Alain Y Kibangou
email: alain.kibangou@univ-grenoble-alpes.fr

Federica Garin
email: federica.garin@inria.fr

Sebin Gracy
email: sebin.gracy@inria.fr

Input and State Observability of Network Systems with a Single Unknown Input

Keywords: Input and state observability, Structural observability, Strong structural observability, Network systems, Cyber-physical security

This paper studies network systems affected by a single unknown input, possibly representing an attack or a failure, to be estimated. The main result is a characterization of input and state observability, namely the conditions under which both the whole network state and the unknown input can be reconstructed from some measured local states. This characterization is in terms of observability of a suitably-defined subsystem, which allows the use of known graphical charactizations of observability of network systems, leading to structural results (true for almost all interaction weights) or strong structural results (true for all non-zero interaction weights). We apply our results to an illustrative example, finding a full characterization of input and state observability of a path graph, affected by a single unknown input and with measurement of a small number of local states.

INTRODUCTION

The physical and the technological worlds around us are made of several entities that interact with each other and then give rise to some complex emerging behaviors. They include social networks, chemical networks, robotic networks, to cite a few. This last decade has seen a significant increase in the number of works devoted to network systems with a particular focus on how to control and/or monitor these networks. Precisely, it is of paramount importance to understand how a given target can be reached by controlling a subset of entities and how to infer the state of the whole network from measurements obtained from a subset of entities. These two questions are clearly related to the issues of controllability and observability, well known by system theorists. Here, we focus on the observability issue, and on linear network systems. Two ways of characterization can be considered. In a first approach, the network is fully known: both the topology and the weights of each interaction are known. In such a case, one can resort to classical matrix rank test tools (Kalman observability matrix or PBH test), but algebraic characterization can become numerically infeasible when the network is large, and also gives little insight into the effects of the network topology. Therefore, graph tools have been derived in the recent literature, mainly for consensus networks, see e.g. [START_REF] Ji | Observability and estimation in distributed sensor networks[END_REF]; [START_REF] Parlangeli | On the reachability and observability of path and cycle graphs[END_REF]; O' Cleary et al. (2013); [START_REF] Kibangou | Observability in connected strongly regular graphs and distance regular graphs[END_REF]. In a second approach, the network is partially known: only the topology is known. The edge weights being unknown, they are considered as free parameters. In this framework, one can seek for structural results (true for almost all choice of free parameters) or strongly structural results (true for all nonzero free parameters). This line of work is the theory of structured systems (see [START_REF] Shields | Structural controllability of multiinput linear systems[END_REF] and [START_REF] Dion | Generic properties and control of linear structured systems: a survey[END_REF]).

Network systems are prone to failures, that can be intentional (from a malicious external entity) or not. Therefore, it is interesting to observe not only the state of the network system, but also a possible unknown input. This problem is known as input and state observability (ISO). Similarly to observability, ISO can be studied either for a fully known network system, with particular numerical values, or seeking for (strongly) structural results with respect to free parameters for interaction weights. For the first approach, there are well-known algebraic characterizations with matrix rank test similar to Kalman and PBH; in Sect. 2 we will recall the main results which will be used for our developments. For the second approach, a characterization of structural ISO has been derived in [START_REF] Boukhobza | State and input observability for structured linear systems: A graph-theoretic approach[END_REF], while no result is known about strong structural ISO. In this paper, we focus on the case where a few agents have their state directly observable, and where a single agent is affected by an unknown input (i.e., by the failure or the attack). The first assumption is quite common in the study of structural properties of network systems, see e.g. [START_REF] Trefois | Zero forcing number, constrained matchings and strong structural controllability[END_REF] and references therein. The second assumption is introduced e.g. in [START_REF] Teixeira | Networked control systems under cyber attacks with application to power networks[END_REF] for detection of cyber attacks. In contrast to the latter work, herein the aim is not to design detection policies but to characterize ISO. Such a characterization can be helpful to select nodes to be monitored, so as to obtain the desired observation without wasting resources in monitoring all nodes.

For this framework with a single unknown input, our main result is Prop. 4, which shows that ISO is equivalent to classical observability of a suitably-defined subsystem.

Using this result and known characterizations of (strong) structural observability, we can characterize both structural ISO and strong structural ISO; the first characterization is simpler than the general one in [START_REF] Boukhobza | State and input observability for structured linear systems: A graph-theoretic approach[END_REF], and the second one is totally new. Then, we apply our results to an illustrative example: we fully characterize structural and strongly structural ISO of path graphs, depending on the number of nodes and on the number and position of the few nodes whose state is observed.

PROBLEM FORMULATION

Consider a network system with N nodes whose interconnections are described by a directed graph G = (V, E), where V = {1, 2, • • • , N } and E ⊆ V × V stand for the vertex set and the edge set respectively. We denote by w ij ∈ R the weight associated with the edge (j, i) ∈ E and by W = [w ij ] the weighted adjacency matrix of G, where

w ij = 0 if (j, i) / ∈ E.
The network dynamics are assumed to be linear, and affected by a scalar unknown input at a node j:

Σ : x(k + 1) = W x(k) + e j,N u(k) y(k) = Cx(k) (1)
where x(k) ∈ R N , u(k) ∈ R and y(k) ∈ R K stand for the network state, the unknown input, and the available measurements at time k, respectively, while e j,N denotes the jth vector of the canonical basis of R N .

From the above set-up, it is clear that we restrict our study to the case where the unknown input affects a single node only. Such a scheme can be used to represent an attack on a node, including denial-of-service or deception attacks on the in-coming communications of node j, see [START_REF] Teixeira | Networked control systems under cyber attacks with application to power networks[END_REF]. We aim at characterizing when it is possible to jointly estimate both the initial conditions of all the nodes of the network and the scalar unknown input, from the direct measurements of the states of a subset O ⊂ V of nodes. More precisely, we are studying input and state observability, defined as follows: Definition 1. System Σ is input and state observable (ISO) if the initial condition x(0) ∈ R N and the input sequence u(0), . . . , u(N -2) can be reconstructed from the measurements y(0), . . . , y(N -1).

The following algebraic characterization is immediate:

Lemma 1. Σ is ISO if and only if the matrix Ψ = [Θ, Γ] has full column rank, where Θ ∈ R KN ×N is the Kalman observability matrix of the pair (W, C), i.e., Θ =    C CW . . . CW N -1    , while Γ ∈ R KN ×(N -1) is defined by Γ =       0 0 • • • 0 Ce j,N 0 • • • 0 CW e j,N Ce j,N • • • 0 . . . . . . . . . . . . CW N -2 e j,N CW N -3 e j,N • • • Ce j,N      
.

The above characterization is similar to Kalman condition, while the following Lemma 2 is the analogue of PBH test. Notice that ISO is equivalent to strong observability (i.e., the possibility to reconstruct the initial state x(0), in the presence of an unknown input) together with left invertibility with delay 1 (i.e., the possibility to reconstruct all inputs up to u(N -2) when observing outputs up to y(N -1), while knowing x(0)). Hence, Thm. 7.17 in [START_REF] Trentelman | Control Theory for Linear Systems[END_REF] together with Thm. 4 in [START_REF] Massey | Inverses of linear sequential circuits[END_REF] give the following characterization. Lemma 2. Σ is ISO if and only if Ce j,N = 0 and the matrix

Φ z = zI N -W -e j,N C 0 has full column rank ∀z ∈ C.
Beyond the study of ISO for a specific W , it is of crucial importance to get results depending on the graph only and not on a particular choice of the edge weights, leading to structural characterization of ISO. Following classical literature on structural controllability and observability, we consider the edge weights (i.e., the entries of W not having beeen fixed to 0) as free parameters, each taking real values, so that each point of R |E| is associated with one particular matrix of the system Σ, while all such matrices share a common structure of imposed zeros.

We will be interested in understanding whether a system Σ is ISO for almost all parameters, or cannot be ISO for any choice of the parameters. Definition 2. Sytem Σ is structurally ISO if it is ISO for all choices of edge weights, except those lying in some proper variety of R |E| .

Inspired by [START_REF] Mayeda | Strong structural controllability[END_REF], we will be also interested in stronger results, which hold true for all choices of non-zero weights, and not only for almost every choice of weights. Indeed, in some cases, it is essential to ensure that system design with some specified pattern of zeros (representing physical absence of links) will never produce a non-observable system. Definition 3. System Σ is strongly structurally ISO if it is ISO for all choices of non-zero edge weights.

Clearly, strong structural ISO implies structural ISO.

CHARACTERIZATION OF INPUT AND STATE OBSERVABILITY

By looking at the algebraic characterization of ISO in Lemma 1, one immediately finds the following necessary conditions for input-and-state observability:

i) Ψ has a number of rows greater than or equal to the number of columns; ii) Θ has full column rank; iii) Γ has full column rank.

Such conditions can be equivalently re-phrased as follows. Proposition 3. The following conditions are necessary for input-and-state observability of Σ: i) C has at least two rows (K ≥ 2); ii) the system (W, C) is observable; iii) Ce j,N = 0.

Proof. The three conditions are equivalent to the abovestated ones. Condition i) is true since Ψ has KN rows and 2N -1 columns. For condition ii), notice that Θ is the Kalman observability matrix of the system (W, C). Condition iii) is equivalent to Γ having full column rank. 2 This proposition, despite its simplicity, can be effectively used to rule out the possibility of ISO in some cases of interest. In particular, condition i) implies that input-andstate observability from a single node i is impossible. Condition ii) excludes all the systems where the corresponding system (W, C) (without unknown input) is not observable. Finally, condition iii) can be interpreted as the necessity to have a direct influence of the input on node(s) whose state is taken into account in measurements (see [START_REF] Teixeira | Networked control systems under cyber attacks with application to power networks[END_REF] for a similar conclusion).

In the sequel, we assume that one can directly measure the states of a set of nodes O ⊂ V, |O| = K ≥ 2 and that the node j affected by the unknown input belongs to O, say O = {j} ∪ Õ. Hence, rows of C are e T k,N , k ∈ O. Without loss of generality we assume that the first row of C is e T j,N . Let S j,N denote the N × (N -1) selection matrix defined by

S j,N = [ e 1,N • • • e j-1,N e j+1,N • • • e N,N
]. We can therefore define the matrices C = S T 1,K CS j,N and W = S T j,N W S j,N . They are obtained by removing from C the first row and the jth column and from W the jth row and column. The following proposition characterizes ISO in terms of observability of a suitable subsystem: Proposition 4. Consider a network system Σ described by (1), with state matrix W , where a scalar unknown input affects node j, and a set O of nodes have their states directly measurable, with j ∈ O and |O| ≥ 2; this corresponds to an observation matrix C with first row e T j,N and other rows e T k,N , k ∈ O \{j}. This system is ISO if and only if the pair ( W , C) is observable, where C (resp. W ) is obtained by removing the first row and the jth column from C (resp. the jth row and jth column from W ).

Proof. We define the two following permutation matrices:

Π 1 =     S T j,N 0 0 S T 1,K 0 e T 1,K e T j,N 0     , Π 2 = S j,N e j,N 0 0 0 1 .
Applying the corresponding permutations to Φ z we get:

Π 1 Φ z Π 2 =     zI N -1 -W S T j,N (zI N -W )e j,N 0 C 0 0 0 1 0 e T j,N (zI N -W )S j,N z -w jj -1     .
Since the last row of this matrix is linearly independent of all other rows, rank(Π

1 Φ z Π 2 ) = rank( Φz ) + 1, with Φz =   zI N -1 -W S T j,N (zI N -W )e j,N C 0 0 1   . Now, the
last column of Φz is linearly independent from all other columns, so that: rank

( Φz ) = rank zI N -1 - W C + 1. As a consequence, rank(Φ z ) = rank(Π 1 Φ z Π 2 ) = rank zI N -1 - W C + 2.
From Lemma 2, we know that the system Σ is ISO if and only if rank(Φ z ) = N + 1, which in turn is equivalent to

rank zI N -1 - W C
= N -1. The latter corresponds to the PBH observability test applied to the system Σ with state matrix W and observation matrix C. 2

Prop. 4 can be interpreted as follows. System Σ can be decomposed in two coupled subsystems:

x(k + 1) = W x(k) + S T j,N W e j,N x j (k) ỹ(k) = C x(k) x j (k + 1) = w jj x j (k) + e T j,N W S j,N x(k) + u(k) y j (k) = x j (k).
Since x j is directly measured, the first system has a known input, so its state can be reconstructed when the pair W , C is observable. Having reconstructed x(k -1) and measured x j (k), one can find u(k) from the second subsystem.

STRUCTURAL AND STRONGLY STRUCTURAL INPUT AND STATE OBSERVABILITY

The richness of Proposition 4 lays in the fact that it turns the more complex problem of ISO into a problem of observability of a related system. In particular, this means that (strong) structural ISO can be assessed by exploiting the rich literature characterizing (strong) structural observability of network systems, and also the results on the dual problem of reachability (or controllability in the continuous-time case). In this section, we will briefly mention some relevant results.

Throughout this section, we consider the system Σ with state x(k) ∈ R n , state matrix W , and observation matrix C. To this system, we associate a graph G = ( Ṽ, Ẽ) which is an induced subgraph of G. Observability of ( W , C) for a given choice of the parameters is characterized by classical results such as Kalman observability matrix having full column rank and PBH test, and is equivalent to reachability of ( W T , CT ) (or its controllability, in the continuous-time case).

Structural observability is defined as the system being observable for at least one choice of the free parameters, or, equivalently, as the system being observable for almost all choices of parameters (i.e., all except those lying in a proper variety of R | Ẽ| ), since observability is known to be a generic property (see [START_REF] Shields | Structural controllability of multiinput linear systems[END_REF] and [START_REF] Dion | Generic properties and control of linear structured systems: a survey[END_REF]). The survey paper [START_REF] Dion | Generic properties and control of linear structured systems: a survey[END_REF] revises some characterizations of structural controllability and observability. Here we will consider a characterization involving a matching in a bipartite graph.

To G, one can associate a bipartite graph B = (U, W, E B ), with U = Ṽ \ Õ and W = Ṽ, constructed as follows: the two vertices in U and W corresponding to a same element v ∈ Ṽ are denoted u v and w v , respectively, and

E B = {(u i , w j ) ∈ U × W s.t. (i, j) ∈ Ẽ} .
With this notation, structural observability of ( W , C) can be characterized as follows:

Proposition 5. The system ( W , C) is structurally observable if and only if the following two conditions are satisfied:

• for all v ∈ Ṽ, in G there exists a path from v to w for some w ∈ Õ ( G is output-connected);

• there exists a matching in B of size n -| Õ|.

Prop. 5 is the same as Thm. 1 in [START_REF] Chapman | On strong structural controllability of networked systems: A constrained matching approach[END_REF], just phrased here for observability instead of controllability. This result can be traced back to [START_REF] Commault | Characterization of generic properties of linear structured systems for efficient computations[END_REF] and [START_REF] Commault | Observability preservation under sensor failure[END_REF].

Strong structural controllability has been studied starting from [START_REF] Mayeda | Strong structural controllability[END_REF]; an interesting discussion of both new and old results can be found in [START_REF] Jarczyk | Strong structural controllability of linear systems revisited[END_REF]. More recently, for the case where columns of the input matrix are of the form e u,N , two equivalent characterizations have been found. A first one is graphical, based on the existence of constrained matchings in two suitably-defined bipartite graphs, see [START_REF] Chapman | On strong structural controllability of networked systems: A constrained matching approach[END_REF]. A second one involves the so-called zero-forcing sets, and is described in [START_REF] Trefois | Zero forcing number, constrained matchings and strong structural controllability[END_REF]. Here we will describe the first one, suitably adapted to deal with observability instead of controllability. We will use the above-described bipartite graph B. We will denote by E loop ⊂ E B the edges of the form {u i , w i }, if any exist; they correspond to self-loops in the graph G.

We will also need a second bipartite graph B × , obtained from B by adding to E B all edges {u i , w i }, i ∈ Ṽ, also for those vertices i not having a self-loop in G. We recall that, given a bipartite graph with vertex sets X , Y and edge set F, a matching M ⊆ F is constrained if there is no other matching involving the same vertex set. In bipartite graphs, the following characteriztion of constrained matching is given in [START_REF] Golumbic | Uniquely restricted matchings[END_REF] Prop. 6 is a rephrasing for observability of the characterization of strong structural controllability in [START_REF] Chapman | On strong structural controllability of networked systems: A constrained matching approach[END_REF], Thm. 5. In this section, we consider an illustrative example: the path graph, namely a graph with V = {1, . . . , N } and (i, j) ∈ E if and only if |i -j| = 1. Fig. 1 shows for example a path graph with 7 nodes. We are considering the undirected path graph, in the sense that all edges appear in both orientations, but we are not enforcing any symmetry in the edge weights. Moreover, we consider that no selfloop is present.

STRUCTURAL INPUT AND STATE OBSERVABILITY OF PATH GRAPHS

For the path graph, in Prop. 9 we give a full characterization of (strong) structural ISO from a small number of nodes (1, 2 and 3 nodes), depending on the position of the observing nodes and on the size N . To obtain this result, we will first characterize (strong) structural observability of a path graph from one and two nodes (Propositions 7 and 8), and then reduce the study of ISO to the observability problem over subgraphs.

Observability being a generic property, 'the system is not structurally observable' means that there exists no choice of the free parameters such that the system is observable, while 'the system is structurally observable' means that it is observable for almost all choices of parameters. The same is true for ISO. Also recall that strong structural observability implies structural observability.

In a path graph, the first and the last nodes will be called terminal nodes, while their neighbors (the second, and the second-last nodes) will be called near-terminal. Proposition 7. Consider a path graph with n nodes, observed by a single node i. The associated network system 1) is not structurally observable, if i is even and n is odd; 2) is strongly structurally observable in two cases: a) i is terminal; b) i is near-terminal and n is even; 3) is structurally observable, but not strongly structurally observable, in all other cases, namely, if i is neither terminal nor near-terminal and one of the following holds: a) i is odd; b) i is even and n is even.

Proof. We use the characterizations from Propositions 5 and 6. First, the graph is output-connected, since it is strongly connected.

Then, we look at the graph B. We notice that it has two disjoint subgraphs B 1 , B 2 : the first one has vertex sets U odd = {u h ∈ U s.t. h odd} and W even = {w h ∈ W s.t. h even}; vice-versa the second one has vertex sets U even and W odd defined accordingly. When i is even and n is odd, it is impossible to find a matching of B of size n-1. Indeed, B 1 has |U odd | = (n + 1)/2 > |W even | = (n -1)/2, and hence no matching of this subgraph can cover all nodes of U odd , also implying that no matching of B can cover all nodes of U. As an example, Fig. 2 illustrates the case with n = 7 and i = 2.

In the cases 2) and 3) we can find a matching of B of size n -1, which is also constrained. When n is even (regardless i being even or odd), we have the following constrained matching: M = {{u h , w h+1 } ∀u h ∈ U odd } ∪ {{u h , w h-1 } ∀u h ∈ U even }, as it is illustrated in Fig. 3.

When n is odd and i is odd, we have the constrained matching M = { {u h , w h+1 } ∀u h ∈ U odd , h < i} ∪ {{u h , w h-1 }∀u h ∈ U odd , h > i}∪{{u h , w h-1 }∀u h ∈ U even }.

By Prop. 5 we can conclude that in cases 2) and 3) the system is structurally observable, since G is outputconnected and there exists a matching in B of size n -1.

Fig. 1 .

 1 Fig. 1. A graph with 7 nodes.

  , Thm. 3.1: a matching M is constrained if and only if there exists a reordering of vertices X = {a 1 , . . . , a |X | } and Y = {b 1 , . . . , b |Y| } such that M = {a 1 , b 1 }, . . . , {a |M| , b |M| } and moreover for all 1 ≤ j < i ≤ |M|, {a i , b j } / ∈ F. With this notation and definitions, we can state the following characterization of strong structural observability. • there exists a constrained matching M × ⊆ E B× of size n -| Õ| and such that M × ∩ E loop = ∅.

	Proposition 6. The system ( W , C) is strongly structurally
	observable if and only if the following two conditions are
	satisfied:
	• there exists a constrained matching M ⊆ E B of size n -| Õ|;

The difference between the two cases is about B × : a constrained matching exists only in case 2), not in case 3). When i is terminal, say i = 1 (the case i = n is obtained symmetrically), we have the constrained matching M × = {{u h , w h-1 }∀u h ∈ U}. When i is near-terminal and n is even, say n = 2 (again, the case i = n -1 is obtained symmetrically), we have the constrained matching

we trivially have M × ∩ E loop = ∅, and so we can conclude by Prop. 6 that the system is strongly structurally observable, in cases 2a) and 2b). On the contrary, in case 3), we can show that there exists no constrained matching of B × covering U. We notice that removing w i disconnects B × in two disconnected subgraphs: B < with vertices U < = {u j , j < i} and W < = {w j , j < i}, and B > with vertices U > = {u j , j > i} and W > = {w j , j > i}. Since i is neither terminal nor near-terminal, Proof. The proof for case 1) is the same as the proof of case 1) in Prop. 7: having two even observer nodes i, i instead of only one does not change B 1 , and hence it is still impossible to find a matching of B covering U .

Cases 2a) and 2b) are immediate from the corresponding cases in Prop. 7, since observability is preserved when adding a second row to C when the system is already observable from a first row. More precisely, adding a second observation node i means removing u i from U in the graphs B and B × of the system with one observation node i. If any (constrained) matching covering U was already present before the removal, then a (constrained) matching exists also after the removal. Hence, from the proof of Prop. 7, we know that a constrained matching of B covering U exists if at least one of i, i is odd, or if both i and i are even and n is even. This, together with G being output-connected, implies that the system is structurally observable in cases 2) and 3) (notice that 3c) implies that one of i and i is odd). Also, this helps us understanding strong observability, which will then depend only on the existence of a constrained matching of B × , having already found a constrained matching for B. Having already proved 2a) and 2b), we will exhibit a constrained matching of B × for the cases 2c) and 2d), and prove that a constrained matching of size n -2 cannot exist in case 3). For the case 2c), say i = i + 1, we have the constrained matching

For the case 2d), say i = i + 2, we have the constrained matching M × = {{u h , w h+1 ∀h < i}} ∪ {{u i+1 , w i+1 }} ∪ {{u h , w h-1 } ∀h > i } (assumptions mentioning even or odd do not matter for finding M × , but they were needed to have a constrained matching M of size n -2 in B).

In case 3), say i < i . There are two vertices w i , w i whose removal separates B × in three disjoint graphs: B < , B middle , and B > , whose vertices have index h < i, i < h < i , and h > i , respectively. The assumptions i, i neither terminal nor near-terminal and i -i ≥ 3 imply that each of the three subgraphs has at least four nodes, in particular it contains a subgraph having nodes u k , u k+1 and w k , w k+1 for some k. Since w i and w i can be matched to at most two nodes in U, any matching of B × covering U contains a perfect matching of one of the three subgraphs B < , B middle , and B > , but this means that the matching cannot be constrained, since such a perfect matching is not unique. 2 Proposition 9. Consider a path graph (without self-loops) with N nodes and a single input in a node j and an observation matrix C with rows e T u,N for u ∈ O. 1) If j / ∈ O, the system is not structurally ISO. 2) If |O| = 1, the system is not structurally ISO.

3) If |O| = 2 and j ∈ O: a) if j is not terminal, then the system is not structurally ISO; b) if j is terminal, then the system is (strongly)

structurally ISO if and only if the system ( W , C) obtained by removing j is (strongly) observable; the system ( W , C) corresponds to a path graph observed by a single node, whose structural observability is characterized in Prop 7. 4) If |O| = 3 and j ∈ O: a) if j is not terminal, then the system W , C has a corresponding graph G consisting of two smaller paths. If both nodes of O \ {j} belong to a same path, then the system is not structurally ISO. If they belong one to each path, then the system is (strongly) structurally ISO if and only if both paths are (strongly) structurally observable; the structural observability of each path can be assessed using Prop 7. b) if j is terminal, the system is (strongly) structurally ISO if and only if the system ( W , C) is (strongly) structurally observable; ( W , C) is a single path graph, observed from two nodes, and its observability is assessed uing Prop 8.

Proof. Items 1) and 2) are from Prop. 3. For the other cases, the main result Prop. 4 reduces ISO to observability of the subsystem ( W , C) obtained by removing j.

Then, cases where the graph G corresponding to the system ( W , C) is not output-connected are excluded, using Prop 5: case 3a), and 4a) with both elements of O \ {j} being in a same path (thus leaving the second path not output-connected). In the remaining cases, (strong) structural ISO boils down to (strong) structural observability of one or two smaller path graph(s), each observed from one or two of its nodes, whose characterization is given in Propositions 7 and 8. 2

CONCLUSION

We have studied linear network systems affected by a scalar unknown input, directly influencing a local state (i.e., one node). We have studied input and state observability, namely the conditions under which both the whole network state and the unknown input can be reconstructed from the measurements of a few local states.

We have fully characterized input and state observability in terms of observability of a suitable subsystem, and then used known results for structural and strongly structural observability to infer corresponding structural results for our problem, in the illustrative example of a path graph.


A Bourouis

E Abtout

Yizengaw

Amory-Mazaudier

M Hamoudi

S Bourouis

A Abtout

E Yizengaw

Sq solar variation at Medea Observatory (Algeria), from 2008 to 2011

Keywords: Transients Sq variations, geomagnetic field, ionosphere, Medea Observatory

est destinée au dépôt

Introduction

It is in 1722 that Graham discovered the existence of diurnal variation of regular terrestrial magnetic field. Later, [START_REF] Stewart | Terrestrial magnetism[END_REF], in an attempt to explain this variation, proposed the existence of an ionized fluid circulating through the field lines of the terrestrial magnetic field. At that time the ionosphere had not yet been discovered. It was in 1901, that Marconi discovered the ionosphere with his transatlantic radio communication. The first soundings of the ionosphere were done by [START_REF] Breit | A Test of the Existence of the Conducting Layer[END_REF]. In 1940 Chapman and Bartels developed the dynamo theory as known today. [START_REF] Chapman | Geomagnetism[END_REF], using the classification given at Innsbruck of the magnetic quiet days, established electric current maps using the magnetic field variations averaged over the five quietest days of one month. This current system was named the Sq current system. The Sq current system is composed by two current vortices (one on each hemisphere), on the sunlit side of the Earth and centered on the local noon. The electric current flow is anticlockwise in the Northern hemisphere, and is clockwise in the Southern hemisphere. The Sq current has two components, internal and external components. While the external component is due to the ionospheric electric currents, the internal component is due to the telluric electric currents. The telluric electric currents are induced by ionospheric electric currents. [START_REF] Gauss | Allgemeine Theorie des Erdmagnetismus. Resultate aus den Beobachtungen des Magnetischen Vereins im Jahre[END_REF] invented a technique that enables the separation of the external and internal fields. Later studies applied the technique to the Sq field [START_REF] Schuster | The diurnal variation of terrestrial magnetism[END_REF][START_REF] Price | New methods for the analysis of geomagnetic fields and their application to the Sq field of 1932-3[END_REF].

Many experimental studies, based on ground magnetic data, were performed on Sq current systems during the International Geophysical Year (IGY) (e.g., [START_REF] Matsushita | On the geomagnetic solar quiet daily variation field during IGY[END_REF][START_REF] Malin | The Sq current system during International Geophysical Year[END_REF]. Later on, [START_REF] Rastogi | Quiet day variation of geomagnetic Hfield at low latitudes[END_REF], [START_REF] Campbell | Annual and semiannual variations of the geomagnetic field at equatorial locations[END_REF] and [START_REF] Vassal | La variation du champ magnétique et ses relations avec I'iélectrojet équatorial au Sénégal Oriental[END_REF] analyzed the geomagnetic field at low latitudes. Campbell & Matshushita (1982) compared Sq currents during quiet and active year. [START_REF] Stening | Longitude and seasonal variations of the Sq current system[END_REF] and [START_REF] Chulliat | On the seasonal asymmetry of the diurnal and semidiurnal geomagnetic variations[END_REF] characterized the longitudinal and seasonal variations of Sq currents. [START_REF] Takeda | Time variation of global geomagnetic Sq field in 1964 and 1980[END_REF] studied seasonal variations of Sq, separately for the solar maximum year of 1964 and the solar minimum year of 1980. [START_REF] Stening | Variations in the strength of the Sq current system[END_REF] and [START_REF] Stening | Day-to-day changes in the latitudes of the foci of the Sq current system and their relation to equatorial electrojet strength[END_REF] analyzed the variation of Sq current strength, and the changes in the position of the foci of Sq current. At middle latitudes, [START_REF] Mayaud | Analyse morphologique de la variabilité jour à jour de la variation journalire "régulière" Sr du champ magnétique terrestre, 2. L e système de courants CM (régions non polaires)[END_REF], [START_REF] Davis | Pocket measurements of Sq currents at midlatitude[END_REF] and [START_REF] Torta | Behavior of the quiet day ionospheric current system in the European region[END_REF], studied the variability of the Sq. More recently, in Africa, Middle East and Asia, [START_REF] Rabiu | Preliminary results from the magnetic field measurements using MAGDAS at Ilorin, Nigeria[END_REF]; [START_REF] Obiekezie | The variability of H component of geomagnetic field at the African sector[END_REF]; [START_REF] Bolaji | Variability of Horizontal Magnetic Field Intensity Over Nigeria During Low Solar Activity[END_REF]; [START_REF] Abbas | Geomagnetic Field Variations at Low Latitudes along 96 • Magnetic Meridian[END_REF] and [START_REF] Pham Thi Thu | Sq field characteristics at Phu Thuy, Vietnam, during solar cycle 23: comparisons with Sq field in other longitude sectors[END_REF], characterized the Sq current system. Satellite data are also used to study Sq current system; recently [START_REF] Pedatella | Seasonal and longitudinal variations of the solar quiet (Sq) current system during solar minimum determined by CHAMP satellite magnetic field observations[END_REF] using CHAMP satellite data, performed the same study as [START_REF] Stening | Longitude and seasonal variations of the Sq current system[END_REF] did during the last solar minimum.

Empirical models derived from ground magnetic data were elaborated by [START_REF] Doumouya | Local time and longitude dependence of the equatorial electrojet magnetic effects[END_REF] on the equatorial electrojet and by [START_REF] Yamazaki | An empirical model of the quiet daily geomagnetic field variation[END_REF], on the Sq. More studies on Sq current systems can also be found (e.g., [START_REF] Chapman | The Earths Magnetism[END_REF], Matsushita & Campbell, 1967[START_REF] Maeda | Variation in geomagnetic field[END_REF][START_REF] Campbell | An introduction to quiet daily geomagnetic fields[END_REF][START_REF] Olsen | The solar cycle variability of lunar and solar daily geomagnetic variations[END_REF][START_REF] Olsen | Geomagnetic Tides and Related Phenomena in Tidal Phenomena[END_REF]. [START_REF] Mazaudier | Electric currents above Saint-Santin, 2,Model[END_REF][START_REF] Vassal | La variation du champ magnétique et ses relations avec I'iélectrojet équatorial au Sénégal Oriental[END_REF] found systematic northward currents over most of the daytime hours; this fact was confirmed from a larger body of data (Mazaudier & Venkateswran, 1985). Such northward asymmetry was previously found by [START_REF] Van Sabben | North-south asymmetry of Sq[END_REF][START_REF] Van Sabben | Magnetosphere currents associated with the N-S, asymmetry of Sq[END_REF]. They found that this asymmetry is particularly strong in the European longitude sector.

More comprehensive studies were performed on the sources of the Sq, i.e. the neutral winds and the ionospheric electric currents. [START_REF] Richmond | Electrodynamics effects of Thermospheric Winds from the NCAR Thermospheric General Circulation Model[END_REF]and [START_REF] Richmond | Ionospheric Electrodynamics: a Tutorial[END_REF] have described the Ionospheric electrodynamics. [START_REF] Hagan | Migrating and non migrating diurnal tides in the middle and upper atmosphere excited by tropospheric latent heat release[END_REF] and 2003 focused their studies on the migrating and non migrating tides in the upper atmosphere, see also [START_REF] Kelley | The Earths ionosphere: Plasma physics and electrodynamics[END_REF], and references therein.

This paper is devoted to the study of the monthly diurnal and seasonal variations of Sq currents using a new dataset from Observatory Medea in Algeria. Where a new magnetometer was installed in 2008 under the umbrella of AMBER project [START_REF] Yizengaw | African Meridian B-field Education and Research (AMBER) Array[END_REF]. The first section presents the data set and data processing technique, and the second section the results. The third section is a discussion of our results at the light of the preceding works with the conclusion.

Data set and data processing

The magnetic observatory at Medea (36.85 • N, 2.93 • E geographic, 27.98 • N, 77.7 • E geomagnetic) in Algeria was installed in 2008 under the umbrella of AMBER project [START_REF] Yizengaw | African Meridian B-field Education and Research (AMBER) Array[END_REF], and is maintained by the Center of Research in Astronomy, Astrophysics and Geophysics (CRAAG, Algeria). In the framework of different scientific programs, United Nations Basic Space Science (UNBSS) International Heliophysical Year (IHY) and International Space Weather Initiative (ISWI), arrays of small instruments were deployed in order to make global measurements of space-physics-related phenomena. The network AMBER (African Meridian B field Education and Research) is one of the arrays of ground-based instruments deployed ini Africa under the IHY program [START_REF] Yizengaw | African Meridian B-field Education and Research (AMBER) Array[END_REF]. One of the objectives of the AMBER array is to address the processes that govern the electrodynamics of the equatorial ionosphere in the African sector. The data set used in this study consists of the hourly values of geomagnetic components, North (X) and East (Y), recorded at Medea Observatory from August 2008 to December 2011. Figure 1 shows the available data; the grey and black correspond respectively to the hourly and the quiet days data. Table 1 gives the number of magnetic quiet days available for each month. The data are sampled at 2Hz (0.5 sec) and are time tagged with an accuracy of 10 -6 s [START_REF] Yizengaw | African Meridian B-field Education and Research (AMBER) Array[END_REF]. Then, these data are averaged to hourly values. The concept of local time (LT) was used throughout the analysis. Algeria is located at (GMT+1) longitudes.

M 0 = M 23 +M 24 +M 01 +M 0 2 4
The hourly departures of M from midnight baseline, (∆M), was obtained by subtracting the midnight baseline values for a particular day from the hourly values for that particular day. The hourly departure is expressed as follow:

∆M t = M t -M 0
where t is the time in hours, ranging from 01:00 to 24:00 LT. The hourly departure is further corrected for non-cyclic variation (Matsushita & Campbell, 1967) . [START_REF] Vestine | The geomagnetic field, its description and analysis[END_REF] have shown that the value at 01.00 LT is different from the value at 24.00 LT. They proposed the correction of this non-cyclic variation by making a linear adjustment on the daily hourly values such that the M value at 01.00 LT, 02.00 LT ... 24.00 LT are consider as N1, N2 ... N24 and taking:

∆ k = N 1 -N 24 23
The linearly adjusted values at these hours are:

N 1 + 0∆ k , N 2 + 1∆ k , .............N 2 4 + 23∆ k
The linearly adjusted values may be expressed as follows:

∆M t (N ) = N t + (t -1)∆ t
The hourly departures corrected for non-cyclic variation give the solar daily variation in X and Y. SqX and SqY denote the solar quiet daily variation in X and Y components during magnetic quiet days from August 2008 to December 2011.

The diurnal variation was obtained by averaging both SqX and SqY hourly values across the five international quiet days of each individual month.

The seasonal variations are also determined by calculating the average value of four months daytime values for each season. These seasonal variations are calculated following Lloyds seasons [START_REF] Lloyd | On Earth-currents, and their connexion with the diurnal changes of the horizontal magnetic needle[END_REF]. The four months of the year are classified based on the three seasons: (1) December solstice or D-Season (January, February, November, and December), (2) Equinox or E season (March, April, September, and October), (3) June Solstice or J-Season (May to August). We have also separated the 2 equinoxes in spring equinox (March, April) and autumnal equinox (September, October).

Figure 2 gives the pattern of the ideal Sq current system (on the left) which is considered as fixed toward the Sun. During the day a magnetic observatory moves below this system of current. At the right of this figure the variations of SqX and SqY observed at the ground level are schematically illustrated.

In the Northern hemisphere SqX is nil at the focus of the Sq system. SqX is positive/Eastward ionospheric electric current above the focus (named model type E, pure maximum, by [START_REF] Mayaud | Analyse morphologique de la variabilité jour à jour de la variation journalire "régulière" Sr du champ magnétique terrestre, 2. L e système de courants CM (régions non polaires)[END_REF]. SqX is negative/Westward ionospheric electric current, below the focus of the Sq System (named model type P, pure minimum, by [START_REF] Mayaud | Analyse morphologique de la variabilité jour à jour de la variation journalire "régulière" Sr du champ magnétique terrestre, 2. L e système de courants CM (régions non polaires)[END_REF]. The SqY, at all latitudes, is positive in the morning (Southward ionospheric electric current) and negative in the afternoon (northward ionospheric electric current). In the Southern hemisphere, it is the contrary as the current flows reverse compared to the northern hemisphere. We must recall here that the main pattern of Sq observed on data is roughly similar to this ideal pattern [START_REF] Pham Thi Thu | Sq field characteristics at Phu Thuy, Vietnam, during solar cycle 23: comparisons with Sq field in other longitude sectors[END_REF][START_REF] De Michelis | On the terms of geomagnetic daily variation in Antactica[END_REF][START_REF] Torta | Behavior of the quiet day geomagnetic variation at Livingston Island and variability of the Sq focus position in the South American-Antarctic Peninsula region[END_REF]. Figure 2: Sq current system schema (after [START_REF] Amory-Mazaudier | Contribution à létude des courants électriques, des champs électriques et des vents neutres ionosphériques des moyennes latitudes, variation régulière et variations perturbées, Études de cas à partir des observations du sondeur á diffusion incohérente de Saint-Santin en relation avec les observations du champ géomagnétique terrestre[END_REF]. The current direction is anticlockwise in the northern hemisphere and clockwise in the Southern hemisphere, at the magnetic equator there is the equatorial electrojet.

On the right side, the diurnal variations of SqX and SqY, observed at ground level for different latitudes are drawn. Medea is located below the focus of the Northern vortex.

Results

Diurnal variations:

Figure 3a shows the daily variation of SqX at Medea for all the available magnetic quiet days from August 2008 to December 2011. This figure illustrates the day-to-day variability of SqX. The day-to-day variability is large, particularly from February to December. This figure highlights that variations of SqX are generally low during the night-time (from 18.00 LT to 00.06 LT) and oscillate between :5 nT and 3 nT. During the daytime we can observe three behaviors: 1) SqX is mainly positive from February to June. SqX reached its maximum, 37 nT, around 15.00 LT on May 25th, 2011. 2) SqX oscillates around zero from October to January. It reached its maximum value of 25 nT around 12.00 LT on October 20th, 2008 and October 27th, 2009 respectively. 3) SqX exhibits negative values in the morning and positive values in the afternoon from July to September. The minimum value, -35 nT, was reached at 10.00 LT on September 19th, 2009 and the maximum value, 43 nT, was observed around 14.00 LT on September 1st, 2011.

Figure 3b is similar to Figure 3a for SqY. The variations of SqY are generally low during the night-time and oscillates between -5 nT and 3 nT. We observe the same pattern for all the days with a maximum in the morning and a minimum in the afternoon, except during February-March 2009, there is apparently no morning maximum. The morning maximum, ∼ 40nT, was observed between 08.00-09.00 LT of April 27 and September 1st, 2011. The minimum value, ∼ -40nT, was observed in the afternoon around 14.00 LT on September 16th, 2011. The variability of SqY is smaller than that of SqX.

Monthly diurnal variations:

Figure 4a shows the monthly diurnal variations of SqX for the 12 months during the years 2009, 2010, 2011 and several months in 2008. The variations of SqX are generally low during the night-time and oscillates between -5nT and 0 nT. During the daytime, we can distinguish the same behaviors as on the individual days (Figure 3a):

1) SqX is mainly positive from February to June, the maximum value of SqX, ∼ 29nT, was observed around 12.00 LT in June 2011. 2) SqX oscillates around zero from October to January, the maximum value of SqX, ∼ 16 nT, was observed around 09.LT in December 2011. 3) SqX is negative in the morning and positive in the afternoon from July to September, the minimum value of SqX, -23 nT, was observed around 10.00 LT in August 2010 and the maximum value, 16 nT, was observed around 14.00 on September 1 st 2011.

The first behavior corresponds to the ideal Sq current (see Figure 2), for a station below the focus, which is the case for Medea. The second behavior corresponds to a station near the focus of the Sq current, which can be slightly above or below the focus. The third behavior observed in summer is different; we will discuss later this point. Figure 4b is similar to Figure 4a for the SqY. All the curves exhibit roughly the same pattern: a positive variation in the morning and a negative variation in the afternoon. This is exactly the behavior of the ideal Sq current. The exception is during FebruaryMarch 2009 when there is apparently no southward current during the morning. The morning maximum values, ∼ 40nT, was observed around 09.00-10.00 LT in June 2011, and the afternoon minimum values, ∼ -37 nT, was observed around 13.00 LT in August 2010.

Figure 5 gives the SqY reversal time for the whole period. The time of the reversal occurs always before noon. This figure shows that the local time of the reversal is in general between 10.00 LT and 11.00 LT during Summer months and after 11.00 LT in winter months. Except in February and Junuary 2009, the reversal time occur before 10.00 LT.

Figure 6 shows the amplitude of the morning maximum (dashed line) and afternoon minimum (block line). In general the amplitude of the afternoon minimum is greater than that of the morning maximum. We observe a seasonal variation: the highest values of the SqY are observed in summer and the lowest values in winter. The amplitude of SqY is increasing from 2009 through 2012, which represents the solar cycle effect.

Seasonal variations

Figure 7 illustrates the seasonal variations of SqX, (left side) and SqY (right side). The SqX exhibits roughly the same pattern for spring, summer and autumn with a maximum of the SqX around 20nT during the day. In summer and autumn, a small negative minimum (few nT) is observed in the morning. During winter the SqX is very small (< 10nT ) and fluctuates. The SqY is similar for all the seasons with a maximum in the morning and a minimum in the afternoon. While the SqX maximum occurred in 2011, the maximum of Sq(Y) occurred earlier, i.e., in 2009 and 2010. It is noted that these seasonal variations are superimposed on the day-to-day variations, which are particularly evident in SqX. The patterns of the seasonal average daily variations are consistent with the Sq current system in the northern hemisphere illustrated in Figure 2. The small amplitude SqX in winter can be understood by the fact that during this season, Medea is closest to the focus of the Sq current system, than during the other seasons.

Figure 8 shows the seasonal variations of SqX (left panels) and SqY (right panels). The variations of SqX in spring, Summer and autumn exhibit the dayside maximum roughly at the same time in 2009 and 2010. However, in 2011, the SqX dayside maximum occurs 2 hours earlier than that of in 2009 and 2010. The seasonal variation in SqY is very large particularly in 2011.

Discussion and conclusion

The Sq variation in the X component observed at the Medea Observatory shows different patterns for different months and seasons (see Figure 3a, Figure 4a, Figure 7 and Figure 8 ). All these differences in shapes do not correspond to the "ideal" Sq current system (Figure 2) but correspond to some other equivalent currents systems represented by three models called R, T and F [START_REF] Mayaud | Analyse morphologique de la variabilité jour à jour de la variation journalire "régulière" Sr du champ magnétique terrestre, 2. L e système de courants CM (régions non polaires)[END_REF]. The left panel in Figure 9 shows the pattern of the R, T and F models and the right panel in the same figure shows the SqX current systems that are caused by each model. The current model R ("ideal Sq system") is a current system with oval or elliptical lines. The short axis is north-south. Its focus is located at mid-latitudes. The model R has an inclination of the small North South axis which is no more directed North South as for the model T. In comparison with the model R, the focus of the model F can be located at higher or lower latitudes. With a displacement of the focus toward the high latitudes, the variation at the focus is nil and is negative from either side. With a displacement of the focus toward low latitudes the effect is positive from either side. The R model causes no variation in SqX component at the latitude of the focus. However, at the either side of the focus, it causes a minimum pure (P-type form) above the focus and maximum pure (form type E) below the focus. At the Medea station, the effect caused by the R model is observed from February to June on diurnal and monthly variations. Indeed, from February to June, the SqX shows a form of type E (i-th maximum pure). From July to September, we observe negative values in the morning and positive values in the afternoon sector. However, in summer and autumn we also observed a secondary minimum. The behavior of the SqX component corresponds to a transition between the type P and type E. This transition corresponds to the effects caused by the T current model. The T current model corresponds to the R model with an inclination of its short axis, which is more North-South. The effect of such a model on the SqX component consists of a maximum followed by a minimum (denoted as T m h ) or a minimum followed by a maximum (denoted as T m l ) [START_REF] Mayaud | Analyse morphologique de la variabilité jour à jour de la variation journalire "régulière" Sr du champ magnétique terrestre, 2. L e système de courants CM (régions non polaires)[END_REF]. In the case of Medea, from July to September, the morphology of the curves corresponds to the Model T. In the morning sector the inclination of the current lines are moved to lower latitudes (i-e T m l ). The monthly average diurnal variations from October to January and the seasonal average Sq variations for the winter can be likened to the effects of R model at the latitude of the focus. This is due to the fact that in winter the focus of Sq current system is very close to the latitude of the Medea station.

The morphology of diurnal, monthly and seasonal variations of the East component, SqY, observed at the observatory Medea (see Figure 3b, Figure 4b, Figure 7 and Figure 8 ), consists of normal type, the "northern type ", with a eastward maximum in the morning followed by an westward minimum in the afternoon. It corresponds to the effect of the ideal Sq current model (Figure 2). The SqY component shows always the same behavior ( Figure 6). Nevertheless there is an asymmetry: the amplitude of the morning maximum is smaller than the amplitude of the afternoon minimum. This asymmetry is more pronounced in summer than in winter. The disappearance of the morning peak is due to the abnormal diurnal pattern of SqY during March 2012) reported on the disappearance of the northern Sq current system in the East African region during February 2009. Also, Yamazaki et al. (2012) reported abnormal Sq behaviors in the American and Asian sectors during FebruaryMarch 2009, following an unusually strong stratospheric sudden warming event. It is worth mentioning here that the abnormal SqY at Medea during FebruaryMarch 2009 could also be associated with the stratospheric sudden warming effect.

The seasonal variation in the zero-crossing time shown in Figure 5 is consistent with the seasonal variation in the local time of the Sq current focus reported by [START_REF] Campbell | Correction to Quiet ionospheric currents of the Northern Hemisphere derived from geomagnetic field records[END_REF]. These observations are due to the shift of the focus of Sq system to high latitudes. The variability of the position of the focus current system Sq has been widely studied. [START_REF] Suzuki | A new analysis of the geomagnetic Sq field[END_REF], using the same data as [START_REF] Mayaud | Analyse morphologique de la variabilité jour à jour de la variation journalire "régulière" Sr du champ magnétique terrestre, 2. L e système de courants CM (régions non polaires)[END_REF] from 68 stations across the globe, showed that the focus of Sq system moves in the northsouth direction through the seasons. [START_REF] Stening | Day-to-day changes in the latitudes of the foci of the Sq current system and their relation to equatorial electrojet strength[END_REF], using data from the Australian and Japanese stations, showed that the focus of the Sq current system moves in the meridional direction from one day to another. By studying the seasonal and annual variations of the Sq current systems, [START_REF] Takeda | Time variation of global geomagnetic Sq field in 1964 and 1980[END_REF][START_REF] Takeda | The correlation between the variation in ionospheric conductivity and that of the geomagnetic Sq field[END_REF] showed that the longitude of the focus of Sq system evolves with the seasons and hours of the day. [START_REF] Malin | The Sq current system during International Geophysical Year[END_REF] highlighted that the longitudinal shift of the northern hemisphere focus is ahead of that of the southern hemisphere by 73 ± 21 minutes.

The afternoon minimum of SqY component in Medea (Figure 6) has a semi-annual variation. This behavior corresponds to the effects of the model with two vortex Sq current system, one in each hemisphere. [START_REF] Yamazaki | Equivalent current systems for the annual and semiannual Sq variations observed along the 210 • MM CPMN stations[END_REF] found that the equivalent Sq current system undergoes semi-annual variations that allow the maintenance of the model with two vortexes. They explained this semi-annual variation is due to diurnal tidal winds. Indeed, it has been widely established that the diurnal variation of the tidal wind plays a major role in the formation of the model of Sq equivalent current system with two vortexes [START_REF] Maeda | Horizontal wind system in the ionospheric E region deduced from the dynamo theory of the magnetic Sq variation, I[END_REF][START_REF] Kato | Horizontal wind system in the ionospheric E region deduced from the dynamo theory of the magnetic Sq variation, II[END_REF]. The morning maximum amplitude has an annual variation. This amplitude is higher in the summer months compared to the winter months. [START_REF] Yamazaki | Equivalent current systems for the annual and semiannual Sq variations observed along the 210 • MM CPMN stations[END_REF] found that the annual variation of Sq current system is characterized by the existence of a single vortex. This uniqueness of the vortex is probably due to the fact that the current intensity is greater in the summer hemisphere than in the winter hemisphere. In the summer, the higher-latitude annual currents are westward in the Northern Hemisphere and eastward in the Southern Hemisphere, and thus the northern Sq current vortex is strengthened and the southern Sq current vortex is weakened. In winter, the higher-latitude annual currents are eastward in the Northern Hemisphere and westward in the Southern Hemisphere, and thus the northern Sq current vortex is weakened and the southern Sq current vortex is strengthened. The annual variation can be explained by seasonal variations of the ionospheric conductivity and the atmospheric tides [START_REF] Mazaudier | Electric currents above Saint-Santin, 2,Model[END_REF]Yamazaki & Yumoto, 2012) and by the fieldaligned currents flowing from the summer to the winter hemisphere in the morning and reversely in the afternoon [START_REF] Fukushima | Electric potential between conjugate points in middle latitudes caused by asymmetric dynamo in the ionosphere[END_REF]. Unfortunately, our two-dimensional approach does not allows us to discuss how much field-aligned currents contribute to the annual variations of Sq.

The amplitude of SqY is increasing from 2009 through 2012, which represents the solar cycle effect (Figure 6). They peaked in 2011, which corresponds to the rising phase of the solar cycle number 24. It is generally accepted that the amplitude variations of the ionospheric electric currents varies linearly with the solar cycle (e.g [START_REF] Olsen | The solar cycle variability of lunar and solar daily geomagnetic variations[END_REF]. In an analysis, on a long series of data from many observatories on the globe, [START_REF] Torta | An inspection of the long-term behaviour of the range of the daily geomagnetic field variation from comprehensive modelling[END_REF] showed that the magnitude of the SqY component varies linearly with the solar cycle. Yamazaki & Yumoto (2012), using a Fourier analysis for the data observed at Kakioka and Gnangara, showed a positive linear correlation between the solar cycle and the annual and semi-annual variations of Sq current system. A strong correlation between the amplitude of Sq and the solar cycle variations has also been demonstrated (e.g Pham et al., 2011). However, this correlation is not the same for all observatories, see table 2 by Pham et al. The three year data from Medea, are not sufficient for the determination of the solar cycle effect.

In conclusion, the analysis of regular variations of Sq at the latitude of Medea helped us to highlight the following features:

-A differences in the diurnal pattern of the northward component SqX at different months and seasons, which corresponds to the effects caused by a transition model R and T. These could be due to the shift of the focus position, deformation of Sq current system and its invasion from an hemisphere to the other.

-An equinoctial asymmetry: The amplitude of SqY in autumn is higher than in spring.

-Abnormal SqY at Medea during February-March 2009 could be associated with stratospheric sudden warming effect.

-A mismatch of the SqY component: the amplitude of the afternoon minimum is greater than that of the morning maximum. The morning-afternoon asymmetry of SqY is probably due to the UT variation of the Sq current system. That is, the Sq current system ∼ 14 UT is stronger than ∼ 8 UT.

These results show that using a single station one can integrate the average movement and deformation of Sq system. However, the results obtained from Medea are not enough on their own to understand the dynamics of the ionospheric electric current systems of the northern hemisphere. The addition of data from observatories in Europe and North Africa may enable us to perform a systematic and comprehensive study of equivalent current system and its transient variations due to solar activity in the sector Africa-Europe. However, using instant mapping the results from all the observatories around the world can indeed contribute to a better representation of Sq current system at the global scale.

Figure 3a :

 3a Figure 3a: The mass plot of diurnal variations of the SqX at Medea for all the available magnetic quiet days from August 2008 to December 2011.

Figure 3b :

 3b Figure 3b: The mass plot of diurnal variations of the SqY at Medea for all the available magnetic quiet days from August 2008 to December 2011.

Figure 4a :

 4a Figure 4a: Monthly mean diurnal variations of the SqX at Medea for each set of the five international quiet days of each month from August 2008 to December 2011.

Figure 4b :

 4b Figure 4b: Monthly mean diurnal variations of the SqY at Medea for each set of the five international quiet days of each month from August 2008 to December 2011.

Figure 5 :Figure 6 :

 56 Figure 5: Times of the reversal of the SqY component at Medea from August 2008 to December 2011.

Figure 7 :

 7 Figure 7: Seasonal variations of SqX (left column) and SqY (right column) at Medea from 2009 to 2011.From top to bottom each row represents the corresponding season for each year 2009, 2010 and 2011 respectively.

Figure 8 :

 8 Figure 8: Seasonal variations of SqX (left column) and SqY (right column) from 2009 to 2011. From top to bottom each panel shows the four seasons of the year. Each figure shows 4 curves corresponding to the variation of either SqX or SqY over the 4 seasons (Winter, Spring, summer and autumn). From top to bottom each row represent the seasonal variation across each individual year

Figure 9 :

 9 Figure 9: Seasonal variations of SqX (left column) and SqY (right column) from 2009 to 2011. From top to bottom each panel shows the four seasons of the year. Each figure shows 4 curves corresponding to the variation of either SqX or SqY over the 4 seasons (Winter, Spring, summer and autumn). From top to bottom each row represent the seasonal variation across each individual year

  2009, probably not seasonal effect. According to Pham et al. (2011), the disappearance of the morning peak was observed at MBour every spring during 1996-2006. At Medea, on the other hand, the disappearance of the morning peak occurred only during March 2009. Thus, this is something different from what Pham et al. (2011) found at MBour. El Hawary & al. (

Table 1 :

 1 Magnetic quiet days available for each month.

	Month	Available quiet Available quiet Available quiet Available quiet
		days (2008)	days (2009)	days (2010)	days (2011)
	January	0	0	5	5
	February	0	4	5	5
	March	0	4	5	5
	April	0	1	5	5
	May	0	5	4	5
	June	0	4	5	1
	July	0	2	5	5
	August	5	4	5	4
	September 4	5	4	4
	October	5	5	2	2
	November 4	3	3	5
	December 3	3	5	5
		21/25	40/60	53/60	51/60
	The baseline value was deduced from the trend determined by the two local midnight levels on
	each side of the day [from 23.00 to 02.00 LT]. The baseline values for the elements used in this
	study are expressed as follows:

NOV 2008 2009 2010 2011

Acknowledgement

we are especially grateful to Akacem Nouredine and Meziane El-Hadi, who have contributed to installing and maintaining the Medea Observatory.

The authors sincerely acknowledge a reviewer whose comments and advices have greatly contributed to improve the final form of this work.