundred (w = 500). Assuming the individual rankings for k tags has been computed, the complexity of this algorithm is O(k).

Scalability Analysis

As noticed by Langville and Meyer [START_REF] Nicole | Survey: Deeper Inside PageRank[END_REF], the number of iterations of PageRank is fixed when both the tolerated error and other parameters are fixed, yielding one hundred for ǫ = 10 -6 (see Section 2). As each iteration consists of the sparse adjacency matrix multiplication, the time complexity of PageRank is linear on the number of edges of the graph. In our case, given a tagged graph G 0 = (N 0 , E 0 , T 0 ), for each tag there is a corresponding subgraph with a known size. Then the total temporal and spatial complexity of the faceted PageRank for all individual tags is

t∈T ′ 0 |E(G 0 (t))| = e∈E0 |T 0 (e)|,
where T ′ 0 := e∈E0 T 0 (e), the complete set of tags. Therefore, if the average number of tags per edge is constant or grows very slowly as the graph grows, then the algorithms in Sections 4.5, 4.6 and 4.7 are scalable, linear on the number of edges of the complete tagged graph. This can be verified empirically on Figure 7, showing that distribution of tags per edges falls quickly, having a mean of 9.26 tags per edge for the YouTube tagged graph and 13.37 for the Flickr tagged graph. These are not heavy-tailed distributions and, since tags are manually added to each uploaded content, we do not expect the average number of tags per recommendation to increase significantly with network growth. In our experiments the computation of all the faceted singleton tag rankings (104, 927 tags) for the video network sample took 211.4 times more time than the single ranking for the complete tagged graph. Meanwhile the photo network sample (283, 093 tags) took 1744.9 times more time.

Our merging algorithms work in real-time because they use only the top w results, where w is a small fixed number like 500 or 1000. Choosing an appropriate w for an application 6 will enable it to store only 6 How to choose a good w is beyond the scope of this paper.

the w top elements of each single-tag facet.

Experimental results

In this section, we compare the behavior of the algorithms presented in Section 4. As a basis of comparison we use two algorithms whose online computation is unfeasible, but which are intuitively reasonable: E-intersection (Section 4.2) and E-union/N -intersection (Section 4.3). In order to quantify the "distance" between the results given by two different algorithms, we use two ranking similarity measures, OSim [START_REF] Taher | Topic-sensitive PageRank[END_REF] and KSim [START_REF] Kendall | A New Measure of Rank Correlation[END_REF][START_REF] Taher | Topic-sensitive PageRank[END_REF]. The first measure, OSim(r 1 , r 2 ) indicates the degree of overlap between the top n elements of rankings r 1 and r 2 . We define the overlap of two sets A and B (each of size n) to be |A ∩ B|/n. The second measure,

KSim(r 1 , r 2 ) = |(u, v) : r ′ 1 , r ′ 2 same order ∀(u, v), u = v| |U |(|U | -1)
where U in the union of all elements in rankings r 1 and r 2 , r ′ 1 is r 1 extended with Ur 2 and r 2 is extended analogously to obtain r ′ 2 . This measure is a variant of Kendall's distance that considers the relative orderings, i.e., counts how many inversions are in a determined top set. In both cases, values closer to 0 mean that the results are not similar and closer to 1 mean the opposite.

Favorite videos network

Samples include all facets of tag pairs {t j , t k } extracted from the 99 most used tags of the network7 . That is, 4851 tag pairs compared with their similarities averaged. For each tag pair the proposed merging algorithms (P R-product, R-sum and τ -Nintersection) were compared with the reference algorithms (E-intersection and E-union/N -intersection) using OSim and KSim to measure the rankings' similarity. Some of the tags are: music, funny, comedy, live, guitar, rock, super, dance, animation, parody, song, mario, game, new, tv, pop, john, love, world.

Table 4 presents a summary of the comparisons for the favorite videos network, where we display aver-aged similarities for different top sizes of ranked users. Figures 8 and9 also show a more detailed summary of results for the OSim metric (because it discriminates different situations better than KSim). The x-axis corresponds to the number of vertices resulting from the basis of comparison algorithm (E-intersection or E-union/N -intersection) and the y-axis to the top number n of vertices used to compute the similarities. The similarity results (between 0 and 1) falling in each of the log-log ranges were averaged. Observe that darker tones correspond to values closer to 1, i.e., more similar results. White spaces correspond to cases for which there are no data, e.g., whenever the y coordinate is greater than intersection size.

Favorite photos network

Experiments with Flickr were similar, top 99 tags paired to form 4851 tag pairs. A small sample of the top 99 tags is: bw, portrait, nature, bravo, sky, blue, water, soe, flower, light, clouds, sunset, red, film, macro, white, landscape, green, girl, blackandwhite.

Discussion

As can be appreciated from Tables 4-5 and Figures 89, the Single Ranking algorithm gave the worst results in most cases.

Since the τ -N -intersection algorithm is based on retaining only the 500 top-ranked users for each tag, it is natural to observe a worse OSim measure than the other algorithms especially for larger than 500node intersections. However, this algorithm gives worse results even for smaller intersections. This fact is explained by the relevance of a large number of recommendations of low-ranked users when computing the PageRank in both the E-intersection and the E-union/N -intersection cases. Also note that the τ -N -intersection approach gave better results on the photo network than in the video network. A possible cause is the assortativeness of photo network (see Figure 4 and Section 3.3). Indeed, since assortativeness implies that users with many recommendations are preferentially recommended by users with also many recommendations, the relevance of low-ranked users in the computation of the centrality measure is lower.

There is a remarkable improvement using algorithm R-sum compared to the other merging al-gorithms when considering the similarity to the E-intersection standard on both networks. Also, the best merging algorithm for the similarity with the second standard E-union/N -intersection is P Rproduct merging algorithm.

Summary

We have proposed different algorithms for merging faceted-rankings of users in collaborative tagging systems which gave results comparable to those of two reasonable standards. We have also analyzed the scalability of this approach.

A prototypic application called Egg-O-Matic is available online [EOM] including ranking merging Rsum to approximate the E-intersection ranking, in a mode called "all tags, same content", and including the ranking merging we called P R-product to approximate the E-union/N -intersection ranking, in a mode called "all tags, any content".

Another step that can be taken to reduce tagdimensionality is clustering to agglomerate them. This work also opens the path for a more complex comparison of reputations, for example by integrating the best positions of a user even if the tags involved are not related (disjunctive queries) in order to summarize the relevance of a user generating content on the web. It is also possible to extend the algorithms in Section 4 to merge of rankings generated from different systems (cross-system ranking) looking to obtain a ranking of users using multiple collaborative tagging systems.

1 :

 1 Example of construction of a tagged graph from a set of contents M 0 and a set of recommendations

Figure 3 :Figure 4 :Figure 5 :

 345 Figure 3: Binned outdegree distribution

Figure 6 :

 6 Figure 6: Example subgraphs of a tagged graph: (a) G(blues); (b) G(jazz); (c) G(blues ∧ jazz); (d) G(blues ∨ jazz).

  k i=1 |E(G(t i ))|, and the time of PageRank and sort algorithms, taking O(m log m), where m = |E(G(t 1 ∧ . . . ∧ t k ))|. Then, the total time complexity for this algorithm is O

  Node C(G(blues)) C(G(jazz)) P R-pr.

Figure 7 :

 7 Figure 7: The distribution of number of tags per edge.

Figure 10 :Figure 11 :

 1011 Figure 10: Photos network: Average similarity to E-intersection

  1 directed graph with tags on the edges, a tagged graph. N is the set of vertices {u 1 , . . . , u n }, E is the set of edges and T (e) is the set of tags {t 1 , . . . , t ke } associated with edge e in E. If e / ∈ E then T (e) := ∅. We shall call a certain set of tags F ⊆ e∈E T (e) a facet.

Table 2 :

 2 . Sizes of the photo tagged graph and some of its subgraphs.

	Graph	vertices	edges
	G	35,210 229,709
	G(blue ∨ f lower)	12,921	20,105
	G(blue)	10,241	11,703
	G(f lower)	7,032	9,566
	G(blue ∧ f lower)	1,551	1,164

Table 5 :

 5 Table 5 as well as Figures 10 and 11 summarize the results. Photos network: Comparison of ranking algorithms

	Average similarity to E-intersection
	Algorithm		OSim|KSim
		top 8	top 16	top 32
	Single	0.08|0.48	0.10|0.50	0.13|0.51
	P R-product 0.36|0.56	0.37|0.58	0.39|0.59
	R-sum	0.53|0.63 0.53|0.64 0.52|0.66
	τ -N -inters	0.15|0.49	0.15|0.51	0.10|0.51
	Average similarity to E-union/N -intersection
	Algorithm		OSim|KSim
		top 8	top 16	top 32
	Single	0.31|0.53	0.34|0.55	0.39|0.56
	P R-product 0.72|0.70 0.78|0.74	0.83|0.79
	R-sum	0.35|0.54	0.42|0.56	0.50|0.59
	τ -N -inters	0.13|0.49	0.12|0.51	0.09|0.51
	Table 4: Videos network: Comparison of ranking al-
	gorithms

Not a multigraph.

In the rest of the paper user or vertex will be used indistinctly to mean a vertex in a tagged graph as an abstraction of the webpage where the user publishes his/her content, and edges or links will be used when referring to edges in a tagged graph build using favorite recommendations.

3 Each content is considered unique, i.e., different users do not upload the same content.

Only the first fifty favorites photos of each user were retrieved.

Some tags like you, video or youtube which give no information were removed from the experiment.


Teresa Lopez-Leon
email: teresa.lopez-leon@univ-montp2.fr

Alberto Fernandez-Nieves

Maurizio Nobili

Christophe Blanc

Nematic-Smectic Transition in Spherical Shells

Keywords: numbers: 61.30.Jf, 61.30.Hn, 64.75.Xc

We study the nematic-smectic phase transition of a thermotropic liquid crystal confined to a spherical shell. Far from the nematic-smectic phase transition temperature, TNS, we observe a configuration with four +1/2 defects, as predicted by theory. Since in this case K1 ≈ K3, the four defects are confined at the thinnest part of the shell to minimize the energy associated to the defect cores. By contrast, near TNS, where K3 ≫ K1, bend distortions become prohibited and the defects organize themselves along a great circle of the sphere, confirming recent theoretical and simulation results. During this structural change, the defects associate in two pairs that behave independently. In the smectic phase, we observe a new configuration displaying curvature walls.

Nematic liquid crystals are typically formed by rodlike molecules displaying orientational order along an average direction, n, called the director. When a nematic is confined to a spherical shell, the director field is necessarily disrupted by the presence of topological defects [1,[START_REF] Kleman | Soft Matter Physics: An Introduction[END_REF]. These systems not only are rich systems to study topological problems, but they also provide a promising route for generating colloids with a valence [START_REF] Nelson | [END_REF], which is an important goal in material science [4]. Different defect structures have been observed, differing in the number and distribution of defects [5]. The associated elastic energy is given by the Franck-Oseen free energy density [START_REF] De Gennes | The physics of liquid crystals[END_REF]:

f = 1 2 K 1 (∇•n) 2 + 1 2 K 2 (n•∇×n) 2 + 1 2 K 3 (n×∇×n) 2 (1)
where K 1 and K 3 are the elastic constants associated to splay and bend distortions. In the limit of a very thin shell, the director can be regarded as a surface field, and thus, the twist term associated to K 2 can be neglected.

When K 1 = K 3 , theory predicts a configuration with four defects arranged in a tetrahedral fashion [START_REF] Lubensky | [END_REF], as shown in Fig. 1(a). In this configuration, each defect has a topological charge s = + 1 2 , reflecting the π rotation experienced by n along a path encircling each defect. This is consistent with a mathematical theorem due to Poincaré and Hopf, which establishes that the total topological charge for a nematic sphere is s = +2 [8]. Interestingly, a new arrangement is expected when K 3 ≫ K 1 , in which the four s = + 1 2 defects are placed along a great circle of the sphere [9,10], as shown in Fig. 1(b). These theoretical expectations, however, have never been experimentally tested nor the effect of the anisotropy in the elastic constants been experimentally explored.

In this letter, we study the effect of the anisotropy in elastic constants K 1 and K 3 by taking advantage of the divergence of K 3 near the nematic-smectic phase transition temperature, T N S [11]. Due to the heterogeneous thickness of the experimental shells, the four s = + 1 2 defects characteristic of the nematic phase appear confined in a small region of the shell and not arranged in a tetrahedral configuration. By decreasing the temperature toward T N S , the defects progressively separate to eventually arrange themselves in a great circle. Below T N S , the translational symmetry of the nematic phase is broken and a smectic shell is formed. To fabricate the liquid crystal shells, we generate double emulsions in a microfluidic glass capillary device [12]. An inner aqueous droplet is encapsulated inside an outer liquid crystal droplet, as schematically shown in Fig. 1-(c), which is in turn dispersed in a continuous aqueous phase. The average thickness of the shell, h, is given by the difference between the outer and inner radii, R-a. In our experiments, R is within the (30 -120)µm range and h is typically (2 -3)% of R. The inner and outer phases contain 1 wt% of polyvinyl alcohol (PVA, M w ∼ 20.000 g mol -1 ), which stabilizes the double emulsion and imposes planar degenerate anchoring to the liquid crystal. The middle phase is 4-n-octyl-4-cyanobiphenyl (8CB), a liquid crystal displaying a nematic-smectic phase transition at T N S = 33.5 • C. Since the inner aqueous solution is slightly denser than 8CB, the inner droplet sinks inside the outer one, making the shell thinner at the bottom and thicker at the top. In terms of thickness, the ex- perimental shells are, thus, axisymmetric with respect to the gravity direction, with a non uniform thickness δ ≈ h(1 -cos θ) that depends on the zenithal angle θ [Fig. 1(c)]. We fabricate and store the shells at 37 • C, where 8CB is in the nematic phase. Due to the divergence of K 3 at T N S , an accurate control of temperature is required to guarantee a quasistatic approach to the nematic-smectic phase transition; we use a heating stage with a Lakeshore controller to regulate the temperature with a precision of 0.01 • C. Additionally, after changing temperature, we wait for 20 minutes to ensure that the system is indeed in an equilibrium state.

Figure 2(a-g) shows the birefringence texture of a nematic shell with four defects as the temperature T is quasi-statically decreased toward T N S . The evolution of the defect positions and director fields is represented in Fig. 2(h-k). When T -T N S 1 • C, the four defects appear confined at the bottom of the shell, as shown in Fig. 2(a). However, as T approaches T N S , defects 1 and 2 start moving away from each other, while defects 3 and 4 only slightly increase their separation, as shown in Figs. 2(a-d). It is only after defects 1 and 2 reach the equatorial plane that defects 3 and 4 significantly move away from each other to eventually reach the equatorial plane too, as shown in Figs. 2(e-g). To quantify the evolution of the defect pairs, we plot the ratio between the angles subtended by defects 1-2 and 3-4 with respect to the center of the shell, α 12 and α 34 , as a function of T -T N S . When T is far from T N S , the defects are almost located at the vertices of a square and α 12 ≈ α 34 , as shown in Fig. 3. Decreasing T causes the elongation of the structure into a rhombus with α 12 > α 34 . This elongation becomes maximum at T -T N S = 0.06. Below this, defects 3-4 progressively migrate toward the equatorial plane. As a result, α 12 /α 34 decreases until eventually the defects are located in a great circle and α 12 ≈ α 34 again, as shown in Fig. 3.

The α 12 /α 34 ratio is closely related to the evolution of K 3 and K 1 with T . Close to T N S , K 3 diverges with respect to K 1 , as shown in the inset of Fig. 3(a), and thus, bend distortions become energetically prohibited. This forces n to align along great circles, which are the counterparts of straight lines on the sphere. In this configuration, schematically represented in Figs. 2(k), the four defects are necessarily located on a great circle. For shells homogeneous in thickness, all great-circle arrangements are degenerated in energy. However, this degeneracy is broken in our experimental shells that are heterogeneous in thickness. In this case, the ground state is achieved when the four defects are located on the equatorial plane, along the only great circle in which δ is uniform.

Far from T N S , when K 3 ≈ K 1 , the four defects are confined at the thinnest part of the shell, as shown in Fig. 2-(a). This results from the non-uniform thickness of these experimental shells [13]. Due to the non-vanishing thickness of the shells, the +1/2 defects are disclination lines that span the shell; to minimize their length, and thus the energy of the system, they group together at the thinner part of the shell. The approximate square arrangement of the defects comes from the approximate isotropy of the elastic constants at high temperatures. In fact, if K 3 /K 1 = 1, the elastic energy associated with any director field is invariant under a π 2 rotation in n, converting bend distortions into splay distortions and vice versa. Therefore, the four defects must be located at the vertices of a square in their equilibrium configuration when

K 3 /K 1 = 1.
At intermediate temperatures, when 1 < K 3 /K 1 < ∞, the energy invariance under a π 2 rotation of n is broken, and thus, the four defects are necessarily arranged in a rhombus structure. Since K 1 does not change much in the temperature range studied, the motion of the defects is mainly controlled by the energy term associated to bend. To explain why the distance between defects 1-2 becomes larger than the distance between defects 3-4 when K 3 /K 1 increases, we investigate how increasing K 3 affects the equilibrium texture shown schematically in Fig. 2(h), where K 3 ≃ K 1 ≡ K. This nematic texture has a free energy that depends on K and the core energies of the defects. Now, we consider an increase of the bend constant from K to K 3 = K + ∆K 3 , while we keep the position of the defects fixed. The additional bend energy density, f b , associated to this increase in the bend constant is mainly controlled by the radius of curvature of the director lines on the sphere. For a longitudinal director field such as the one shown in Fig. 2(k), f b is zero since the integral curves of the director field are great circles, whereas f b = ∆K 3 cot 2 θ/2R 2 for a latitudinal texture [START_REF] Vitelli | We use here the covariant approach to nematic elasticity developed[END_REF]. In Fig. 2(h), the curvature of the director lines is given by the locations of the defects, which allow us to split the sphere into two caps. In the lower cap, defined by the position of defects 3 and 4 [shaded region in Fig2(h)], the magnitude of f b is given by f b V (1) , where V (1) and V (1) are the volumes corresponding to the regions considered. In spherical coordinates,

V (1) = α34/2 0 2πR 2 h(1 -cos θ) sin θdθ ≈ πR 2 hα 4
34 /64 and V (2) ≈ 4πR 2 h. In the limit of small angular distances, we obtain F ∼ α -2 12 , the system will spontaneously increase α 12 as K 3 increases, as we observe experimentally. At T N S , the shell becomes smectic. In the smectic phase, the molecules assemble in equispaced layers perpendicular to n. From a geometric point of view, the nematic texture shown in Fig. 2(k) can be readily converted into a simple smectic texture that preserves the four s = + 1 2 defects [15]. The resulting smectic texture is schematically represented in Fig. 4(a), n is aligned along great circles and the bi-dimensional layers follow latitude lines. Experimentally, however, the texture develops a series of additional distinctive patterns in time. Just after the phase transition, a set of longitudinal lines form on the thicker part of the shell along the direction of the former director field, as shown in Fig. 4(b). These lines connect defects 1 and 2 and divide the shell into several crescent domains; one of those domains is highlighted in Fig. 4(b). As the sample is rotated between cross-polarizers, the extinction angle periodically changes from -13 • to +13 • from one crescent domain to the next one, as shown in the inset of Fig. 4(b). Since n tilts in opposite symmetric directions in adjacent domains, the longitudinal lines can be identified as curvature walls [16]. We note that the resultant zig-zag modulation of the smectic layers reminds of the patterns observed after a Helfrich Hurault instability, as schematically indicated in Fig. 4(c).

After formation of these curvature walls, we observe that a secondary set of curvature walls form, but these have a much smaller angular distortion, ≈ 1 • , than the They extend across each domain perpendicularly to the tilted director field and give rise to a dark/light striped pattern inside domains, as shown in Fig. 4(d). The transition ends with the formation of primary curvature walls at the lower hemisphere, as shown in 4(e).

The description provided up to this point implicitly considers that n is uniform through the shell thickness, which is equivalent to a concentric stacking of the bidimensional structure of Fig. 4(c). Within this picture, the curvatures walls would stem from the significant splay distortions in Fig. 4(a), which are released via the dilation of the layers at the walls [START_REF] De Gennes | The physics of liquid crystals[END_REF]. Alternatively, the walls could result from the non-zero thickness of our experimental shells, as explained bellow.

In a thick shell, where the inner and outer spheres have significantly different radii, a strict planar anchoring can be satisfied on the inner sphere but not on the outer one; this results from the fact that having strong planar anchoring on a curved surface is only compatible with a bulk smectic texture without defects or layer dilations if the smectic layers grow outwardly from the convex side of the surface [15]. As a result, the smectic layers filling the shell are tilted at the outer surface violating the preferred planar anchoring and entailing an anchoring energy cost. To illustrate this, we construct a local bulk smectic texture from a surface texture with planar anchoring in Fig. 4(f); the result is a tilt angle, ψ, at the outer surface. For typical R = 119µm and h = 4µm, we obtain a typical tilt angle ψ = arccos(a/R) ≈ 15 • .

It is tempting to think that this supplementary anchoring energy can be decreased if the smectic layer is progressively twisted from the inner sphere to the outer sphere in the way shown in Fig. 4(g). The twist angle is φ = 13 • at the outer sphere and φ = 0 at the inner sphere. Note that this twist implies a slight dilation of the layers via curvature walls. The zig-zag of the smectic layers at the surface sphere implies these are longer than the corresponding smectic layers at the inner sphere; this corresponds to a larger effective layer separation, d ′ = d/cosφ, with d is the spontaneous thickness of the layer [see Fig. 4(h)]. The number of smectic layers on both surfaces is the same when d ′ /d = R/a or equivalently when cos φ = a/R. For typical values of R and a, we obtain a twist angle at the outer surface of φ = 15 • , consistent with our experimental observations. Such an arrangement, however, also induces a tilt in the layers at the outer surface. Accommodating φ = 0 at the inner surface and φ = 13 • at the outer surface without layer dilation entails tilting the smectic layer by an angle ψ, with maximal value given by tan ψ = h/(p tan Φ), where p is the width of the corresponding crescent domain [see Fig. 4(i)]. Using the experimental values p ≈ 15 µm, φ ≈ 13 • and h ≈ 4µm), we obtain ψ = 45 • , implying there is also an important penalty in anchoring energy in the case of the considered twisted structure. Detailed energy calculations are required in order to elucidate the ultimate origin of the observed curvature walls.

Nematic shells undergo significant configurational changes as K 3 /K 1 is varied. When K 3 /K 1 ≈ 1, the four s = + 1 2 defects appear confined in the thinnest part of the shell, approximately located at the vertices of a square. By contrast, when K 3 /K 1 ≫ 1, the four defects relocate on the equatorial plane, confirming recent theoretical and simulation expectations [9,10]. The transition between these two limiting configurations proceeds continuously through a series of equilibrium states in which the defects typically arrange in a rhombus whose diagonals ratio depends on the value of K 3 /K 1 . This offers new possibilities for controlling the position of defects, and eventually, if these are functionalized, the directionality of the interactions between shells. New types of defects, not observed in shells before, appear when the nematic order is replaced by smectic order. At this point, we observe the formation of primary and secondary curvature walls, whose origin we have discussed by considering two possible scenarios. Our results open the way to rigorous theoretical calculations and exemplify the richness of behaviors that can be obtained when the elasticity of the material or the degree of order in the shell is changed.

T.L-L. thanks the European Marie Curie Program (IEF-236091). A.F-N. thanks the National Science Foundation (DMR-0847304).

FIG. 1 :

 1 FIG. 1: Arrangement of four s = + 1 2 disclinations on a bi-dimensional shell (a) in a tetrahedral configuration when K1 ≈ K3, and (b) along a great circle when K1 ≪ K3. (c) Cross-section of an experimental shell with non-uniform thickness.

FIG. 2 :

 2 FIG. 2: (a-g) Cross-polarized images of a 8CB nematic shell as temperature T decreases toward the nematic-smectic phase transition temperature TNS. (h-k) Evolution of the director field and defect structure during the process, in which K3/K1 changes between 1 < K3/K1 < ∞. Each panel shows the director field on the lower (left) and upper (right) hemispheres.

≈

  ∆K 3 cot 2 (α 34 /2)/2R 2 , while for the remaining part of the sphere, f(2) b ≈ ∆K 3 cot 2 (α 12 /2)/2R 2 .The corresponding energies, F

≈

  πh∆K 3 α 2 34 /32 and F (2) b ≈ 8πh∆K 3 /α 212 . The motion of the defects must be driven by the relaxation of this additional bent energy. Since F

FIG. 3 :

 3 FIG. 3: (a) Ratio between the angular distances α12 (between defects 1-2) and α34 (between defects 3-4) as a function of T -TNS. The inset shows the variation of K3 and K1 versus T -TNS [11]. (b) Geometric definition of α34.

FIG. 4 :

 4 FIG. 4: (a) Longitudinal texture in a bi-dimensional smectic shell. The schematic shows the director field (discontinuous lines) and smectic layers (continuous lines) on the upper hemisphere. (b) Birefringence texture observed in experimental smectic shells, where curvature walls divide each hemisphere in crescent domains. The inset in (b) shows that n is tilted in opposite directions in adjacent domains. (c) Director field corresponding to the texture shown in (b), where the walls provoke a zig-zag modulation of the smectic layers. (e) Secondary curvature walls provoking a secondary optical modulation inside crescent domains. (d) Primary curvature walls at the thinner part of the shell. (f) Local 3-dimensional smectic texture resulting from the bi-dimensional one shown in (a). (g) Possible 3-dimensional texture in which the smectic layers twist through the shell thickness. (h) In this configuration, the effective layer thickness d ′ increases at the outer sphere compensating the effect of the different curvatures of the inner and outer spheres. (i) This texture implies a large tilt angle ψ with respect to the outer sphere normal.


Book review : Youth unemployment and employment policy: a global perspective Niall O'Higgins, Geneva: International Labour Office, 2001. pp. xii212. Price: 27.50 Swiss Francs (paper).

Gérard LASSIBILLE IREDU-CNRS Université de Bourgogne

Publié dans Economics of Education Review, 21, 2002, pp.394-395 This book considers the issue of youth unemployment in various countries (industrialized, in transition and developing) and provides concrete policy recommendations to promote employability of young people.

The first part of the book documents the "problem" of youth employment. After describing the incidence of youth unemployment among OECD countries in chapter one, the book analyzes in chapter two the characteristics of youth employment. Based on international data (gathered by OECD and ILO, as well as information extracted from specific country report studies) the chapter shows that unemployment is not evenly spread among young people. Indeed, the international comparisons indicate clearly that unemployment is significantly higher among young people with low levels of education as well as the disabled, ethnic minorities and, to some extent, among young women. The foregoing picture suggests that youth employment policies should not be aimed at young people as a whole, but rather should con-centrate on those particular segments of the population. Chapter three sheds light on both the causes and consequences of youth unemployment. The available cross country experience shows that youth unemployment mainly reflects poor macroeconomic performance; as a consequence emphasis must be placed on creating sufficient growth and improving the employment content of growth. Other factors like the rise in the relative size of the youth labour force, are also important in explaining the incidence of unemployment among young people. However, in most countries, sustained economic growth can overcome many of the problems associated with the rapid growth of the youth labour market. In contrast, international data indicate that there is no strong evidence that youth unemployment is closely linked to youth/adult relative wage rates. Persistent and high unemployment among the youth are a major cause of concern. More attention should be paid to redressing the imbalance in job opportunities available to disadvantaged young people, due to the fact that both the incidence and the duration of unemployment are particularly important among certain groups of the population.

The second part of the book looks at the policy response to youth unemployment across different countries. Chapter four is a brief discussion on the usefulness of an overall youth policy. It describes different initiatives and recommendations adopted at international level for improving youth employment policy (such as the "World Programme of Action for Youth to the Year 2000 and Beyond" adopted by the United Nations General Assembly in 1995, and the 1998 "Guideline for Employment Policy" adopted by the European Commission) arguing that national educational and employment policies have to be coordinated. Chapter five focuses on the role of monitoring and evaluation of the design and implementation of any new policy and programme, aimed at reducing youth unemployment. Improving labour market information and data collection, particularly in developing countries, and creating a system of continuous monitoring and evaluation, are essential elements for effective responses to the youth employment problem. Chapter six analyzes the impact of a change in the minimum wage on youth employment. Based on a wide range of studies carried out in industrialized and developing countries on this topic, the author shows that increasing youth wages does not necessarily lead to lower youth unemployment. According to international experiences such policies have generally a small or zero effect on youth employment and there is little or no reason for adopting such a measure, although it may have other useful effects on income distribution. Chapter seven looks at the role of education and training systems in reducing youth unemployment. Indeed, the chapter examines the organization of education systems according to several aspects (role of certification, sources of finance, school vs work-based training, sequential vs dual systems); the chapter also analyzes the ability of the German dual system to facilitate the transition from school to work, and discusses the extent to which such a system could be adopted by other countries. Chapter eight examines the role of active labour market policies in reducing youth employment. The chapter focuses on three kinds of policies most commonly used across countries: a) policies designed to promote wage employment; b) policies encouraging self-employment among the young and c) policies aimed at disadvantaged young people. Indeed, experiences from several countries (The UK, Jamaica, The USA, Chile) are documented. Chapter nine discusses the role workers' and employers' organizations may play in the design and implementation of youth employment policies. For example, the involvement of social partners in determining the curricula of courses within the dual apprenticeship system, or in formulating and implementing active labour market policies, is likely to improve the relevance and effectiveness of programmes aimed at facilitating the transition of young people from school to work. Finally chapter ten summarizes the main policy recommendations discussed throughout the book.


domain: sdv sde

text:
Jean-Claude Krapez
email: krapez@onera.fr

Albert Olioso
email: olioso@avignon.inra.fr

A combination of temperature, vegetation indexes and albedo, as obtained by airborne hyperspectral remote sensing, for the evaluation of soil moisture 1

Keywords: soil moisture, soil wetness, remote sensing, triangle method, vegetation index, hyperspectral, thermal infrared

Temperature as obtained by infrared remote sensing of bare soil or vegetation covered soil during day depends on soil water content ranging from the surface down to the root-zone. The main reason is that evaporation and plant transpiration cools down the soil surface and the leaves. However surface temperature does not only depend on soil moisture. It has to be combined with other measured variables. In this work we start with the classical combination between temperature and a vegetation index and we explore the benefit of adding the albedo or the cellulose absorption index to retrieve the surface soil moisture. The correlation between the inferred Soil Vegetation Wetness Index and the true moisture content is analyzed based on the thermal infrared and visible-NIR hyperspectral images recorded during Hymap 2007 campaign over Camargue (France).

Introduction

Information on soil moisture is of first importance for irrigation scheduling and for improving crop yield, but also in water management in the climate change context and for the evaluation of the anthropic ecosystem impact. Remote sensing with earth observation instruments has been considered for a few decades to provide the requested information over large areas and at various scales.

Vegetation and soil temperatures have long been recognized as an indicator of water availability. As a matter of fact, under proper water supply, the transpiration cools the leaves towards air temperature. Water is exchanged through the stomata, i.e. the pores in the leaf epidermis. On the opposite, under water stress condition, the plant closes its stomata to limit transpiration. This leads to foliage temperature increase. Similarly, the soil surface temperature is inversely correlated with water content due to evaporation. Additionally, soil drying induces a reduction of soil thermal inertia which also leads to soil temperature increase.

Thermal sensors used for environment monitoring on board of satellite or airplane have a spatial resolution ranging from a few meters to a couple of kilometres. The consequence is that in the footprint corresponding to a pixel, the surface is seldom homogenous: most pixels are a mixture of vegetation and soil. The radiance reaching the elementary detector is thus a combination of the radiance coming from leaves and from soil. As a consequence, the brightness temperature depends on temperature and emissivity of both soil and vegetation, and on the vegetation cover fraction. Part for this reason, the problem of evaluating the soil moisture is underdetermined when only considering the brightness temperature: a same brightness temperature value can be associated with a large range of soil moisture. It is thus necessary to add other observation data to reduce this underdetermination and finally to reduce the uncertainty of the inferred moisture.

A method proposed a few years ago consists in jointly analyzing the temperature and a vegetation index which is used as a proxy of the vegetation cover. Such index could be the classical NDVI (Normalized Difference Vegetation Index) which corresponds to the normalized difference between the signals recorded in the red and in the near infrared (typically at about 0.65 μm and 0.8 μm). This index is indeed strongly correlated with the vegetation cover: it ranges from about 0.1 for bare soil to values close to 0.8 for dense and green vegetation, depending on the wavelength choice. It is also correlated with foliage density which is expressed by LAI, Leaf Area Index (total surface of leaves per unit surface). However NDVI is also sensitive to the soil spectral characteristics. Other indexes were proposed for being less dependant on soil background [START_REF] Rondeaux | Optimization of soil adjusted vegetation indices[END_REF]. Among them we considered TSAVI (Transformed Soil Adjusted Vegetation Index) in this work (its definition is given l a t e r i n E q . [ 3 ] ) . O t h e r v e g e t a t i o n i n d e x e s d e s i g n e d f o r p r o v i d ing a proxy for vegetation cover could also be used. For this reason we will denote VI for the chosen vegetation index.

When plotting the two-dimensional distribution of temperature and vegetation index (T-VI) corresponding to an area with well distributed vegetation cover and moisture content, one gets a triangular of trapezoidal cloud of points (see Fig. 1). The four vertices correspond to the extreme conditions of a crop: the bottom vertices (low VI) correspond to dry and saturated bare soil, whereas the upper vertices (High VI) correspond to well-irrigated and water-stressed full cover vegetation [START_REF] Moran | Estimating Crop water deficit using the relation between surface-air temperature and spectral vegetation index[END_REF]. The left limit is called the "wet" (or "cold") edge whereas the right limit is called the "dry" (or "warm") edge. It is usually admitted that the soil dryness linearly increases when moving from the left ("wet") edge to the right ("dry") edge. For this reason, a soil moisture index was defined from the relative distance of a given point to the dry edge (a constant VI). Similarly, it is argued that the vegetation stress increases from the "wet" edge to the "dry" edge. A consequence is that the position of a particular point relatively to these extreme lines could also be a measure of the evapotranspiration flux at the time of the remote sensing test. Soil moisture increases from the dry edge (on the right) to the wet edge (on the left) [START_REF] Moran | Estimating Crop water deficit using the relation between surface-air temperature and spectral vegetation index[END_REF].

Different methods were devised for defining the position of the wet and dry edges. The empirical method uses the experimental T-VI distribution and linear edges are set to fit its border [START_REF] Sandholt | A simple interpretation of the surface temperaturevegetation index space for assessment of surface moisture status[END_REF]. Iso-moisture straight lines are then distributed linearly between the wet edge and the dry edge. Another method is based on the use of Penman-Monteith equation for evapotranspiration (which is derived from an energy budget equation) for expressing the difference between surface temperature and air temperature for the four extreme cases that define the vertices of the trapezoidal T-VI plot [START_REF] Moran | Estimating Crop water deficit using the relation between surface-air temperature and spectral vegetation index[END_REF]. As before, iso-moisture straight lines are distributed linearly between the wet edge and the dry edge. Still another method is based on the use of a SVAT (Soil Vegetation Atmosphere Transfer) model in order to evaluate the radiation flux, the convective flux and the water/vapor fluxes between soil, vegetation and atmosphere [START_REF] Carlson | An interpretation of methodologies for indirect measurement of soil water content[END_REF]. The SVAT model allows building iso-moisture lines as well as iso-evaporation lines to which the T-VI distribution can then be compared. After inversion, one retrieves a moisture map of the overflown area.

These methods were compared in [START_REF] Krapez | Comparison of three methods based on the temperature-NDVI diagram for soil moisture characterization[END_REF] where some improvements were also proposed. We highlighted the possibility to infer not only a map of the surface moisture but also an estimation of the root-zone moisture.

Nevertheless the concept of exploiting only T-VI (eventually together with some meteorological data like in the methods described in [START_REF] Moran | Estimating Crop water deficit using the relation between surface-air temperature and spectral vegetation index[END_REF] and [START_REF] Carlson | An interpretation of methodologies for indirect measurement of soil water content[END_REF]) is based on a simple representation of the thermal and radiative properties of the complex soil/vegetation structure and of the exchanges therein. However, a large number of parameters, in addition to soil moisture, are influencing the apparent soil/canopy temperature. Furthermore these parameters may vary between the different vegetation types and the different soil types that are present in the scene. Spreading the temperature distribution along only one dimension like the VI is not enough for efficiently separating the sensitivities and for allowing an unambiguous determination of soil moisture. In addition to temperature and vegetation index, the authors in [START_REF] Chauhan | Spaceborne soil moisture estimation at high resolution: a microwave-optical/IR synergistic approach[END_REF] suggested to consider surface albedo as retrieved from visible-near IR reflectance signals. As a matter of fact the albedo α has a high influence on the net radiation flux and as such has a significant impact on temperature. Furthermore, with nowadays hyperspectral instruments, it is an easy task to evaluate with a good precision this optical property. In [START_REF] Chauhan | Spaceborne soil moisture estimation at high resolution: a microwave-optical/IR synergistic approach[END_REF] the identification process was divided in two phases. The first phase is a learning phase which consists in a fitting procedure between the optical remote sensing data T, NDVI, α on one hand and Mo on the other hand, where Mo is the reference moisture inferred from low-resolution microwave remote sensing. The regression law is a polynomial limited to second order terms:

+ + + + + + + + + = α α α α [1]
Each observable variable T*, NDVI*, α* is obtained first by downscaling the optical data to the microwave resolution and then by normalizing it with the corresponding minimum and maximum values. High resolution moisture maps were then constructed by applying the quadratic relation in Eq. [1] to the raw, i.e. high resolution, optical images.

We propose to take into account the albedo in a different way which avoids referring to microwave data for the learning process. The idea is merely to extent the procedure defined for 2D T-VI plots in [START_REF] Sandholt | A simple interpretation of the surface temperaturevegetation index space for assessment of surface moisture status[END_REF] to 3D T-VI-α plots. Dry and wet edges now become dry and wet 2D surfaces in the 3D representation. We define the Soil-Vegetation Wetness Index (SVWI) from the relative distance between the considered pixel and the dry surface.

When introducing the albedo, it is expected to better separate areas according to the absorbed solar radiation. We also looked for separating vegetated areas according to the fraction of green vegetation versus senescent vegetation. A suitable index for quantifying the fraction of senescent vegetation is CAI (Cellulose Absorption Index). As a matter of fact, a broad absorption band near 2.1 μm appears in all compounds possessing alcoholic -OH groups such as sugars, starch and cellulose. This absorption band appears in the reflectance spectra of dry plant residues and a cellulose absorption index (CAI) based on the reflectance in three bands -two on the shoulders at 2 μm and 2.2 μm and one at 2.1 μm (cellulose-lignin absorption maximum) was proposed in [START_REF] Daughtry | Discriminating Crop Residues from Soil by Shortwave Infrared Reflectance[END_REF]:

( ) 1 . 2 2 . 2 0 . 2 2 5 . 0 ρ ρ ρ - + = CAI [ 2 ]
where ρ i is the reflectance at wavelength i.

A linear unmixing approach using Normalized Difference Vegetation Index (NDVI) and the Cellulose Absorption Index (CAI) proved indeed to be efficient in providing an estimation of fractional cover of photosynthetic vegetation (fPV), nonphotosynthetic vegetation (fNPV) and bare soil [START_REF] Guerschman | Estimating fractional cover of photosynthetic vegetation, non-photosynthetic vegetation and bare soil in the Australian tropical savanna region upscaling the EO-1 Hyperion and MODIS sensors[END_REF].

In the same manner as before with albedo, we propose to use CAI as an additional observation parameter and to exploit the three-dimensional T-VI-CAI distribution features to infer a soil moisture map.

The aim of this study is to analyse the benefit of adding a third index to the classical T-VI empirical method for soil moisture mapping [START_REF] Sandholt | A simple interpretation of the surface temperaturevegetation index space for assessment of surface moisture status[END_REF]. The proposed procedures will be applied on remote sensing data obtained during the HyEurope 2007 campaign over Camargue (France) with HyMap and ATM sensors operated by DLR. HyMap is a hyperspectral imager with 128 bands from 0.45 μm to 2.5 μm whereas ATM is a multispectral imager with 11 bands from 0.42 μm to 13 μm.

Measurement campaign

Airborne remote sensing campaign

During the HyEurope 2007 campaign, a HyMap hyperspectral sensor (HyVista Corporation Website, 2011) was operated by DLR, Oberpfaffenhofen, Germany, on board of a Do228 airplane over several sites in Europe (DLR HyEurope Campain, 2010). Seventeen flight lines were flown on June 23 rd over Camargue region, south of France.

HyMap is a spectroradiometer utilizing diffraction gratings and four 32-element detector arrays (1 Si, 3 liquid nitrogen-cooled InSb). It measures radiation in 128 spectral bands between 0.45 μm and 2.48 μm. It is a whisk-broom type scanner, i.e. a mirror scans across the airplane path, reflecting light towards the four arrays which collect a spectrum one pixel at a time. The angular width is 61.3° with 512 pixels in each line (2 mrad across track IFOV and 2.5 mrad along track IFOV). At the 1950 m mean flight altitude the swath was thus 2.3 km and the resolution was at nadir about 4 m x 5 m.

Together with HyMap was flown the multispectral sensor Daedalus AADS 1268 " A i r b o r n e T h e m a t i c M a p p e r " ( A T M ) . I t i s a l s o a w h i s k -b r o o m t y p e scanner. Measurement is performed in ten bands from 0.42 μm to 2.23 μm plus in one large thermal infrared band from 8 to 13 μm. The angular width is 86° with 716 pixels in a line. The swath was thus 3.6 km and the nadir pixel dimension of approximately 5 m.

Data from HyMap and ATM were orthorectified and resampled at 4 m x 4 m resolution by DLR. Fluctuations of altitude were taken into account in the geometric correction. An atmospheric correction was also applied to HyMap data, but not to ATM data. The ATM thermal channel was nevertheless calibrated absolutely in temperature units (brightness temperature).

Channel 16 (0.675 μm) and 26 (0.819 μm) from HyMap were combined to provide a vegetation index map of the flown area. A spectrum recorded over bare soil was used to evaluate the soil line coefficients a,b which are necessary for computing TSAVI (Transformed Soil Adjusted Vegetation Index) [START_REF] Rondeaux | Optimization of soil adjusted vegetation indices[END_REF]. Channel 11 (8-13 μm) from ATM provided the brightness temperature map. Modtran4 simulations for atmospheric radiation and transmission were performed at ONERA in order to convert at-sensor brightness temperature to on-ground true temperature as described in [START_REF] Lagouarde | Experimental Study of Brightness Surface Temperature Angular Variations of Maritime Pine (Pinus pinaster) Stands[END_REF]. For this purpose an estimation of the effective emissivity map was necessary. Effective emissivity was computed from a weighted average of soil emissivity and vegetation emissivity according to the model described in [START_REF] Valor | Mapping Land Surface Emissivity from NDVI: Application to European, African, and South American Areas[END_REF] which takes into account mean values for vegetation emissivity and for bare soil emissivity. Vegetation emissivity was taken from [START_REF] Coll | Validation of temperature-emissivity separation and split-window methods from TIMS data and ground measurements[END_REF] whereas mean soil emissivity was evaluated from measurements performed on saturated and dry soil samples taken from six different fields in flight line 2 [START_REF] Lesaignoux | Influence of surface soil moisture on spectral reflectance of bare soil in the 0.4 -15 μm domain[END_REF]. Standard deviation of altitude fluctuations was 3.7 m. Its impact on soil temperature evaluation was estimated to be less than 0.02 K; this correction was thus discarded.

For present analysis we focused on flight line n°2 which was flown at 10h20 UTC over a region with crops of wheat, alfalfa (flowering plant cultivated as forage crop), meadows, rice and vineyards and some bare soil fields (average coordinates: N 43.629° E 4.477°).

In-situ measurements

In-situ measurements were performed by INRA in a series of fields in the central part of flight line n°2. In particular, a series of soil samples were extracted in the 0-5 cm upper soil layer on June 25 th f o r g r a v i m e t r i c w a t e r c o n t e n t m e a s u r e m e n t (5 samples per spot, 3 spots per field or per crop type) (see Table 1).

In the following we will also focus the discussion on three other types of crops: harvested wheat, tilled wheat and rice (see Table 2).

Table 1. List of in situ water content gravimetric measurements and mean moisture value (% weight).

Table 2. Additional fields without in situ moisture measurement.

Temperature and Vegetation Index maps

Temperature map as obtained with Daedalus ATM sensor is reported in Fig. 2 left. Corresponding TSAVI (Transformed Soil Adjusted Vegetation Index) map as obtained with Hymap sensor is reported in Fig. 2 right. The color scale was chosen to represent high vegetation cover in green and bare soil in yellow. In general, the highest temperature values are observed over bare soils. Especially cold plots correspond to irrigated rice fields.

Moisture identification from 2D T-VI distribution

When plotting the Temperature-Vegetation Index data for whole flight line, one gets the distribution in Fig. 3. It is roughly trapezoidal with an apex at high vegetation index (high vegetation cover). The drawing of the straight lines which define the wet and dry edges as described in [START_REF] Sandholt | A simple interpretation of the surface temperaturevegetation index space for assessment of surface moisture status[END_REF] is rather subjective. Here we chose a more objective approach based on statistics: for each TSAVI interval of width 0.05, a low temperature limit and a high temperature limit are defined from the 1.5% and the 98.5% percentile. The results are then interpolated to provide the blue and red curves, respectively wet edge and dry edge in Fig. 3. By this way the distribution is well bounded; in the same time, possible outliers are rejected.

At each Vegetation Index level, the moisture index is defined as the relative distance to the dry edge. We call it Soil Vegetation Wetness Index (SVWI). Except for very few points, it ranges between 0 and 1. In Fig. 4 this moisture index is compared to the measured moisture (in % weight) for the 18 spots listed in Table 1. The dots are not arranged along a straight line. This imperfect correlation has several explanations:

-The results for meadow fields are those which are farthest from the regression line (the three dark blue dots in the upper part). Meadow spots actually present high TSAVI values and it is well known that temperature shows a decreasing sensitivity to soil water content for increasing values of Leaf Area Index (i.e. at the top of the T-VI distribution). Another possible reason is that high VI areas in flight line 2 may offer a too narrow range of soil moisture (the dry edge would be underestimated at the top of the trapezoid); in such situation the present empirical approach for moisture evaluation would lead to a biased wetness index.

-In-situ measured moisture is in % weight whereas SVWI is linked to volume water content: indeed meadow soil has generally a lower density than bare soil.

-In-situ measurements were performed two days after the remote sensing campaign.

Anyway, when excluding the meadow data, the correlation between SVWI and gravimetric moisture is reasonably good: the correlation coefficient R 2 reaches 0.61.

Moisture identification from 3D distributions T-VI-α α α α or T-VI-CAI

When considering the albedo or the Cellulose Absorption Index (CAI) as an additional observable parameter, one gets the 3D distributions reported in Fig. 5. When adding a third parameter, the wet edge and the dry edge become bounding surfaces.

Similarly to the previous case, we defined by computing the 1.5% temperature percentile and the 98.5% temperature percentile values over cells obtained after discretizing the VI-α plane, respectively the VI-CAI plane. Cellulose Absorption Index (bottom). The "wet" surface and the "dry" surface are defined to stick to the distributions from bellow and from top.

The correlation coefficient R 2 between SVWI and gravimetric moisture content slightly diminishes to 0.57 when using albedo whereas it increases to 0.69 when using CAI (see Fig. 6). Therefore introducing the albedo doesn't seem to provide a significant improvement, at least for present types of crops. The use of the Cellulose Absorption Index (CAI) seems to be more promising. This may be due to the fact that this index has the ability to separate senescent vegetation, like ripe wheat and harvested wheat, from bare soils and from green vegetation. Different classes of surfaces which present highly different processes regarding evapo-transpiration but which previously merged in the classical T-VI plot, are now separated in the T-VI-CAI plot.

The proposed Wetness Index map as obtained from the analysis of the T-VI-CAI distribution is plotted in Fig. 7. A strong correlation is found between the wetness index and the crop type: with only a few exceptions, the rice fields are the wettest, and the driest soil is observed in alfalfa fields and ploughed fields. For wheat fields, soil moisture decreases when going from ripe vegetation field, to tilled field and then to harvested field. Bare soils and tilled wheat fields present similar wetness index.

Conclusion

Adding a third observation parameter like CAI or albedo to temperature and vegetation index which are used in the classical "triangle method" can bring the advantage of discriminating areas having different moisture content but otherwise presenting the same T-VI values. A better correlation is thus expected between the calculated moisture and the measured one. For the analyzed crops, the addition of CAI proved to be efficient, contrarily to albedo. The number of reference data used for the correlation analysis was however limited. Other remote sensing databases should be considered to further assess the benefit of the proposed method.

Figure 1 .

 1 Figure 1. Principle of soil moisture mapping from a Temperature-Vegetation Index (T-VI) plot. The vertices of the trapezoidal distribution correspond to: (1) wellwatered full cover, (2) non-transpiring full cover, (3) wet bare soil, (4) dry bare soil. Soil moisture increases from the dry edge (on the right) to the wet edge (on the left)[START_REF] Moran | Estimating Crop water deficit using the relation between surface-air temperature and spectral vegetation index[END_REF].

Figure 2 .

 2 Figure 2. Map of vegetation index TSAVI (left) and of temperature (right) recorded over flight line 2 (about 2.5 km x 8.3 km).

Figure 3 .

 3 Figure 3. Temperature-Vegetation Index plot with the location of wet edge (in blue) and the dry edge (in red). The results for the 27 spots listed inTable 1 and 2 are represented with dots with scaled color.

Figure 4 .

 4 Figure 4. Comparison between the Soil Vegetation Wetness Index (SVWI) computed from remote sensing data and the in situ measured moisture. SVWI was obtained by using the 2D T-VI distribution in Fig. 3.

Figure 5 .

 5 Figure 5. Distribution of temperature, Vegetation Index and either albedo (top) or

Figure 6 .

 6 Figure 6. Comparison between the Soil Vegetation Wetness Index (SVWI) computed from remote sensing data and the in situ measured moisture. SVWI was obtained by using the T-VI-CAI distribution in fig. 5-bottom.

Figure 7 .

 7 Figure 7. Soil Vegetation Wetness Index (SVWI) map computed from remote sensing data by exploiting the T-VI-CAI distribution in fig. 5-right (high water content is represented in blue).

  Table 1 and 2 are represented with dots with scaled color.

Acknowledgements

The authors are indebted to DLR, Oberpfaffenhofen, Germany, for providing the HyEurope 2007 remote sensing data. The authors would like to thank INRA team for ground truth measurements, A. Lesaignoux and S. Fabre from ONERA for providing the soil spectra and the French Programme National de Télédétection Spatiale (INSU -PNTS) for providing funding for the acquisition of ground truth data in Camargue. This research is within the framework of ONERA Enviro project.