C Zaroli
email: c.zaroli@unistra.fr

J.-J Lévêque

B S A Schuberth

Z Duputel

G Nolet

Global S-wave tomography using receiver pairs: an alternative to get rid of earthquake mislocation

Keywords: Earthquake source observations, Body waves, Seismic tomography

Global seismic tomography suffers from uncertainties in earthquake parameters routinely published in seismic catalogues. In particular, errors in earthquake location and origin-time may lead to strong biases in measured body wave delay-times and significantly pollute tomographic models. Common ways of dealing with this issue are to incorporate source parameters as additional unknowns into the linear tomographic equations, or to seek combinations of data to minimize the influence of source mislocations. We propose an alternative, physicallybased method to desensitize direct S-wave delay-times to errors in earthquake location and origin-time. Our approach takes advantage of the fact that mislocation delay-time biases depend to first order on the earthquake-receiver azimuth, and to second order on the epicentral distance. Therefore, for every earthquake, we compute S-wave differential delay-times between optimized receiver pairs, such that a large part of their mislocation delay-time biases cancels out (for example origin-time fully subtracts out), while the difference of their sensitivity kernels remains sensitive to the model parameters of interest. Considering realistic, randomly distributed source mislocation vectors, as well as various levels of data noise and different synthetic Earths, we demonstrate that mislocation-related model errors are highly reduced when inverting for such differential delay-times, compared to absolute ones. The reduction is particularly rewarding for imaging the upper-mantle and transition zone. We conclude that using optimized receiver pairs is a suitable, low cost alternative to get rid of errors on earthquake location and origin-time for teleseismic direct S-wave traveltimes. Moreover, it can partly remove unilateral rupture propagation effects in cross-correlation delay-times, since they are similar to mislocation effects.

I N T RO D U C T I O N

Teleseismic body waves are sensitive to the physical properties of the media through which they propagate, so that they contain a lot of information on the 3-D structure of the Earth's interior. How to extract the most relevant structural information from seismograms is still an open question. Global seismic tomography encompasses numerous ways to turn seismic wave traveltime anomalies into velocity anomalies in the mantle. Although global tomographic models could further be refined using recent theoretical developments in seismic wave propagation (e.g. [START_REF] Dahlen | Fréchet kernels for finitefrequency traveltimes-I. Theory[END_REF][START_REF] Tromp | Seismic tomography, adjoint methods, time reversal and banana-doughnut kernels[END_REF][START_REF] Fichtner | Full waveform tomography for upper-mantle structure in the australasian region using adjoint methods[END_REF], we believe that significative improvements could also come from an improved exploitation of the continuously expanding worldwide network of digital seismometers. Indeed, the number of permanent and temporary stations on land (or islands) has grown during the last decade, as has the number of Ocean Bottom Seismometers (OBS). The recent development of submarines Mobile Earthquake Recording in Marine Areas by Independent Divers (MERMAIDS), designed to record seismic waves under water will also further increase the amount of available seismic data (e.g. [START_REF] Simons | A future for drifting seismic networks[END_REF].

Since its premisses, global seismic tomography has suffered from errors in the earthquake parameters routinely published in seismic catalogues, including clock drifts, earthquake mislocations, focal mechanism errors, and, for cross-correlation delay-times, the effects of rupture propagation. In this study, we focus on errors in earthquake location and origin-time which can lead to strong biases in measurements of body wave delay-times, and significantly pollute tomographic models; we also point out the similarity of mislocation effects with unilateral rupture directivity effects. Source locations often have errors of the order of 10 km in each direction, though larger errors or biases may exist in some oceanic regions away from seismic networks (e.g. [START_REF] Kennett | Traveltimes for global earthquake location and phase identification[END_REF][START_REF] Shearer | Improving global seismic event locations using sourcereceiver reciprocity[END_REF]. When no station is available near the earthquake, it is well known that depth accuracy of shallow events can be affected by the strong trade-off between depth and origin-time (e.g. [START_REF] Nolet | A Breviary of Seismic Tomography[END_REF]). [START_REF] Bolton | Traveltimes of P and S from global digital seismic networks: implication for the relative variation of P and S velocity in the mantle[END_REF] estimate delay-time errors for P and S waves caused by earthquake mislocation errors to be 0.6-1.2 and 1.6-2.5 s, respectively, assuming a typical depth uncertainty of 10 km, at an epicentral distance of 70 • , and for an epicentral mislocation vector of length 10-20 km.

Let us recap the most widely used strategies to deal with uncertainties in earthquake location and origin-time in global body wave tomography. A first approach consists in seeking combinations of data such that the influence of source errors subtracts out (e.g. [START_REF] Kuo | Lateral heterogeneity and azimuthal anisotropy in the north atlantic determined from SS-S differential travel times[END_REF][START_REF] Woodward | Global upper mantle structure from long-period differential travel times[END_REF][START_REF] Woodward | Constraints on the large-scale structure of the Earth's mantle[END_REF][START_REF] Paulssen | On PP-P differential travel time measurements[END_REF][START_REF] Houser | Shear and compressional velocity models of the mantle from cluster analysis of longperiod waveforms[END_REF]. One can invert for the delay-time difference of two seismic phases, that is S and ScS, recorded at same receiver i such that:

δt S i -δt ScS i = ⊕ [K S i (r) -K ScS i (r)
]δ ln V S (r)d 3 r, where K denotes some traveltime sensitivity kernel. Though these observables become insensitive to errors in origin-time, there may remain some residual mislocation biases for short epicentral distances i (e.g. up to ±1.8 s for an horizontal mislocation of 18 km), or there may be some lack of sensitivity to model parameters of interest for large i (depending on the kernel difference, K S i -K ScS i ). Another limitation is that one cannot combine all seismic phases in this way, for a typical global shear wave data set. For instance, many direct S phases cannot be combined with other phases, such as ScS or SS, because unavailable or not measured for the same source-receiver geometry (e.g. [START_REF] Montelli | A catalogue of deep mantle plumes. new results from finite-frequency tomography[END_REF][START_REF] Houser | Shear and compressional velocity models of the mantle from cluster analysis of longperiod waveforms[END_REF]. A second, popular approach is to incorporate corrections to the published source parameters (location and origin-time) as additional unknowns into the linear system of tomographic equations (e.g. [START_REF] Nolet | A Breviary of Seismic Tomography[END_REF]. This approach has the disadvantage that data cannot always discriminate between modifying the source parameters and changing some of the velocity anomalies (in particular in the source region). Another, more formal approach consists in mathematically desensitizing the linear tomographic equations to errors in source parameters (e.g. [START_REF] Pavlis | The mixed discrete-continuous inverse problem: application to the simultaneous determination of earthquake hypocenters and velocity structure[END_REF][START_REF] Spencer | Travel-time inversion for simultaneous earthquake location and velocity structure determination in laterally varying media[END_REF][START_REF] Masters | A shear-velocity model of the mantle[END_REF]. Basically, for each event, one considers that the travel-time residuals d can be expressed as: d = Gm + Ah, where G is the matrix containing the projection of the corresponding sensitivity kernels on the model grid, m represents the unknown 3-D velocity anomalies, A is a matrix containing the earthquake-receiver geometry and h is a four-components vector that describes a perturbation in location and origin-time of the earthquake. [START_REF] Masters | A shear-velocity model of the mantle[END_REF] show that one may seek linear combinations of the data to render them insensitive to the event location, by computing a projector matrix P such that PA is zero. The major issue with this projection method is that each new time-residual (Pd) is a linear combination of all the original time-residuals, so that the new sensitivity matrix (PG) is no longer sparse, and [START_REF] Masters | A shear-velocity model of the mantle[END_REF] report that it causes computational difficulties for massive inverse problems. It may also be difficult to intuitively apprehend the sensitivity of such algebraically combined data to some particular model parameters.

In this study, we present a physically based, low cost alternative to desensitize teleseismic long-period direct S-wave delaytimes to errors in earthquake location and origin-time. Our approach consists in computing, for each event, differential S delaytimes between receiver pairs (i, j), and solving for: δt S i -δt S j = ⊕ (K S i (r) -K S j (r))δ ln V S (r)d 3 r. We will discuss how to efficiently select optimized receiver pairs (i, j), so that the differential delaytimes δt S i -δt S j become nearly insensitive to source mislocations, while the differences of sensitivity kernels K S i -K S j remain sensitive to model parameters of interest. Our motivations are to take advantage of: (1) our knowledge that mislocation delay-time biases depend to first order on the earthquake-receiver azimuth, and to second order on the epicentral distance and (2) the continuously expanding worldwide broad-band seismic networks, which we believe can make it possible to routinely use such optimized receiver pairs in global body wave tomography. The benefits of our approach will be illustrated with several tomographic tests using realistic synthetic data sets, biased by randomly distributed source mislocation vectors. We will use two different synthetic earth models (Gaussian Random Field and Geodynamic models) of shear velocity anomalies distributed in the whole-mantle to produce statistically relevant structural time residuals. Our synthetic data will also include various levels of noise. Our goal will therefore be to show that mislocationrelated model errors are highly reduced when inverting for such differential delay-times, δt S i -δt S j , compared to absolute ones, δt S i .

S Y N T H E T I C DATA W I T H R E A L I S T I C S TAT I S T I C S

Receivers and earthquakes

Our approach aims at taking advantage of the rapidly expanding worldwide seismic networks. [START_REF] Houser | Shear and compressional velocity models of the mantle from cluster analysis of longperiod waveforms[END_REF] report that, in the last decade, it has become common to record an earthquake on more than 200 worldwide broad-band seismometers. In order to set up our synthetic experiment, therefore, we consider a dense, realistic spatial distribution of 738 receivers and 144 earthquakes, as shown in Fig. 1. Earthquake locations corresponding to events with a body wave magnitude 5.5 ≤ m b ≤ 6.5 are extracted from the Harvard GCMT catalogue. Receiver locations correspond to real locations of permanent and temporary broad-band seismometers, including OBS.

Teleseismic S traveltimes

Unlike, for example, [START_REF] Lou | AIMBAT: a python/matplotlib tool for measuring teleseismic arrival times[END_REF], we aim at working with the framework of finite-frequency tomography (e.g. [START_REF] Dahlen | Fréchet kernels for finitefrequency traveltimes-I. Theory[END_REF][START_REF] Zaroli | Frequency-dependent effects on global S-wave traveltimes: wavefront-healing, scattering and attenuation[END_REF][START_REF] Mercerat | Comparison of ray-and adjoint-based sensitivity kernels for body-wave seismic tomography[END_REF][START_REF] Zaroli | An objective rationale for the choice of regularisation parameter with application to global multiple-frequency S-wave tomography[END_REF][START_REF] Mercerat | Cross-borehole tomography with correlation delay times[END_REF]. Thus, the ith datum δt i corresponds to the time-lag maximizing the cross-correlation of an observed S-waveform, u obs i (t), with its corresponding ray-theoretical synthetic waveform, u syn i (t) and over the time-window [t 1 , t 2 ]:

δt i (T ) = τ ∈ R, t 2 t 1 u syn i (t)u obs i (t -τ )dt = max . ( 1
)
The observed and synthetic waveforms are filtered around a central period T prior to cross-correlation measurements (e.g. [START_REF] Zaroli | Frequency-dependent effects on global S-wave traveltimes: wavefront-healing, scattering and attenuation[END_REF], so that the time-delay is frequency-dependent. Because teleseismic S waves often have their maximum of energy around 20 s period, we shall use T = 20 s throughout this study. Following [START_REF] Dahlen | Fréchet kernels for finitefrequency traveltimes-I. Theory[END_REF], the linear problem, to be solved for m(r), is:

δt i (T ) = ⊕ K i (r; T )m(r)d 3 r . ( 2
)
The volume ⊕ = V i (T ) is limited to the region where the amplitude of the finite-frequency sensitivity (Fréchet) kernel K i (r; T ) is significant (e.g. [START_REF] Zaroli | An objective rationale for the choice of regularisation parameter with application to global multiple-frequency S-wave tomography[END_REF], and m(r) represents unknown 3-D shear velocity perturbations, δ ln V S (r), with respect to the 1-D reference velocity model IASP91 [START_REF] Kennett | Traveltimes for global earthquake location and phase identification[END_REF], at each point r in the medium. The linearity of eq. ( 2) is guaranteed over a wide range of anomaly amplitudes [START_REF] Mercerat | Comparison of ray-and adjoint-based sensitivity kernels for body-wave seismic tomography[END_REF], which will be central to our method for relating differential delay-times to differences of finite-frequency kernels (cf. eq. 10). Formally, one should not write an equality sign in eq. ( 2), since the measured timelags are affected by effects other than the 3-D structure (cf. eq. 3), but it is a common way to do so (e.g. [START_REF] Nolet | A Breviary of Seismic Tomography[END_REF]). The dependence on the period T will be dropped hereafter for ease of notation. We aim at computing realistic S-wave time-residuals δt i , where i denotes the receiver index for each earthquake. Considering our source-receiver geometry, the total number of absolute delay-times δt i we have, for direct S phases, is N abs = 54 652. After correction for physical dispersion due to intrinsic anelastic processes, δt i can be expressed as:

δt i = δt 3D i + δt N i + δt X i , (3)
where δt 3D i , δt N i and δt X i are residual-times caused by 3-D shear velocity anomalies, measurement noise, and earthquake mislocation (including clock drift), respectively.

To compute the structural delay-times δt 3D i , one needs to design a 3-D synthetic earth model, that we refer to as m true . This true-model should contain 3-D shear velocity anomalies throughout the entire mantle, and should allow us to yield structural data with realistic statistics. We generate a true-model using a Gaussian random field (GRF) with an exponential correlation function (correlation length ∼400 km), as shown in Fig. 2. Note that the seismic heterogeneities in our true-model are characterized by both short and long wavelengths, with some sharp discontinuities. Fig. 2(b, shows the histogram of shear velocity anomalies (δ ln V S ), which follows a normal distribution with mean 0 per cent and standard deviation 1 per cent. Structural delay-times are therefore computed as:

δt 3D i = ⊕ K i (r)m true (r)d 3 r . ( 4
)
As shown on Fig. 2(b, right-hand panel), the histogram of δt 3D i also follows a normal distribution, N (μ 3D , σ 3D ), with mean μ 3D = 0 s and standard deviation σ 3D = 3.1 s. Our distribution of syn-thetic structural delay-times is therefore in agreement with [START_REF] Bolton | Traveltimes of P and S from global digital seismic networks: implication for the relative variation of P and S velocity in the mantle[END_REF], who found a standard deviation of 3.2 s for real data. In Section 4.3.1, another true-model input (more 'Earth-like') will be considered to compute a second set of structural time-residuals for further testing our method.

To compute the noise-related residual-times δt N i , we randomly draw N abs samples from a normal distribution, such that:

δt N i ∼ N (μ N , σ N ), (5)
with mean μ N = 0 s and standard deviation σ N = 0.7 s, according to the realistic estimates derived by [START_REF] Zaroli | Frequency-dependent effects on global S-wave traveltimes: wavefront-healing, scattering and attenuation[END_REF] and [START_REF] Bolton | Traveltimes of P and S from global digital seismic networks: implication for the relative variation of P and S velocity in the mantle[END_REF] for ∼20 s dominant period teleseismic S waves. In Section 4.3.2, a 'pessimistic' case consisting in twice noisier data (i.e. σ N = 1.4 s) will also be considered for better quantifying the expected benefits from using receiver pairs. To compute a global set of mislocation time-residuals, δt X i , we first need to randomly generate mislocation vectors between true (i.e. exact) and original (i.e. false) source locations: δx s = x true s x 0 s . As seen in Section 1, [START_REF] Bolton | Traveltimes of P and S from global digital seismic networks: implication for the relative variation of P and S velocity in the mantle[END_REF] estimate that S delays due to event mislocation should follow a normal distribution with mean 0 s and standard deviation 1.6-2.5 s, corresponding to epicentral mislocation vectors of length δ 10-20 km, respectively. Therefore, we need to generate 144 triplets of mislocation parameters (δx, δy, δz), that are compatible with those realistic statistics. One may a priori assume that δx and δy follow normal distributions with null means and same standard deviations, that is: σ x = σ y . For the horizontal mislocation δ to be on average 10 km, we have to set: σ x = σ y 8 km. It is well known that, for teleseismic body wave delay-times, errors in horizontal mislocation (δx, δy) tend to dominate the error budget in the mislocation time-residuals, because the errors in origin time (δt 0 ) and depth (δz) tend to mutually cancel out on average (e.g. [START_REF] Bolton | Traveltimes of P and S from global digital seismic networks: implication for the relative variation of P and S velocity in the mantle[END_REF][START_REF] Nolet | A Breviary of Seismic Tomography[END_REF]. As detailed in Section 3, we aim at computing differential S delays for receiver pairs, which intrinsically are insensitive to the origin-time. Thus, for sake of simplicity we discard the errors in origin-time from the earthquake parameters, but compensate for this by making sure that the horizontal mislocation (δx, δy) remains on average predominant. To do this, we assume that the vertical mislocation (δz) follows a normal distribution with null mean and standard deviation: σ z σ x /2. This choice, which is related to the amount of compensation between the errors in origin-time and source depth, is not crucial since using optimized receiver pairs will also largely remove the errors in source-depth (cf. Section 3). At this stage, we randomly draw 144 triplets (δx, δy, δz) from the three normal distributions N (0, σ x ), N (0, σ y ) and N (0, σ z ). Mislocation time-residuals are computed at all receivers i, for each event, as:

δt X i = t 1D i x true s -t 1D i x 0 s , ( 6
)
where t 1D i (x true s ) and t 1D i (x 0 s ) denote the S phase ray-theoretical traveltimes, in the 1-D earth model IASP91, from true and original source locations, respectively, to receiver i's location. Traveltimes t 1D i are computed with the TauP Toolkit [START_REF] Crotwell | The TauP toolkit: flexible seismic travel-time and ray-path utilities[END_REF]. This allows us to generate an 'optimistic' set (i.e. δ 10 km) of 54 652 mislocation time-residuals, that follows a normal distribution with null mean and standard deviation σ X 1.4 s [close to the 1.6 s estimated by [START_REF] Bolton | Traveltimes of P and S from global digital seismic networks: implication for the relative variation of P and S velocity in the mantle[END_REF]]. This 'optimistic' set of 144 mislocation vectors δx s (δx, δy, δz) is then multiplied by two in order to generate a second 'pessimistic' set (i.e. δ 20 km) of mislocation time-residuals, that follows a normal distribution with null mean and standard deviation σ X 2.8 s [close to the 2.5 s estimated by [START_REF] Bolton | Traveltimes of P and S from global digital seismic networks: implication for the relative variation of P and S velocity in the mantle[END_REF]].

Influence of source mislocation on traveltimes

In practice, mislocation residual-times, δt X i , are commonly encountered in earthquake seismology (e.g. [START_REF] Kikuchi | Inversion of complex body waves[END_REF][START_REF] Zhan | Rupture complexity of the 1994 Bolivia and 2013 sea of Okhotsk deep earthquakes[END_REF]. For a teleseismic S-wave recorded at receiver i, one can write:

δt X i ≈ - ||δx s || 2 c × cos i , ( 7
)
where ||δx s || 2 = ||x true s x 0 s || 2 denotes the Euclidean norm of the mislocation vector between true and original source locations, c = V S (x 0 s ) denotes the shear wave phase velocity in IASP91 within the source region, and i is the angle between δx s and the ray path at x 0 s . A bit of geometry leads to:

cos i = cos i 0,i cos η + sin i 0,i sin η cos ϕ i , ( 8
)
where i 0,i is the ray take-off angle at x 0 s , η is the angle of δx s with respect to the vertical axis, and ϕ i = ϕ -ϕ i is the azimuth difference between δx s (azimuth ϕ) and the departing ray at x 0 s (earthquakereceiver azimuth, ϕ i ). Thus, using eqs ( 7) and ( 8), one can predict the effect of source mislocation on direct S phase delays. Fig. 3 shows an example of mislocation residual-times, δt X i , for direct S waves recorded at receivers i with epicentral distance i ∈ [28 • , 99 • ], for an earthquake located in the Indian Ocean with mislocation parameters (δx, δy, δz) = (18.3, -8.3, 2.5) km. The residual δt X i varies from -3.2 s to +2.1 s, which is quite a large range of variation compared to structural residual-times (cf. Section 2.2). Fig. 3 shows that δt X i is dominated at first order by a sinusoidal-like dependence on the earthquake-receiver azimuth ϕ i , though it also depends at second order on the epicentral distance i (i.e. on the ray take-off angle i 0, i ). Indeed, the mislocation budget is usually dominated by its horizontal component (cf. Section 2.2), and in the case of a purely horizontal mislocation (η = π/2), the eq. ( 8) leads to δt X i ∝ sin i 0,i cos ϕ i . The effect of an error in depth location only depends on the epicentral distance, since for a purely vertical mislocation (η equal to 0 or π ) the eq. ( 8) leads to δt X i ∝ cos i 0,i . Moreover, note that unilateral rupture directivity effects on crosscorrelation delay-times can lead to a similar sinusoidal-like pattern (cf. Appendix). In the following, we show that source mislocation (and unilateral rupture propagation) effects can partly be removed by using well-chosen receiver pairs. 7) and ( 8), at 28 • and 99 • epicentral distance ( i ), respectively. ϕ i denotes the earthquake-receiver azimuth. (b) Circles denote an epicentral distance of 28 • and 99 • , and the blue star represents the earthquake epicentre.

A R AT I O N A L E F O R O P T I M I Z E D R E C E I V E R PA I R S

Our idea is to compute, for each event, differential delay-times of S phases simultaneously recorded at some pairs of receivers (i, j), that is: δt ij = {δt i -δt j }. Using eq. ( 3), such differential data can be formally expressed as:

δt i j = δt 3D i -δt 3D j + δt N i -δt N j + δt X i -δt X j . ( 9
)
The linear tomographic problem (eq. 2) becomes:

δt i j = ⊕ (K i (r) -K j (r))m(r)d 3 r. ( 10
)
We want that the new mislocation term, δt X i j = {δt X i -δt X j }, subtracts out for well-chosen (optimized) receiver pairs (i, j). Using eqs ( 7) and ( 8), one can also predict the effect of source mislocation on the differential residual-time δt X i j , for a couple of receivers (i, j):

δt X i j ≈ - ||δx s || 2 c × (cos i -cos j ). ( 11
)
To minimize these mislocation residual-times, we propose to focus on receiver pairs with similar earthquake-receiver azimuths and epicentral distances, yet with different data sampling of the Earth. Indeed, to cancel out the mislocation term δt X i j , one needs cos i → cos j , meaning that the two receivers (i, j) should have similar earthquake-receiver azimuths and epicentral distances (i.e. ray takeoff angles). However, doing so would lead to a pair of receivers so close to each other that the differential kernel would go to zero for a large number of model parameters. Since two receivers should neither be too close nor too far, we found after some trials that, for S wave time-residuals measured at T = 20 s period, a relevant compromise is to select receiver pairs (i, j) such that:

{|ϕ i -ϕ j | → 0 • and | i -j | → 35 • }. ( 12
)
Let ξ ij = |cos icos j | denotes the term in δt X i j that should be reduced for relevant receiver pairs; note that ξ ij ranges from 0 to 2 for random pairs (i, j). According to our criterion (eq. 12), an 'ideal' receiver pair (i, j) with, for instance, {ϕ i = ϕ j , i = 50 • , j = i + 35 • }, leads to: ξ ij = 0.046 for a purely vertical mislocation, and 0 ≤ ξ ij ≤ 0.121 for a purely horizontal one. As expected, such a receiver pairing should allow us to strongly reduce the mislocation-related errors in differential data. Fig. 4 shows an example of the difference of two finite-frequency kernels, K i -K j , for the same ideal receiver pair (

ϕ ij = |ϕ i -ϕ j | = 0 • , and ij = | i -j | = 35 • ).
The differential kernel remains sensitive to velocity anomalies in a large part of the mantle, though it may sometimes be weaker at shallow depth (depending on the anomaly size and its location). For example, we compare the relative sensitivity of absolute versus differential data to some shear velocity anomaly located in the source vicinity. The situation is sketched in Fig. 4, where a square-shaped anomaly of 200 km edge is depicted within the transition-zone. It shows that such an anomaly may become less visible when taking delay-time differences, though this should not prevent us from using receiver pairs in global tomography (cf. Section 4.2).

In this study, for each receiver i we select its best-partner receiver j such that:

j = {k ∈ [1; N sta ], W ik (ϕ ik , ik ) = max}, ( 13
)
where N sta is the number of stations for which the target seismic phase has been measured, and where the weight functional W i j should lead us to select the most adequate receiver pairs (i, j) according to our criterion (eq. 12). We found that, for our data, the weights can be set as follows to meet our needs:

⎧ ⎪ ⎨ ⎪ ⎩ W i j = W ϕ i j × W i j W ϕ i j = e -(ϕ i j √ ln 2/C 1 ) 2 W i j = e -((35-i j ) √ ln 2/C 2 ) 2 -e -(35 √ ln 2/C 2 ) 2 1-e -(35 √ ln 2/C 2 ) 2 , ( 14
)
where:

C 1 = 25 • ; C 2 = 35 • , if ij ≤ 35 • , and C 2 = 35/3 • , otherwise.
As illustrated in Fig. 5(a), the weights W ϕ i j and W i j are built to reach their maxima for ϕ ij = 0 • and ij = 35 • , respectively. Note that the √ ln 2 factor causes the weight W ϕ i j , for instance, to decrease by a factor of 2 as ϕ ij increases by C 1 . (d-f) corresponding zoom-in near the source region, respectively. One sees that K i -K j can become less sensitive to the structure on the source side, as illustrated with the averaged kernel sensitivity to a square-shaped velocity anomaly of 200 km length inside the transition zone. Kernels are computed at T = 20 s period; units are s km -3 . Finding an optimum and more general criterion for pairing receivers is beyond the scope of this study, whose main goal is to quantify the possible advantages of using receiver pairs in global S-wave tomography. The crucial point is to meet the physical basis of our criterion (eqs 12-14), namely the need to build pairs such that their differential kernels tend to be insensitive to the errors in origin-time and source location, while remaining sensitive to the mantle structure of interest. Thus, we only need an empirical ex-pression for the weight functional, W i j , to automate the selection of receiver pairs based upon our a priori physical expectations of what relevant pairs should be.

{ i = 50 • , j = i + 35 • , ϕ i = ϕ j }. (a-c) K i , K j , K i -K j ,
Considering our earthquake-receiver geometry (Fig. 1), we report that 96 per cent (resp., 99.5 per cent) of all our best-partner receiver pairs (i, j) are characterized with weights W ϕ i j and W i j both superior to 0.8 (resp., 0.6). We did not discard the very small number of pairs with too low weights to facilitate further comparisons of by guest on September 26, 2014 http://gji.oxfordjournals.org/ Downloaded from tomographic models based on either absolute or differential data. For example, Fig. 5(d) shows all the receiver pairs corresponding to the same earthquake as in Fig. 3. Here, we only select the bestpartner pairs, for each event, to keep the number of differential data (N diff = 45 875) comparable to the number of absolute ones (N abs = 54 652). However, if significantly increasing the number of differential data does not represent a prohibitive computational issue, it could be interesting to consider several partners i for each receiver j, provided that their weights W i j are greater than some threshold (cf. Section 4.3.3). Doing so would increase the number of differential data, and hopefully could lead to a better recovery of the coherent structural information through the inversion process.

G L O B A L T O M O G R A P H Y U S I N G R E C E I V E R PA I R S

The inverse problem

The linear direct problem for either the absolute or the differential delay-times can be written in the usual formulation:

d = Gm, (15)
where d (of size N) and m (of size M) denote vectors of (absolute or differential) data and model parameters, respectively. The G matrix represents the projection of sensitivity kernels (or kernel differences) onto the model grid. Following [START_REF] Zaroli | An objective rationale for the choice of regularisation parameter with application to global multiple-frequency S-wave tomography[END_REF], we use a data-driven, irregular, model parametrization (spherical triangular prisms and spherical layers) and analytical ray-based finitefrequency traveltime sensitivity kernels. We assume that the prior covariance matrices of the data, C d , and of the model parameters, C m , follow Gaussian probability functions, such that the optimum estimate of model parameters, m, can be obtained by minimizing (e.g. [START_REF] Tarantola | Inversion of traveltimes and seismic waveforms[END_REF]:

f (m) = (Gm -d) t C -1 d (Gm -d) + m t C -1 m m. ( 16
)
Doing so leads to solving for m a system of normal equations:

G C 1 2 d C -1 2 m m = d 0 . ( 17
m(λ) = {m ∈ R M , ||d -Gm|| 2 2 + λ 2 ||m|| 2 2 = min}. ( 18
)
There is plenty of regularization strategies to find an appropriate damping value for the model solution m(λ), though they are rarely fully objective when applied to real data whose uncertainties are often just best guesses (e.g. [START_REF] Hansen | The use of the L-curve in the regularization of discrete ill-posed problems[END_REF][START_REF] Montelli | Global P and PP traveltime tomography: rays versus waves[END_REF][START_REF] Boschi | On the relevance of born theory in global seismic tomography[END_REF][START_REF] Nolet | A Breviary of Seismic Tomography[END_REF][START_REF] Zaroli | An objective rationale for the choice of regularisation parameter with application to global multiple-frequency S-wave tomography[END_REF]. Since in our case we know what is the true-model, m true , the most natural definition for an optimal damping value, λ opt , is:

λ opt {m(λ)} = {λ ∈ R + , ||m(λ) -m true || 2 = min}, (19)
Table 1. Notations for particular models, m(λ), and their associated optimal damping values, λ opt {m(λ)}.

m(λ) Data Mislocation λ opt {m(λ)} A 0 (λ) δt 3D+N i " no misloc. " λ A 0 D 0 (λ) δt 3D+N i j - λ D 0 A 1 (λ) δt 3D+N+X i " σ X = 1.4 s " λ A 1 D 1 (λ) δt 3D+N+X i j - λ D 1 A 2 (λ) δt 3D+N+X i " σ X = 2.8 s " λ A 2 D 2 (λ) δt 3D+N+X i j - λ D 2
where m true denotes the vector of true-model parameters (projection of m true onto the model grid). The model solution m(λ opt ) is the closest, from the 2 -norm point of view, to the true-model. The damping derived from eq. ( 19) is similar to what a subjective choice could have yield, for instance based upon an L-curve analysis. Using our knowledge of the true-model to infer the damping value is thus not a crucial point. However, using this truly optimal damping will allow us to objectively perform a further, fruitful error analysis in Section 4.2.

Results and proof-of-concept

In the context of using direct S waves in global tomography, our goal is to show that it is beneficial to invert for differential rather than absolute delay-times, provided that earthquake mislocations are of the order of 10-20 km. Let us define notations for particular tomographic models and their associated optimal damping values (cf. Table 1). Subscript k will alternatively refer to three mislocation cases:

⎧ ⎨ ⎩ k = 0 ⇔ " no misloc. " k = 1 ⇔ " σ X = 1.4 s " k = 2 ⇔ " σ X = 2.8 s " . ( 20
)
Let us consider the 'absolute models', A k (λ), and the 'differential models', D k (λ), resulting from inversions (λ denotes some damping value) of absolute and differential S delay-times affected or not by source mislocations, respectively:

A k (λ) ⇐ δt i = δt 3D i + δt N i if k = 0 δt i = δt 3D i + δt N i + δt X i if k = {1; 2} . ( 21
)
and

D k (λ) ⇐ δt i j = δt 3D i j + δt N i j if k = 0 δt i j = δt 3D i j + δt N i j + δt X i j if k = {1; 2} . ( 22
)
One can define their associated optimal damping values, for k = {0; 1; 2}, as follows:

λ A k = λ opt {A k (λ)} λ D k = λ opt {D k (λ)} . ( 23
)
Our goal therefore consists in showing that, for k = {1; 2}, the differential models D k (λ D k ) are less differing from the true-model m true , when compared to the absolute models A k (λ A k ), meaning that errors on earthquake locations are generating (significantly) less errors into the model solutions if one inverts for such differential (δt ij ) rather than absolute (δt i ) residual-times.

Reference absolute model

First, note that the best tomographic solution within our reach, if S delay-times were free of mislocation biases (i.e. k = 0), would require to use absolute rather than differential data, in order to fully exploit the kernels. The corresponding reference absolute model, A 0 (λ A 0 ), hence represents the best achievable tomographic model given our data geometry. The term {m true -A 0 (λ A 0 )} therefore represents the basic errors for any absolute model, and is related to two factors: (1) limited data coverage and (2) applied regularization to deal with measurement noise. Fig. 6 displays at several depths through the mantle a model comparison between A 0 (λ A 0 ) and m true . Since only teleseismic S phases are used, A 0 (λ A 0 ) mostly differs with m true at shallow depth (upper-mantle, transition-zone), and preferentially beneath oceanic regions (lower data coverage). At greater depth (mid lower-mantle and deeper), model differences largely decrease.

Extra errors for absolute and differential models

We are then interested in comparing the extra errors for absolute and differential models, that is: ). Note that the extra errors for differential models do not vary much for both mislocation cases, which is an evidence for the model insensitivity to source mislocations when using such differential data. We conclude that it is very rewarding to invert for differential, rather than absolute, S delay-times, provided that receivers could appropriately be paired (according to eq. 12) and mislocation residual-times would statistically be characterized by σ X = 1.4-2.8 s.

{A k (λ A k ) -A 0 (λ A 0 )} and {D k (λ A k ) -A 0 (λ A 0 )},

Contributors to extra errors

Last but not least, we are interested in (quantitatively) identifying the contributors to the extra errors for absolute and differential models. This can easily be achieved if we realize that the extra errors can be decomposed, for absolute-and differential models, such that:

(λ A 1 ) -A 0 (λ A 0 )}, (b) {A 1 (λ A 1 ) -A 0 (λ A 1 )}, (c) {A 0 (λ A 1 ) -A 0 (λ A 0 )}, (d) {D 1 (λ D 1 ) -A 0 (λ A 0 )}, (e) {D 1 (λ D 1 ) -D 0 (λ D 1 )}, (f) {D 0 (λ D 1 ) -D 0 (λ D 0 )} and (g) {D 0 (λ D 0 ) -A 0 (λ A 0 )}.
(λ A 2 ) -A 0 (λ A 0 )}, (b) {A 2 (λ A 2 ) -A 0 (λ A 2 )}, (c) {A 0 (λ A 2 ) -A 0 (λ A 0 )}, (d) {D 2 (λ D 2 ) -A 0 (λ A 0 )}, (e) {D 2 (λ D 2 ) -D 0 (λ D 2 )}, (f) {D 0 (λ D 2 ) -D 0 (λ D 0 )} and (g) {D 0 (λ D 0 ) -A 0 (λ A 0 )}.
A k (λ A k ) -A 0 (λ A 0 ) = {A k (λ A k ) -A 0 (λ A k )} +{A 0 (λ A k ) -A 0 (λ A 0 )} , ( 24
) and ⎧ ⎨ ⎩ D k (λ D k ) -A 0 (λ A 0 ) = {D k (λ D k ) -D 0 (λ D k )} +{D 0 (λ D k ) -D 0 (λ D 0 )} +{D 0 (λ D 0 ) -A 0 (λ A 0 )} , ( 25
)
respectively, and where k = {1; 2}. The situation is illustrated in Figs 7 and8. On the one hand, eq. ( 24) tells us that the extra errors for absolute models result from two terms: (1) The first term {A k (λ A k ) -A 0 (λ A k )} is due to adding mislocation biases in the data, while keeping the same regularization parameter; (2) The second term {A 0 (λ A k ) -A 0 (λ A 0 )} is due to increasing the regularization parameter, to deal with mislocation-biased data. The extra errors for absolute models are dominated by the first term (mislocation) as shown in Figs 7(b) and 8(b). A non-negligible contribution may also come from the second term (damping) in case of strong mislocation biases, in particular at the top of mid lower-mantle, cf. Fig. 8(c). Indeed, to deal with large mislocation biases (k = 2) in absolute data, one has to significantly increase the regularization parameter (λ A 2 λ A 0 ). On the other hand, eq. ( 25) tells us that the extra errors for differential models result from three terms: (1) The first term {D k (λ D k ) -D 0 (λ D k )} is due to adding mislocation biases in the data, while keeping the same regularization parameter; (2) the second term {D 0 (λ D k ) -D 0 (λ D 0 )} is due to increasing the regularization parameter, to deal with mislocation-biased data and (3) the third term {D 0 (λ D 0 ) -A 0 (λ A 0 )} is due to the use of kernel differences, which can itself be decomposed into the sum of a (predominant) first part {D 0 (λ D 0 ) -A 0 (λ D 0 )} involving a lack of sensitivity to some model parameters and a second part {A 0 (λ D 0 ) -A 0 (λ A 0 )} involving the need for more stringent damping to deal with enlarged noise-related errors in differential data (cf. Section 4.3.2). The extra errors for differential models are essentially not influenced by the first term (mislocation) as shown in Figs 7(e) and 8(e). This result confirms our appropriate receiver pairing for getting rid of a large part of mislocation effects in differential delay-times. There is therefore no need to impose some extra damping for differential models to deal with mislocation-related errors (λ D k ≈ λ D 0 ), and the second term (damping) of the extra errors for differential models is very weak, cf. Figs 7(f) and 8(f). Finally, the dominant term of the extra errors for differential models is the third one (kernel difference), as shown in Figs 7(g) and 8(g). Even though using kernel differences necessarily involves a lack of sensitivity to some model parameters, our results (Section 4.2.2) clearly show that the overall balance is largely in favour of differential data via the use of optimized receiver pairs.

Discussion

In the following, we aim at discussing several points that the reader could wonder about before applying our approach to some real global body wave data set. We also mention some perspectives towards fully getting rid of earthquake mislocation using multiple receiver pairs.

Earth-like true-model

First, one may wonder whether using a more Earth-like true-model, instead of a Gaussian random field (GRF) model, could modify our conclusions? For instance, it is well known that 3-D velocity anomalies inside the (real) mantle are not uniformly distributed (e.g. higher amplitudes in the upper-mantle). Thus, we have done supplementary calculations with another, more Earth-like, true-model denoted m true Geody . We used the shear wave velocity structure corresponding to the high-resolution geodynamic model S09-M2-Q by [START_REF] Schuberth | Thermal versus elastic heterogeneity in high-resolution mantle circulation models with pyrolite composition: high plume excess temperatures in the lowermost mantle[END_REF][START_REF] Schuberth | Synthetic seismograms for a synthetic earth: long-period P-and S-wave traveltime variations can be explained by temperature alone[END_REF], here referred to as Geody. This geodynamic model relies on three assumptions: (1) Large-scale flow structure related to plate motion history; (2) Radial three-layer (lithosphere, upper and lower mantle) viscosity profile in agreement with postglacial rebound and geoid observations and (3) Isochemical whole mantle flow with pyrolite composition. Shear wave velocities are computed by converting the temperature field of the mantle to elastic parameters and density using thermodynamic models of mantle mineralogy. Fig. 10 shows this second true-model, with respect to IASP91, after projection onto our own model parametrization. It compares well to global tomographic models in terms of spectral characteristics and magnitude of velocity anomalies, though it cannot correctly predict the exact location and pattern of structure in the deep mantle. In addition, it shows slightly different, yet realistic, S-wave delay-time statistics (cf. Fig. 10b). Note that m true Geody is characterized by large provinces in the mantle with very weak amplitudes [δ ln V S (r) ≈ 0], as shown on the tomographic crosssections in Figs 10(c) and (d). Such a feature precludes to analyse the extra errors for differential models related to the use of kernel differences. This is the reason why we first used the GRF truemodel, with amplitudes uniformly distributed in the whole mantle. We now compute a new set of structural delay-times as:

δt 3D i, Geody = ⊕ K i (r)m true Geody (r)d 3 r , ( 26
)
and then a new set of absolute time residuals:

δt i,Geody = δt 3D i, Geody + δt N i + δt X i . ( 27
)
Finally, using exactly the same receiver pairs (source-receiver geometry is identical), we compute the extra errors for the new absolute and differential models, as shown in Figs 9(b) and (d). The extra errors are, again, significantly larger for absolute models than for differential models, for the two considered mislocation cases. We conclude that our results do not depend much on the input truemodel, provided that it (grossly) reproduces the statistics of real data.

Noisy data

The noise-related errors are enlarged when taking delay-time differences (σ N is multiplied by a factor of √ 2), so that the inversion of differential data may require more stringent regularization parameter to suppress their induced model errors (cf. Section 4.2.3). Therefore, one may wonder whether the total extra errors could be larger in differential rather than absolute models for different levels of noise in the data. We have already shown that there was no reason to be worried about this hypothesis in the (realistic) case of σ N = 0.7 s. Here, we aim at verifying that our results still hold in a more pessimistic case where σ N = 1.4 s. We have then multiplied by two the original noise-related delay-times:

δt N i, Pessi = δt N i × 2 . ( 28
)
Then, using this new set of δt N i, Pessi we have recomputed all the extra errors for the absolute-and differential models, with the same mislocation regimes (σ X = 1.4 and 2.8 s) and true-model inputs (GRF and Geody). The new results are shown in Figs 9(e)-(h), and clearly demonstrate the interest of using receiver pairs with much noisier data. In particular, it still works fairly well in the worst-case scenario, that is weak mislocation-related errors (σ X = 1.4 s) and large noise-related errors (σ N = 1.4 s), as shown in Figs 9(e) and (f). We conclude that the use of optimized receiver pairs in global Swave tomography should be beneficial for real data applications, no matter whether the earthquake location catalogue is fairly accurate (σ X = 1.4 s) or less (σ X = 2.8 s), and the level of noise in the data set is relatively weak (σ N = 0.7 s) or large (σ N = 1.4 s).

Sparse receiver coverage

We have previously shown that, with our synthetic coverage, using all the pairs of best-partner receivers (eqs 12-14) was highly beneficial to decrease the mislocation-related errors in the model space. When dealing with present-time data, source-receiver geometry may be sparser, mainly because not all stations are recording simultaneously. It is thus likely that some of those best-partner pairs could not be relevant enough for getting rid of source mislocation. This should not be a brick wall that prevents us from using all other appropriate pairs, and the best option would therefore be to invert for a mix of absolute (δt i ) and differential (δt ij ) delay-times. The related question that then arises is how to set up a criterion indicating when to switch to absolute data. This could be done by setting a threshold value on weights W ϕ i j and W i j ensuring that both the azimuth and distance criteria of eq. ( 12) are 'fairly' met. What could be these minimal weights? Formally answering this question is beyond the scope of this study. However, we can get a first idea from our synthetic case. In this successful case, 99.5 per cent of the selected pair weights, W ϕ i j and W i j , are greater than 0.6, and 96 per cent of them are greater than 0.8. Note that W ϕ i j and W i j being greater than 0.8 corresponds to: ϕ ij ≤ 14 • and 22 • ≤ ij ≤ 42 • , respectively. As mentioned in Section 3, using a threshold criterion, instead of a best partner criterion, may significantly increase the number of selected differential data. Although it could lead to a better recovery of the coherent structural information, it would also lead to heavier computational issues. The appropriate minimal values for W ϕ i j and W i j should thus result from some compromise between the amount of differential data and the degree of compatibility with eq. ( 12).

Other seismic phases

Figs 7(a) and 8(a) are a reminder that earthquake mislocations may represent severe limitations to the final resolution of global S-wave tomographic models. This is particularly true for the upper-mantle and transition zone, mainly in regions where earthquakes occur. Removing such bias may be crucial, for instance when jointly inverting for body wave and surface wave, so that body waves do not bring biased informations in regions of common data sensitivity (uppermantle and transition-zone). Although we chose to focus on direct S waves in this study, one could also desensitize direct P wave delaytimes using similar receiver pairs. [START_REF] Bolton | Traveltimes of P and S from global digital seismic networks: implication for the relative variation of P and S velocity in the mantle[END_REF] report that mislocation effects are even more troublesome for P-wave tomography, so that the benefits from using receiver pairs could even potentially be greater. Future work will also consist in tuning our criteria for efficiently combining receivers having recorded other kind of seismic phases (e.g. ScS, SS).

Towards fully getting rid of mislocation using multiple receiver pairs

We aim at showing that, for each event, it is theoretically possible to fully get rid of source mislocation effects, yet sampling the regions of interest, by combining S phases recorded at four well-chosen receivers. Consider two receiver pairs (i, j) and (k, l) such that:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ i ≈ j ≈ k ≈ l |ϕ i -ϕ k | ≈ 180 • |ϕ j -ϕ l | ≈ 180 • i = j = k = l . ( 29
)
Using eqs ( 8) and ( 11), one can demonstrate that the data combination {δt ij + δt kl } leads to fully get rid of errors in source location and origin-time, that is:

δt X i j + δt X kl ≈ 0 . ( 30
)
To maximize the sensitivity of such multiple kernel differences, {K i -K j + K k -K l }, to model parameters, one should favour receiver pairs (i, j) and (k, l) such that:

|ϕ i -ϕ j | ≈ |ϕ k -ϕ l | → 90 • . ( 31
)
A dense enough receiver coverage would be required to find four receivers (i, j, k, l) verifying eqs ( 29) and (31). In order to exploit by guest on September 26, 2014 http://gji.oxfordjournals.org/ Downloaded from the most relevant information in mislocation-biased data, the best approach would therefore consist in inverting for a mix of absolute delays (δt i ), simple differential-delays (δt ij ), and multiple differential delays (δt ij + δt kl ), depending upon the actual source-receiver geometry to be dealt with. We postpone a more formal study on the feasibility of using such multiple receiver pairs in global body wave tomography (e.g. weights definition, sensitivity of multiple kernel differences).

CONCLUSION

From its first applications, global seismic tomography has suffered from uncertainties in earthquake parameters, including clock timedrift, earthquake mislocation, and, for cross-correlation delay-times, the effects of rupture propagation. In this study, we have focussed on errors in earthquake location and origin-time. Teleseismic Swave residual-times are commonly affected by mislocation biases with standard-deviation σ X ∼ 2 s, while their 3-D structural part corresponds to σ 3D ∼ 3 s. Thus, earthquake uncertainties can represent severe limitations to the improvement of global tomographic models. In this study, we have presented an alternative, physicallybased method to desensitize teleseismic long-period direct S-wave delay-times to errors in earthquake location and origin-time. Our approach takes advantage of the fact that mislocation delay-time biases depend to first order on the earthquake-receiver azimuth, and to second order on the epicentral distance. For each event, differential delay-times are computed between well-chosen receiver pairs. We have shown the feasibility of selecting receiver pairs such that differential data become nearly insensitive to source mislocation parameters (e.g. origin-time fully subtracts out), while the (finitefrequency) kernel differences remain sensitive to model parameters of interest. We found that a good compromise is to favour receiver pairs with similar earthquake-receiver azimuths, and whose epicentral distances differ by ∼35 • . Considering realistic, randomly distributed source mislocation vectors, as well as various levels of data noise and different synthetic Earths, we have shown that mislocation-related model errors could highly be reduced when inverting for such differential delay-times, compared to absolute ones.

In particular, we have shown how much it could be rewarding in the upper-mantle and transition-zone. We conclude that using optimized receiver pairs is a suitable, low cost alternative to get rid of errors on earthquake location and origin-time for teleseismic direct S-wave traveltimes. Moreover, it can partly get rid of unilateral rupture propagation effects in cross-correlation delay-times, since they are similar to mislocation effects. The same benefits should hold for teleseimic direct waves. Finally, using receiver pairs could help us to better exploit the weak finite-frequency effects (e.g. wavefronthealing) recently observed on teleseismic S-wave cross-correlation delay-times (e.g. [START_REF] Zaroli | Frequency-dependent effects on global S-wave traveltimes: wavefront-healing, scattering and attenuation[END_REF], and lead to improved imaging of small-scale 3-D velocity anomalies in future global tomographic models.

A P P E N D I X : S O U RC E P RO PA G AT I O N, C RO S S -C O R R E L AT I O N T I M E -R E S I D UA L S , A N D R E C E I V E R PA I R S

Since the beginning of the era of digital instrumentation, a popular way to efficiently measure seismic time-residuals has been to apply cross-correlation techniques (e.g. VanDecar & Crosson 1990). Cross-correlation measurements are affected by source propagation effects that should be dealt with, no matter whether the data inversion is based on ray-theoretical or finite-frequency approaches. A natural way to deal with source propagation is to include it in the computation of synthetic seismograms. However, it is often not possible to do so because source kinematics is poorly known in most cases. It turns out that the way we deal with source mislocations in this study (through the use of receiver pairs) could also be efficient for rejecting a large part of unilateral rupture propagation effects in teleseismic cross-correlation body wave delay-times. The purpose of this appendix is to illustrate this point. First, let us consider for a given earthquake the observed and synthetic waveforms of a direct S phase recorded at receiver i, denoted by u obs i (t) and u syn i (t), respectively. In finite-frequency tomography (e.g. [START_REF] Dahlen | Fréchet kernels for finitefrequency traveltimes-I. Theory[END_REF], for instance, the correlation delay-time is defined as the time-lag maximizing the cross-correlation function between u obs i (t) and u syn i (t) over some time-window (cf. eq. 1). The cross-correlation function γ of two signals s 1 (t) and s 2 (t) is defined in the spectral domain such that: i + δt X i is the simple correlation delay-time due to 3-D shear velocity anomalies, noisemeasurement and source mislocation, respectively (cf. Section 2.2). However, even in a very smooth earth model with no significant diffraction, the shape of the waveform can be influenced by the kinematics of the source if the fault length L is large. In the case of unilateral rupture propagation (e.g. [START_REF] Aki | Quantitative Seismology[END_REF], one can write the observed seismic waveform in the spectral domain such that:

U obs i (ω) = U obs i (ω) sin X i (ω) X i (ω) e iX i (ω) X i (ω) = ω L 2 1 v -cos i c , ( A3
)
where i is the angle between the ray direction to receiver i and the direction of rupture propagation, v is the rupture velocity and c is the local shear velocity within the source region. For such an unilateral rupture, one can write the cross-correlation function of the observed and synthetic waveforms in the spectral domain as:

γ U obs i ; U syn i (ω) = | ṁ(ω)| 2 sin X i (ω) X i (ω) e iω(δt i +δt R i ) , ( A4
)
where δt R i = L 2 ( 1 v -cos i c ). Since the phase shift in the exponential term of the cross-correlation is proportional to ω, it is a pure traveltime error, independent of frequency. Therefore, the actual cross-correlation delay-time δt i at receiver i can be expressed as:

δt i = δt i + δt R i . (A5)
The angle i can be viewed as analogous to the angle i in eq. ( 8), if one replaces the mislocation vector by the vector of rupture propagation. Similarly to source mislocation effects, the unilateral rupture propagation residual-times, δt R i , thus depend to first order on the earthquake-receiver azimuth, and to second order on the epicentral distance-assuming that the source propagation is pre-dominantly horizontal (cf. Section 2.3), which should be verified for a certain number of events with m b ∼ 6. The differential delay-times for a couple of receivers (i, j) can therefore be written as:

δt i -δt j = {δt i -δt j } + δt R i -δt R j , (A6)
where the differential effect of rupture propagation becomes:

δt R i -δt R j = - L 2c × (cos i -cos j ). (A7)
Note that eq. ( A7) is highly similar to eq. ( 11). Therefore, using optimized receiver pairs (i, j), as defined in Section 3 to deal with mislocations, would also lead to minimize the term (cos icos j ), and thus to almost cancel out the effect of unilateral rupture propagation in differential cross-correlation delay-times { δt i -δt j }, in addition to get rid of a large part of errors in origin-time and location. Determining whether this azimuthal part is dominant in source propagation effects would require to elaborate tomographic tests similar to those conducted in Section 4. However, to do so would require a good knowledge of the statistics of rupture propagation for a significant set of worldwide earthquakes, which is not currently available.

Figure 1 .

 1 Figure 1. Spatial distribution of receivers (triangles) and earthquakes (stars). Black dashed line: tectonic plates.

Figure 2 .

 2 Figure 2. First true-model input, m true (Gaussian random field).

Figure 3 .

 3 Figure 3. An example of mislocation residual-times, δt X i , for direct S phases measured at receivers i (cf. eq. 6). Mislocation parameters are: (δx, δy, δz) = (18.3, -8.3, 2.5) km, corresponding to an horizontal mislocation δ ∼ 20 km. (a) Solid and dashed black lines are theoretical mislocation residual-times computed using eqs (7) and (8), at 28 • and 99 • epicentral distance ( i ), respectively. ϕ i denotes the earthquake-receiver azimuth. (b) Circles denote an epicentral distance of 28 • and 99 • , and the blue star represents the earthquake epicentre.

Figure 4 .

 4 Figure 4. Illustration of the sensitivity corresponding to the difference of finite-frequency kernels, for direct S waves, in an 'ideal' case where: { i = 50 • ,

Figure 5 .

 5 Figure 5. (a-c) Illustration of the weight-functionals, W i j (ϕ i j , i j ) = W ϕ i j (ϕ i j ) × W i j ( i j ), used for selecting the optimized receiver pairs (i, j), for each event, where ϕ ij = |ϕ i -ϕ j |, and ij = | ij |. (d) The best-partner receiver pairs are shown with black solid lines.

)

  We use simple covariance matrices for the data, C d = σ 2 d I N (uniform data errors), and for the model, C m = σ 2 m I M , where I N and I M are identity matrices of size N × N and M × M, respectively. The value of λ = σ d /σ m allows us to regularize the problem by damping the model norm. In our experience, a simple regularization parameter (damping, λ) is sufficient to obtain smoothed model solutions since finite-frequency kernels integrate over a large volume (several Fresnel zones). For each damping value, λ, we use LSQR (e.g.[START_REF] Paige | LSQR: an algorithm for sparse, linear equations and sparse least squares[END_REF][START_REF] Grunberg | Conception d'une méthode de maillage 3D parallèle pour la construction d'un modèle de Terre réaliste par la tomographie sismique[END_REF]) to find the corresponding model solution, m(λ). The LSQR algorithm is an iterative row action method that converges to solution:

Figure 6 .

 6 Figure 6. Comparison of the first true-model input (Gaussian random field) with its corresponding reference absolute model. (a) A 0 (λ A 0 ), (b) m true , (c) m true -A 0 (λ A 0 ). Note that in order to visually enlarge the illustrative results of our approach, we do not use a whole Earth projection in Figs 6-8 (geographical considerations are not within our scope), but we show the statistics of our results over the entire mantle in Fig. 9.

  respectively, where k = {1; 2}. The word extra refers to the supplementary model errors (in addition to the basic errors) involved by using mislocation-biased data and/or differential kernels. Figs 7(a, d) and 8(a, d) show tomographic pictures of the extra errors for absolute and differential models, and their histograms are shown in Figs 9(a) and (c). We report that, while the true-model varies in amplitude (up to) ↑ ±3.5 per cent, the extra errors for absolute models can locally be very significant within the upper-mantle (↑ ±3 per cent, if k = 1; ↑ ± 4 per cent, if k = 2) and transition-zone (↑ ±2.5 per cent, if k = 1; ↑ ±3 per cent if k = 2), though they are weaker at the top of mid lower-mantle (↑ ±1 per cent, if k = 1; ↑ ±2 per cent if k = 2). Meanwhile, we report that the extra errors for differential models are significantly lower within the upper-mantle (↑ ±2 per cent, if k = {1; 2}), transition-zone (↑ ± 2 per cent, if k = {1; 2}), and at the top of mid lower-mantle (↑ ±1 per cent, if k = {1; 2}

Figure 7 .

 7 Figure 7. Case of low mislocation-bias (σ X = 1.4 s): comparison of the extra errors for absolute-and differential models (cf. eqs 24-25). (a) {A 1(λ A 1 ) -A 0 (λ A 0 )}, (b) {A 1 (λ A 1 ) -A 0 (λ A 1 )}, (c) {A 0 (λ A 1 ) -A 0 (λ A 0 )}, (d) {D 1 (λ D 1 ) -A 0 (λ A 0 )}, (e) {D 1 (λ D 1 ) -D 0 (λ D 1 )}, (f) {D 0 (λ D 1 ) -D 0 (λ D 0 )} and (g) {D 0 (λ D 0 ) -A 0 (λ A 0 )}.

Figure 8 .

 8 Figure 8. Case of high mislocation-bias (σ X = 2.8 s): comparison of the extra errors for absolute and differential models (cf. eqs 24-25). (a) {A 2(λ A 2 ) -A 0 (λ A 0 )}, (b) {A 2 (λ A 2 ) -A 0 (λ A 2 )}, (c) {A 0 (λ A 2 ) -A 0 (λ A 0 )}, (d) {D 2 (λ D 2 ) -A 0 (λ A 0 )}, (e) {D 2 (λ D 2 ) -D 0 (λ D 2 )}, (f) {D 0 (λ D 2 ) -D 0 (λ D 0 )} and (g) {D 0 (λ D 0 ) -A 0 (λ A 0 )}.

Figure 9 .

 9 Figure 9. Normalized histograms of the extra errors for absolute and differential models, computed over the whole mantle and shown for three depth ranges. Dark-grey corresponds to |A k (λ A k ) -A 0 (λ A 0 )|, and light-grey to |D k (λ D k ) -A 0 (λ A 0 )|, where k = {1; 2} denotes the two mislocation regimes, σ X = {1.4; 2.8} s, respectively. Two different true-model inputs are considered, Gaussian random field (GRF) and Geodynamical (Geody), as well as two regimes of noise-related errors, σ N = {0.7; 1.4} s. A logarithmic scale is used to facilitate the comparison of extra errors.

Figure 10 .

 10 Figure 10. Second true-model input, m true Geody (Geodynamical).

  γ [S 1 ; S 2 ](ω) = S 1 (ω)S * 2 (ω), (A1)where S(ω) = FT {s(t)} denotes the Fourier transform of signal s(t), and * denotes the complex conjugate. If one considers a rupture history m(t) giving rise to a source time function ṁ(t) in the far-field, with spectrum denoted as ṁ(ω), we have: and predicted travel times, and δt i = δt 3D i + δt N

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A C K N O W L E D G E M E N T S

The authors thank the SEISGLOB ANR 2011 Blanc SIMI 5-6-016-01 and the ERC (Advanced grant 226837). This research has benefited from stimulating discussions with O. Lengliné and A. Maggi. The authors thank X. Chen and two anonymous reviewers for helpful reviews that improved the original manuscript.


Fateh Saci
email: sacifateh@yahoo.fr

Boualem Aliouat
email: boualem.aliouat@unice.fr

Do strategic partnerships create value? "The empirical case of SBF 250 firms"

Strategic partnerships are theoretically now considered as real levers of value creation. This value creation takes different forms 1 . The objective of our research, after analyzing the results of empirical works focused on the acquisitions announcements impact on the stock market performance and also their actual value creation in the long-term, is to check if strategic partnerships create market value and the actual value. Our final results show that in the short term the announcement of a strategic partnership has a negative impact on performance as opposed to the announcement of a merger-acquisition, while other hand on the long-term, there is no positive impact (neutral impact) of strategic partnerships on financial performance.

We explain this result by the phenomenon of "creation of compensatory value" in the context of a strategic and financial plan

Keywords

Strategic partnerships, market value, compensatory value 1 This value creation takes different forms, essentially strategic value, substantial value, institutional value and financial value (including shareholders).

Introduction

Since the 1980s, regardless of their size or their business sectors, companies grow increasingly by acquisitions, mainly by strategic partnerships (PS) [START_REF] Paturel | Stratégie de croissance externe[END_REF]. These have multiplied to the point of looking like a fairly common form in strategic business options. This trend is explained by several reasons relating to the intentions of stakeholders concerned whether, as noted by some authors, to financial motivations of shareholders, substantial or institutional [START_REF] Aliouat | Vers un pilotage des alliances stratégiques par des méthodes perceptives de la valeur. 18 ième Conférence AIMS[END_REF].

(Jacquot and Koehl, 1998) have made the same goals and motivations in the case of cooperation in general. For both authors, the motivations are often provided by environmental explanations (increased competition, market globalization, technological change, internationalization of markets, etc..), economic interpretations (economic and organizational efficiency) or cultural interpretations and cognitive (socio-cultural factors, plans of interpretation of the actors, vision, ...).

Today, with the new economic situation and the multiplication of financial crises, external growth strategies including strategic partnerships2 and mergers and acquisitions3 have become two of the most appropriate ways to deal with any internal or external crisis but also to cope with intense competition, integrate a new market, reduce or minimize costs, maximize profits, a complete range, expand into new markets, acquire technology or know-how to obtain scarce resources, maintain certain resources, innovation, etc.. (Jacquot and Koehl, 1998). In short, preservation strategies, capture, creation or production of the current or new value according to [START_REF] Paturel | Pour une refonte du paradigme de la création de valeur[END_REF].

However, with financial globalization, firms are more focused on shareholder wealth, while the latter have experienced strong growth in liquidity and increase profits. This is why practitioners and researchers interested more in this area.

The search for the maximization of shareholder wealth which is central focus of this competition to the large size as specified [START_REF] Albouy | Théorie, applications et limites de la mesure de la création de valeur[END_REF] In his article, Theory, Applications and limitations of measuring value creation: "not very long ago, the main objective was racing for turnover, critical size, market share, even in annual net profit. (…).The creation of shareholder value would not be a fashion or a matter of communication as many subjects of management? ".

Literature review on strategic partnerships

The analysis of work on the subject of our research enabled us to notice a real dissonance between the results of previous work. Especially with those who argue for the creation of real financial value, and those who advocate for the perception of value.

Our research focuses on measuring the creation of financial value and shareholder value even though some authors as (Pesqueux, 1990), (Hoareau and Teller, 2001) and (Caby Hyrigoyen, 2001) consider that the accounting and financial information is based on utilitarian references that emphasize market values at the expense of existential values (partnership values).

But despite these criticisms on quantitative criteria of performance measurement, we must not forget that the survival of a company can only be a consequence of the financial results effectively measured, that is [START_REF] Gilbert | Comment évaluer la performance RH ? Question universelle, réponses contingentes[END_REF] affirmed.

So this is what motivated us to focus more on studies carried out on the measurement of value creation on quantitative criteria, but after examining them, we have noticed that there are few studies which were conducted on the actual financial performance of external growth strategy, including strategic alliances and also few comparative studies on these two types of acquisitions where the primary focus of our research which contributes to a better understanding of questions concerning the motivations to conclude strategic alliances and mergers and acquisitions, and to conduct a comparative study.

The results so far are controversial and uncertain, therefore, the interest in identifying the nature of the impact of strategic alliances and mergers and acquisitions on the financial performance where the second interest is to measure the impact strategic alliances and mergers and acquisitions on financial performance (reduced to the creation of financial value).

The choice of this thesis is not a coincidence, but rather a motivation to answer questions of [START_REF] Aliouat | Les stratégies de coopération industrielle[END_REF]. Aliouat, after analyzing the results of empirical studies on the impact of acquisitions announcements on market value where he has seen the impact was positive, made the following two questions:

-For a more or less distant horizon, would the market performance of acquisitions coincide with the actual performance? -Does the confirmation of hypotheses formulated and verified by the authors remain valid for the case of alliances?

Hence the formulation of the following problem:

What is the impact of a strategic partnership on the financial performance of the company?

The answer to this problem requires the response to the two following research questions:

-Does the announcement of a strategic partnership have an impact on the market valuation?

-Does the conclusion of a strategic partnership have an impact on the real financial value in the long term?

Theoretical framework

To answer our two research questions we absolutely must refer to some particular theoretical current, the theory of market efficiency, the agency theory, the theory of transaction costs and the theory of resources.

We have formulated 1 hypothesis on the impact of the announcement on the stock market valuation from the theory of market efficiency (Fama, 1965) and agency theory (Jensen and Ruback effect, 1983). And for H2 to H6 assumptions based on the theory of transaction costs [START_REF] Coase | nature of the firm[END_REF], Williamson, 1985) and the theory of resources (RBV) (Penrose, 1959); [START_REF] Wernerfelt | A Resource-based View of The Firm[END_REF] and [START_REF] Barney | Firm Resource and Sustainable Competitive Advantage[END_REF]. These two theoretical currents meet competitive strategy described by [START_REF] Batsch | Finance et stratégie[END_REF].

Previous empirical work

Given that there is little practically few research on the financial performance of strategic partnerships, our research builds on previous studies including strategic alliances and mergers and acquisitions that show positive and negative effects.

Regarding work on the market performance, in the case of M & A either in France or beyond, we notice a positive impact in contrast to strategic alliances. For work on actual financial performance, the results are rather mixed either for Strategic Alliances (SA) or Mergers and

Acquisitions (M & A).

The following tables list the main previous research:

Results

References on the market performance

Mergers and Acquisitions (M & A)

Positif impact

In France: [START_REF] Saci | La création de valeur des alliances stratégiques et des fusions-acquisitions : justification comparative par le modèle de mesure de la valeur financière « cas des sociétés du SBF 250[END_REF] ; [START_REF] Hubler | Alliances, acquisitions et valorisation boursière[END_REF] Anglo-xaxons works : (Dodd, 1980) ; (Travlos, 1987) ; (Huang and Walkling, 1987) ; (Doukas and Travlos, 1988); (Jennings and Mazzeo, 1991) ; (Markides and Oyon, 1998) Strategic Alliances (SA)

Negative impact or neutral

In France: [START_REF] Saci | La création de valeur des alliances stratégiques et des fusions-acquisitions : justification comparative par le modèle de mesure de la valeur financière « cas des sociétés du SBF 250[END_REF] ; [START_REF] Hubler | Alliances, acquisitions et valorisation boursière[END_REF] Anglo-xaxons works : (Mac Connel and Nontell, 1985) ; Finnerty and All, 1986) ; (Lee and Wyatt, 1990); (Koh and Venkatramen, 1991) ; (Reuer and Miller, 1997) ;

(
(Jacquot and Koehl, 1998) ; [START_REF] Das | Impact of Strategic Alliances on Firm Valuation[END_REF])

Results

References on the actual financial performance

Mergers and Acquisitions (M & A)

Mixed results [START_REF] Saci | La création de valeur des alliances stratégiques et des fusions-acquisitions : justification comparative par le modèle de mesure de la valeur financière « cas des sociétés du SBF 250[END_REF] ; (Paik, 2005); [START_REF] Camerlynck | Pre-Acquisition Profile of Privately Held Companies Involved in Take-Overs: An Empirical Study[END_REF]; [START_REF] Yook | The Measurement of Post-Acquisition Performance Using EVA[END_REF]; [START_REF] Park | Prior Performance Characteristics Of Related And Unrelated Acquirers[END_REF]; (Silhan and Howard, 1986) ; (Perdreau, 1998); (Harrison and al., 1991);

Strategic Alliances (SA)

Mixed results [START_REF] Saci | La création de valeur des alliances stratégiques et des fusions-acquisitions : justification comparative par le modèle de mesure de la valeur financière « cas des sociétés du SBF 250[END_REF] ; (Paik, 2005); [START_REF] Combs | Explaining Interfirm and Performance: Toward a Reconciliation of Predictions from The Resource-based View and Organizational Economics[END_REF] ;

(Lorenzoni and Lipparini, 1999) ; (Reuer and Miller, 1997) ; (Deeds and Hill, 1996) ; [START_REF] Park | The Effect of Strategic alliances on Performance; A Study of International Airline Industry[END_REF]; (Baum and al., 2000) ; (Park and Dong-Sung, 1997)

Research model, assumptions and definition of variables

Research model

We adopt the following model: Therefore, these findings led to the following hypothesis.

Assumptions or hypothesis

H1: There is a positive relationship between the announcement effect of a strategic partnership and market value.

H2: The increase in the margin on operating income has a positive impact on financial performance in a strategic partnership.

H3: The increase in turnover realized abroad has a positive impact on financial performance in a strategic partnership.

H4: The increase in annual sales (turnover) has a positive impact on financial performance in a strategic partnership.

H5: Increased variation sales abroad to total sales has a positive impact on financial performance in a strategic partnership.

H6: The increase in the number of employees has a negative impact on financial performance in a strategic partnership.

Definition of variables

The choice of variables is done referring to the work of some authors as [START_REF] Saci | La création de valeur des alliances stratégiques et des fusions-acquisitions : justification comparative par le modèle de mesure de la valeur financière « cas des sociétés du SBF 250[END_REF] and [START_REF] Hubler | Alliances, acquisitions et valorisation boursière[END_REF] to measure the market performance where they chose the RAM as dependent variable and the presence / absence as explanatory variable and the research work of [START_REF] Saci | La création de valeur des alliances stratégiques et des fusions-acquisitions : justification comparative par le modèle de mesure de la valeur financière « cas des sociétés du SBF 250[END_REF] who has chosen ROA, ROE, PBR as dependent variable and the operating profit margin on the sales realized abroad, the change in turnover realized abroad as compared to total turnover and number of employees as explanatory variables and the work of (Harrison and al., 1991) where they chose ROA as dependent variable and capital, debt, the intensity of research and development and general and administrative expenses of sale, as explanatory variable. We have also used our personal network and that of the laboratory. In this respect, we contacted several researchers and specialists.

Dependent variables Explanatory variables

We have established a sample of 48 strategic partnerships selected from the SBF 250 French companies (on Euronext Paris) and we used historical market and financial data between 1997 and 2012. For our study is made on a longitudinal period of seven (07) years, three (03) years before the strategic partnership, the year of the strategic partnership and three (03) years after the strategic partnership, (Cornett and Tehranian, 1992).

Descriptive statistics

Ranking of strategic partnerships (SP) according to current nomenclature of NYSE Euronext Paris: It is very important to analyze accounting data and financial ratios before to calculating their variation. This allows us to have an overview of the characteristics of our sample noting that ROA, ROE and PBR variables are in percent (%), variables, OPPRMAR, COSTGS, FSALES, SALES are in millions of euros (€) FSALES_SALES as coefficient and N is the number of employees.

Methodology and data analysis

To answer to our problem, our choice is based on epistemological positivism based on a quantitative approach by adopting a hypothetical-deductive approach (assumptions, tests).

To analyze the data, we used the Student 4 test for the study of events and tests on panel data estimation on STATA and RATS.

In our work, the study of the measure of value creation relates only to the extent of creating financial value. We then propose the results of the correlation study and those of the three regressions models developed for measuring the creation of real value. 4 The Student test was performed on the results calculated using the market model

In order to have a solid basis for our analysis, we referred to earlier work [START_REF] Saci | La création de valeur des alliances stratégiques et des fusions-acquisitions : justification comparative par le modèle de mesure de la valeur financière « cas des sociétés du SBF 250[END_REF]; [START_REF] Barber | Detecting Long-Run Abnormal Stock Returns: The Empirical Power and Specification of Test Statistics[END_REF] and [START_REF] Hubler | Alliances, acquisitions et valorisation boursière[END_REF] for measuring the market value creation and work of [START_REF] Saci | La création de valeur des alliances stratégiques et des fusions-acquisitions : justification comparative par le modèle de mesure de la valeur financière « cas des sociétés du SBF 250[END_REF]; [START_REF] Park | Prior Performance Characteristics Of Related And Unrelated Acquirers[END_REF]; [START_REF] Yook | The Measurement of Post-Acquisition Performance Using EVA[END_REF]; (Harrison and al., 1991); [START_REF] Camerlynck | Pre-Acquisition Profile of Privately Held Companies Involved in Take-Overs: An Empirical Study[END_REF], and the use of databases DATASTREAM,

Thomson One Banker and Diane to collect as much data about the companies in our sample.

Triangulation of methods allows the confirmation of the empirical contribution of our thesis.

Using the method of event study and regression method for estimating panel data, analyzing different data highlights this triangulation methods.

These different ways of triangulation, namely theoretical triangulation, triangulation of data sources and triangulation methods allow to increase the validity and reliability of the analysis [START_REF] Miles | Analyse des données qualitatives[END_REF]. They are intended to ensure a better quality of analysis by crossing data and studies.

Correlation tests

Table 3 shows the pairwise correlations of the variables in our study. These measures show the quality of our model and the absence of collinearity problems of independent variables. 5) show that the time series of annual returns have negative skewness for both variables ROA and ROE to explain. But we find a positive skewness for the dependent variable PBR.

For the explanatory variables, we observe a negative skewness for the explanatory variables, SALES, FSALES and N, except for OPPRMAG COSTGS and where we see positive skewness.

Concerning stationarity test of Dickey Fuller (ADF) on strategic partnerships, we find that all variables, either to explain or explanatory are stationary (S) for each variable in Table 5.

In summary, from the results of (Jarque Bera, 1980), we find mixed results. These findings will also be confirmed with the study of regression below.

Student tests

The possible significance of AR, AAR and ACAR according to tests performed and measured by standard tests (1%, 5% and 10%).

The objective of this test is to measure the impact of strategic alliances and effects of mergers and acquisitions on the market performance of French concerned companies. The announcements of strategic partnerships in general, the analysis of the AAR shows a significant and negative impact on the third and seventh day after the announcement (-0.197 and -0.080, respectively), but actually the reactions are positive, since the AAR increases.

Throughout the event window, we note that the average abnormal returns (AAR) down to -5 j = j = -1, with j = 0, a positive market reaction is noted (the passage of RA -0.57 to 0.45 between j = -1 and j = 0). From j = 0 to j = 7, we notice positive reactions in general. Then, we notice some positive reactions, sometimes negative.

If a negative impact on stock prices is associated with strategic partnerships, it seems that this is the form of partnership it himself that his announcement is the consequent of the market reaction.

AAR ACAR

Anticipating the stock market annones of strategic partnerships is indeed on seeing the results of our research. A significant negative impact is recorded days (03) days after the announcement, and average abnormal returns (AAR) increases.

Referring to the research of [START_REF] Saci | La création de valeur des alliances stratégiques et des fusions-acquisitions : justification comparative par le modèle de mesure de la valeur financière « cas des sociétés du SBF 250[END_REF] and [START_REF] Hubler | Alliances, acquisitions et valorisation boursière[END_REF] and the conclusion of our results, we can say that the information on the outcome of strategic partnerships managed gradually to large financial players part before the actual announcement to the public by the press. And therefore the information is already taken by financial actors to calculate stock prices and this is why we see just a single positive market reaction on the day (j = 3) (0.548%)

for AR contrary to the results of mergers and acquisitions, we find average abnormal returns are generally positive.

We have kept the advice of [START_REF] Hubler | Alliances, acquisitions et valorisation boursière[END_REF] on the enlargement of the event window to see if average cumulative abnormal returns continue to be degraded, and noting that [START_REF] Hubler | Alliances, acquisitions et valorisation boursière[END_REF] chose an event window [j -10, j +10].

For this, we chose to expand our event window to j+20 after the announcement of the event to see the evolution of the average cumulative abnormal returns. We find that the average cumulative abnormal returns (ACAR) are read in less stable.

After analyzing our results by sector of activity, we find positive reactions to Sector 1 and 7, respectively automotive and tourism sector.

Regression tests

3.

Analysis of the results (Student's t test and estimation of panel data)

In the short term

Negative impact of a strategic partnership on market performance. We note that our results on the market performance of strategic partnerships are consistent with those of [START_REF] Saci | La création de valeur des alliances stratégiques et des fusions-acquisitions : justification comparative par le modèle de mesure de la valeur financière « cas des sociétés du SBF 250[END_REF] and (Hubled and Meschi, 2000) on the stock market performance of strategic alliances.

In the long term

Strategic partnerships (SP) have no impact on creating real financial value of the company.

Compared to our assumptions, we reject the hypothesis suggesting the existence of a positive relationship between the conclusion of the Strategic partnership and the creation of financial value, because we have only H2 is validated.

Discussion

Discussion of results on the creation of market value

Our results, in which market quotations are used to measure short-term performance, suggest throughout that the financial markets seem to be indifferent to announcements of strategic partnerships. However, in large part, mergers and acquisitions are seen as good news by the financial actors [START_REF] Saci | La création de valeur des alliances stratégiques et des fusions-acquisitions : justification comparative par le modèle de mesure de la valeur financière « cas des sociétés du SBF 250[END_REF] and [START_REF] Hubler | Alliances, acquisitions et valorisation boursière[END_REF].

Strategic partnerships are not seen as good news in general if we refer to the logic of [START_REF] Hubler | Alliances, acquisitions et valorisation boursière[END_REF].

However, when we analyzed our results by sector, we found that for the automotive sector

(1), the impact of the announcements on the SP market value is positive and significant. The same results are observed, but with less importance in the aerospace and transportation sectors

(3) tourism ( 7) and the distribution and industry sector (8).

Which is difficult to explain without further details. (Koh and Venkataraman, 1991) have observed weak positive market reactions (but not statistically significant) to the announcement of strategic alliances, but provided no explanation for this observation, and it may also be due to the business. Similarly, (Das and Seng, 1988) in their work "Impact of Strategic Alliances on firm valuation", have found a positive impact on the technological alliances ads, while marketing alliances are seen as neither good, nor bad.

Discussion of results on the creation of real value

The comparative reading of the theoretical literature and the empirical one shows that strategic partnerships are practices that can generate both positive and negative effects. This observation led us to assume that the combined effect of the alignment of interests and inefficiencies own strategic partnerships is likely to create a net neutral effect on creating financial value of the company.

Through empirical studies cited below concerning external growth strategies, including strategic alliances and mergers and acquisitions, we find notice in one hand, the impact of strategic partnerships on creating financial value is neutral on the other hand, the results of [START_REF] Saci | La création de valeur des alliances stratégiques et des fusions-acquisitions : justification comparative par le modèle de mesure de la valeur financière « cas des sociétés du SBF 250[END_REF] and that of [START_REF] Yook | The Measurement of Post-Acquisition Performance Using EVA[END_REF], who has worked on the development of the EVA after the acquisitions operations indicate that the improved operating performance is ruined by the costs capital of large premiums paid to the target company, and therefore, acquisitions do not create real economic gains for shareholders of the acquiring company.

We stipulate that strategic partnerships create financial value, but it is destroyed in part by the costs incurred by them as well as other negative effects [START_REF] Yook | The Measurement of Post-Acquisition Performance Using EVA[END_REF].

In conclusion, the absence of the impact of the strategic partnership (SP) on financial performance can be explained as follows:

-The impact of strategic partnership (SP) may be combined to give a total net neutral effect on performance;

-The negative effects absorb the potential gains from strategic partnership (SP) (creating compensatory value);

-As we integrated the different costs, we can explain this neutrality of the performance by what may call the creation of compensatory value: companies anticipate a decrease in their income (due to the loss of a contract for example) will result in lower revenues or a possible new charge and to compensate for these losses, the companies are for strategic alliances and mergers and acquisitions (financial and strategic intent);

-Intentions to use strategic partnerships (SP) are ultimately intentions other than financial intentions, as our results show that the creation of financial value is minimal, or even neutral (financial intent minima).

Conclusion

The analysis of the problem of measuring the financial value creation through strategic partnerships (SP) remains a field of study rarely treated in a French context, including the extent of creating long-term value. This finding is one of the main interests that motivated this research whose objective was to contribute to a better understanding of the financial value creation of strategic partnerships (PS) of French listed companies.

The contributions of our research: On the theoretical contributions. We have conducted a study on the actual performance of strategic partnerships (PS); we proposed a research model with new variables and finally a theoretical framework with four streams. In terms of methodological contributions: We conducted a study on two different periods and with an econometric approach based on two types of tests.

Search terms: sample and limited period of study; unavailability of details of the financial statements and the absence of moderating variables.

Financial

  Performance = creation of financial value = constant + variable Strategic Partnership (PS) + control variables Hence the following linear regression for the Strategic Partnership (PS): PF it = ß 0 + ß 1 *APSP + ß 2 * OpPrMargin + ß 3 * CostGS + ß 4 * Fsales + ß 5 * Sales + ß 6 Fsales_Sales + ß 7 NPART + ß 8 N + ß 9 S +ε.

  of the sample, data selection and descriptive statisticsIn the case of our study, we have used accounting and financial data (secondary data), including Thomson One Banker, Datastream, Diane; reference documents of companies (published and updated on the website of the Autorité des marchés financiers (AMF)); the companies' annual reports (available on their website); economic and trade press (including "les Echos" years between 1997 and 2011, Boursier.com, Capital.fr, Usine Nouvelle, Expansion, Le Figaro, the Point.fr ...), websites (via Google).

Figure 1 :Figure 2 :

 12 Figure 1: AAC 5 and ACAR 6 on ads strategic partnerships (PS)

Table 1 :

 1 Summary of belonging PS nomenclature according to NYSE Euronext Paris

		(Source: the author)
	Type of market	Number of strategic partnerships (SP)	%
	CAC 40	24	50%
	CAC Next 20	6	13%
	CAC Mid 100	12	25%
	CAC Small 90	6	13%
	TOTAL	48	100%

Table 2 :

 2 Number of PS operations by business segment (sectors)

	Code Sector		Number PS	%	Amount turnover	%
						in M €
	1	Automobile		2	4%	483,403163	2%
	2	Pharmaceutical	and	1	2%	2332,62614	8%
	3	chemistry Aerospace end transport	5	10% 1560,18239	5%
		Telecom,	Internet,
	4	Electronics, IT and services	17	35% 5882,51112	19%
	5	Banking and insurance	6	13% 16345,2892	53%
	6	Energy and construction	3	6%	2446,81391	8%
	7	Tourism		3	6%	976,602792	3%
	8	Distribution et Industry	11	23% 959,124617	3%
		TOTAL		48	100% 30986,5533	100%

Table 3 :

 3 Correlations

		ROA	ROE	PBR	Opprmar	Costgs	Fsales	Sales	FSales-	N
	ROA	1.0000							sales
	ROE	0.6163	1.0000
	PBR	0.3094	0.2380	1.0000
	Opprmar	0.6514	0.3733	0.1734	1.0000
	Costgs	0.1053	0.1050	0.0419	0.0001	1.0000
	FSales	0.2443	0.2908	0.1790	0.1966	0.3160	1.0000
	Sales	0.2710	0.3044	0.1790	0.2635	0.3412	0.7607	1.0000
	FSales_sal	-0.0802 -0.0302 -0.0316	-0.0854	-0.0033 0.0096 -0.0057	1.0000
	es N	0.1396	0.2097	0.1588	0.0937	0.3680	0.4955	0.6335	0.0050	1.000
										0

Table 4 :

 4 Study of normality and stationarity on market performanceIn summary, from the results of Jarque Bera (1980), we find a negative impact of the announcement of a strategic partnership. These results will be logically confirmed with the study of the Student test below.

	Our
	result
	s,

varia bles for measuring the creation of market value, reported in the above table

(Table 4

) show that the time series of abnormal returns have a negative skewness in the case of strategic partnerships. This result proves that the lower average yields are more frequent. In our case the AR of SP has a kurtosis equal to 36.890 and Skewness equal to 2149. But we have the same result, a negative skewness CAR. Unlike AAR and the ACAR that have a positive skewness.

Table 5 :

 5 Study of normality and stationarity on the actual performance

	Variables	Statistics	Strategic Partnership
	ROA	Mean	4.301
		Std. dev.	4.773
		Skewness	-0.228
		Kurtosis	4.082
		JB	236.241
		ADF	S
	ROE	Mean	9.698
		Std. dev.	23.949
		Skewness	-7.337
		Kurtosis	78.302
		JB	88852.359
		ADF	S
	PBR	Mean	1.449
		Std. dev.	1.703
		Skewness	6.083
		Kurtosis	54.980
		JB	44392.262
		ADF	S
	OpPrMar	Mean	8.743
		Std. dev.	6.296
		Skewness	0.328
		Kurtosis	1.213
		JB	26.678
		ADF	S
	COSTGS	Mean	5.756
		Std. dev.	32.265
		Skewness	12.014
		Kurtosis	184.016
		JB	417578.459
		ADF	S

Table 6 :

 6 AR, CAR, AAR, and ACAR on ads strategic partnerships (PS)

	Days (j) AR (%)	CAR (%)	TEST (AR)	T	AAR (%)	TEST (AAR)	T	ACAR (%)	TEST (ACAR)	T
	-5	.0601931 .0582295 .2850749		.116459 .1805525		.1144954 .2672494
	-4	-.0616915 .0182558 .1383906		.0547675 -.1630754 .1327513 .2594624
	-3	-.7215893 -.1667054 -1.132249 -.6668218 -1.796621 -.0339542 -.0581251
	-2	.0580746 -.1217494 -1.070238 -.6087472 .1730509		-.1557036 -.2419179
	-1	-.5786754 -.1979038 -1.705535 -1.187423 -1.496962 -.3536074 -.4933199
	0	.452451 -.104996 -.8965339 -.7349716 1.130403		-.4586033 -.5714083
	1	-.3353783 -.1337938 -1.228098 -1.07035 -1.08328		-.5923971 -.6694549
	2	.3420577 -.0809214 -.7899211 -.7282922 .7950878		-.6733185 -.7045912
	3	.5487469 -.0179545 -.1885502 -.1795453 1.653899* -.691273 -.6755386
	4	.0425398 -.0124551 -.1410752 -.1370055 .1331324		-.703728 -.647401
	5	-.5374336 -.0562033 -.6118272 -.6744391 -1.633564 -.7599313 -.6578853
	6	.0322036 -.0494027 -.5687747 -.6422355 .1150106		-.809334 -.6641843
	7	.5612679 -.0057834 -.0683457 -.0809677 1.992271* -.8151174 -.6375855
	8	-.2949542 -.0250615 -.2768484 -.3759218 -.3592318 -.8401788 -.6274576
	9	.1427862 -.014571 -.1638355 -.2331356 .4740328		-.8547497 -.6115652
	10	.1502961 -.0048729 -.0542289 -.0828395 .6455482		-.8596226 -.5887003
	11	.3332254 .0139103 .1705383		.2503858 .9535367		-.8457122 -.5565771
	12	.1545414 .021312 .2769374		.4049273 .6373369		-.8244003 -.5229534
	13	-.248061 .0078433 .1002955		.1568663 -.8954688 -.816557 -.4984786
	14	-.1423991 .0006889 .0090371		.0144672 -.3505415 -.815868 -.4807289
	15	-.0475303 -.0015029 -.0193567 -.0330629 -.1689149 -.8173708 -.4648626
	16	.0823157 .0021414 .026367		.0492528 .2490563		-.8152294 -.4481397
	17	.3844216 .0180698 .2466611		.4336744 .9874286		-.7971596 -.4251653
	18	.3829418 .0326646 .4710311		.8166162 1.270825		-.7644949 -.3971072
	19	.1519085 .0372509 .5319845		.9685246 .4293601		-.7272439 -.3674848
	20	-.4506675 .0191799 .266479		.517857 -1.326625 -.708064 -.347745

Table 7 :

 7 Results of regressions PS (before the strategic partnership)

The success of a partnership based on respect for different steps[START_REF] Wahyuni | Managing international strategic alliance relationships[END_REF]

Despite the crisis of 2007, the same year the number of M & A has peaked and it is from these data that one can reasonably conclude that the F & A may be responses to address a crisis or rather an opportunity intended to benefit fragile business targets. According to final figures provider Thomson Financial data published Thursday, January 3, 2008.

In case in which the company enters into a strategic partnership, reading linear regressions before and after the strategic partnership shows that there are some effects more or less significant after the partnership by making a comparison between the two tables of regressions before and after the strategic partnership.

We find for the dependent variable ROA, a significant positive impact for OpPrMargin, CostGS and Sales with significance levels respectively 1%, 5% and 10%.

Another significant positive impact for the predictor variable OpPrMargin to explain ROE (coefficient = 2.350) for a significance level of 1.

Finally, for the dependent variable PBR, we are not seeing any impact of the explanatory variables.

Research perspectives:

-A study with other variables to explain and more explanatory variables (out, R & D costs, longevity, patents, etc..) to refine the analysis of the financial impact;

-A two-dimensional study taking into account two levels of analysis within a single empirical study (complete our study with interviews with managers of the companies in our sample);

-A study to ascertain possible causal relationship between a strategic partnership (SP)

and the creation of financial value realized (effect on sequence comparisons).