R G Novikov
email: novikov@math.univ-nantes.fr

The ∂-approach to monochromatic inverse scattering in three dimensions

We discuss a method for monochromatic inverse scattering in three dimensions of [R. Novikov 2005] and implemented numerically in [Alekseenko, Burov, Rumyantseva 2007]. This method is obtained as a development of the ∂-approach to inverse scattering at fixed energy in dimension d ≥ 3 of [Beals, Coifman 1985] and [Henkin, R.Novikov 1987] and involves, in particular, some results of [Faddeev 1965[Faddeev , 1974] ] and some ideas of the soliton theory (in particular, some ideas going back to [Manakov 1976] and [Dubrovin, Krichever, S.Novikov 1976]). Our studies go back also, in particular, to [Regge 1959].

Introduction Consider the Schrödinger equation

-∆ψ + v(x)ψ = Eψ, x ∈ R d , d ≥ 2, E > 0, (1.1)
where v is a sufficiently regular function on R d with sufficient decay at infinity.

(1.2)

For equation (1.1) we consider the scattering amplitude f (k, l), where (k, l) ∈ M E ,

M E = {k, l ∈ R d : k 2 = l 2 = E}, E > 0. (1.3)
For definitions of the scattering amplitude see formula (1.4) below and, for example, [F3], [FM]. The scattering amplitude f arises, in particular, as a coefficient with scattered spherical wave e i|k||x| /|x| (d-1)/2 in the asymptotics of the wave solutions ψ + (x, k) describing scattering of incident plan waves e ikx for equation (1.1):

ψ + (x, k) = e ikx + c(d, |k|) e i|k||x| |x| (d-1)/2 f ¡ k, |k| x |x| ¢ + o ¡ 1 |x| (d-1)/2 ¢ as |x| → +∞, (1.4)
where k ∈ R d , k 2 = E, c(d, |k|) = -πi(-2πi) (d-1)/2 |k| (d-3)/2 . Given v, to determine f one can use, in particular, the following integral equation

f (k, l) = v(k -l) - Z R d v(m -l)f (k, m)dm m 2 -k 2 -i0 , (1.5) where v(p) = ¡ 1 2π ¢ d Z R d
e ipx v(x)dx, p ∈ R d .

(1.6)

In addition, (1.5) is an equation for f (k, •) on R d for each fixed k ∈ R d , k 2 = E, and the scattering amplitude arises as the restriction f ¯ME . We consider the following monochromatic inverse scattering problem for equation (1.1):

Problem 1.1 Given f on M E at fixed E > 0, find v on R d (at least approximately, but sufficiently stably for numerical implementations).

Note that the Schrödinger equation (1.1) at fixed positive energy can be considered also as the acoustic equation at fixed frequency (see, for example, Section 5.2 of [HN] and Section 4 of the present article).

Therefore, Problem 1.1 is also a basic problem of the monochromatic ultrasonic tomography. Actually, the creation of effective reconstruction methods for inverse scattering in multidimensions (and especially in three dimensions) was formulated as a very important problem many times in the mathematical literature, see, for example, [Gel], [F3], [Gro]. In particular, as it is mentioned in [Gro]: "For example, an efficient inverse scattering algorithm would revolutionize medical diagnostic, making ultrasonic devices at least as efficient as current X-ray analysis". The works [START_REF] Novikov | The ∂-approach to approximate inverse scattering at fixed energy in three dimensions[END_REF], [START_REF] Alexeenko | Solution of three-dimensional acoustical inverse scattering problem II. Modified Novikov algorithm[END_REF] can be considered as recent steps to this ojective.

In [START_REF] Novikov | The ∂-approach to approximate inverse scattering at fixed energy in three dimensions[END_REF] we assume that v ∈ W n,1 s (R d ) for some n ∈ N, n > d -2, and some s > 0, (1.7)

where

W n,1 s (R d ) = {u : Λ s ∂ J u ∈ L 1 (R d ) for |J| ≤ n}, (1.8a)
where

J ∈ (N ∪ 0) d , |J| = d X i=1 J i , ∂ J u(x) = ∂ |J| u(x) ∂x J 1 1 . . . ∂x J d d , Λ s w(x) = (1 + |x| 2 ) s/2 w(x), x ∈ R d . Suppose, first, that kvk n,1 s = max |J|≤n kΛ s ∂ J vk L 1 (R d ) (1.8b)
is so small for fixed n, s, d and E 0 > 0 that the following well-known Born approximation

f (k, l) ≈ v(k -l), (k, l) ∈ M E , E ≥ E 0 , (1.9)
where f is the scattering amplitude for equation (1.1) and v is the Fourier transform of v (see (1.5),(1.6)), is completely satisfactory. Then Problem 1.1 (for fixed d ≥ 2 and fixed

E ≥ E 0 ) is reduced to finding v from v on B 2 √ E , where B r = {p ∈ R d : |p| < r}, r > 0.
(1.10)

This linearized monochromatic inverse scattering problem can be solved by the formula

v(x) = v lin appr (x, E) + v lin err (x, E), v lin appr (x, E) = Z B 2 √ E e -ipx v(p)dp, v lin err (x, E) = Z R d \B 2 √ E e -ipx v(p)dp, (1.11) x ∈ R d , E ≥ E 0 . In addition, if v ∈ W n,1 0 (R d
) for some n > d and kvk n,1 0 ≤ C (where we use definitions (1.8a), (1.8b)), then

|v(p)| ≤ c 1 (n, d)C(1 + |p|) -n , p ∈ R d , (1.12a)
and, therefore,

|v lin err (x, E)| ≤ c 2 (n, d)CE -(n-d)/2 , x ∈ R d , E ≥ E 0 , (1.12b)
where c 1 (n, d), c 2 (n, d) are some positive constants. Thus, in the Born approximation (1.9) (that is in the linear approximation near zero potential) we have that:

(1) f on M E stably determines v lin appr (x, E) of (1.11) and (2) the error

v lin err (x, E) = v(x) -v lin appr (x, E) = O(E -(n-d)/2 ) in the uniform norm as E → +∞ for n-times smooth v in the sense (1.7), n > d. (In particular, v lin err = O(E -∞ ) in the uniform norm as E → +∞ for v of the Schwartz class on R d .)
For general nonlinearized case for d = 3 analogs of (1.9), (1.11), (1.12b) are given in [START_REF] Novikov | The ∂-approach to approximate inverse scattering at fixed energy in three dimensions[END_REF] (and implemented numerically in [START_REF] Alexeenko | Solution of three-dimensional acoustical inverse scattering problem II. Modified Novikov algorithm[END_REF]). These results are presented below in Sections 2, 3 and 4. However, before this some additional remarks may be in order.

Remark 1.1. One can see that v lin appr (x, E) at fixed E > 0 is not a precise reconstruction of v, in general. On the other hand, if v satisfies (1.7) and, in addition, is compactly supported or exponentially decaying at infinity, then v on B 2 √ E uniquely determines v on R d \B 2 √ E (at fixed E > 0) by an analytic continuation and, therefore, in the Born approximation (1.9) (for d ≥ 2) f on M E (at fixed E ≥ E 0 ) uniquely determines v on R d . However, in contrast with v lin appr the latter determination is not sufficiently stable for direct numerical implementation.

Remark 1.2. The works [START_REF] Novikov | Reconstruction of a two-dimensional Schrödinger operator from the scattering amplitude at fixed energy[END_REF], [START_REF] Novikov | Multidimensional inverse spectral problem for the equation -∆ψ + (v(x) -Eu(x))ψ = 0, Funkt. Anal. i Pril[END_REF], [START_REF] Novikov | The inverse scattering problem on a fixed energy level for the twodimensional Schrödinger operator[END_REF], [START_REF] Novikov | The inverse scattering problem at fixed energy for the three-dimensional Schrödinger equation with an exponentially decreasing potential[END_REF] give, in particular, uniqueness theorems and precise reconstructions for the problem of finding v on R d from f on M E (at fixed E > 0 for d ≥ 2) for the general nonlinearized case. These results are global for d ≥ 3 (and assume that v is sufficiently small in comparison with fixed E > 0 for d = 2). However, these reconstructions of [START_REF] Novikov | Reconstruction of a two-dimensional Schrödinger operator from the scattering amplitude at fixed energy[END_REF], [START_REF] Novikov | Multidimensional inverse spectral problem for the equation -∆ψ + (v(x) -Eu(x))ψ = 0, Funkt. Anal. i Pril[END_REF], [START_REF] Novikov | The inverse scattering problem on a fixed energy level for the twodimensional Schrödinger operator[END_REF], [START_REF] Novikov | The inverse scattering problem at fixed energy for the three-dimensional Schrödinger equation with an exponentially decreasing potential[END_REF] are not sufficiently stable for direct numerical implementation (because of the nature of Problem 1.1 explained already for the linearized case (1.9) in Remark 1.1). Actually, any precise reconstruction of v, where v ∈ C n (R d ) and supp v ∈ B r for fixed n ∈ N and r > 0, from f on M E at fixed E > 0 is exponentially unstable (see [Mand]). In the present text (as well as in [START_REF] Novikov | Rapidly converging approximation in inverse quantum scattering in dimension 2[END_REF], [START_REF] Novikov | Approximate inverse quantum scattering at fixed energy in dimension 2[END_REF], [START_REF] Novikov | The ∂-approach to approximate inverse scattering at fixed energy in three dimensions[END_REF]) we do not discuss such reconstructions in detail.

Remark 1.3. In [HN] it is shown, in particular, that if Λ s v ∈ L ∞ (R d , R) for some s > d, where Λ is the weight of (1.8), and if d ≥ 2, then for any fixed E and δ, where 0 < δ < E, the scattering amplitude f on ∪ λ∈[E-δ,E+δ] M λ uniquely determines the Fourier transform v on B 2 √ E . However, unfortunately, this determination of [HN] involves an analytical continuation and, therefore, is not sufficiently stable for direct numerical implementation.

Remark 1.4. In [Ch] an efficient numerical algorithm for the reconstruction from multi-frequency scattering data was proposed in two dimensions, but [Ch] gives no rigorous mathematical theorem.

Remark 1.5. On the other hand (in comparison with results mentioned in Remarks 1.2, 1.3, 1.4) in [START_REF] Novikov | Rapidly converging approximation in inverse quantum scattering in dimension 2[END_REF], [START_REF] Novikov | Approximate inverse quantum scattering at fixed energy in dimension 2[END_REF] we succeeded to give stable approximate solutions of nonlinearized Problem 1.1 for d = 2 and v satisfying (1.7), n > d = 2, with the same decay rate of the error terms for E → +∞ as in the linearized case (1.9), (1.11), (1.12b) (or, more precisely, with the error terms decaying as O(E -(n-2)/2 ) in the uniform norm as E → +∞). In [START_REF] Novikov | Rapidly converging approximation in inverse quantum scattering in dimension 2[END_REF], [START_REF] Novikov | Approximate inverse quantum scattering at fixed energy in dimension 2[END_REF] we proceed from the fixed-energy inverse scattering reconstruction procedure developed in [START_REF] Novikov | Construction of a two-dimensional Schrödinger operator with a given scattering amplitude at fixed energy[END_REF], [GM], [START_REF] Novikov | Reconstruction of a two-dimensional Schrödinger operator from the scattering amplitude at fixed energy[END_REF], [START_REF] Novikov | The inverse scattering problem on a fixed energy level for the twodimensional Schrödinger operator[END_REF] for d = 2. In turn, the works [START_REF] Novikov | Construction of a two-dimensional Schrödinger operator with a given scattering amplitude at fixed energy[END_REF], [GM], [START_REF] Novikov | Reconstruction of a two-dimensional Schrödinger operator from the scattering amplitude at fixed energy[END_REF], [START_REF] Novikov | The inverse scattering problem on a fixed energy level for the twodimensional Schrödinger operator[END_REF] are based on some ideas and approaches of the solition theory (in particular, on some ideas and approaches going back to [M1], [DKN], [M2], [ABF], [START_REF] Novikov | Finite-zone, two-dimensional, potential Schrödinger operators. Explicit formulas and evolution equations[END_REF], [START_REF] Novikov | Finite-zone, two-dimensional Schrödinger operators. Potential operators[END_REF]) and on some results of [F1], [F2], [F3], [GN]. The work [START_REF] Novikov | Rapidly converging approximation in inverse quantum scattering in dimension 2[END_REF] was stimulated also by the "onedimensional" works [MNPSF] and [HeNo]. The reconstruction procedure of [START_REF] Novikov | Construction of a two-dimensional Schrödinger operator with a given scattering amplitude at fixed energy[END_REF], [GM], [START_REF] Novikov | Reconstruction of a two-dimensional Schrödinger operator from the scattering amplitude at fixed energy[END_REF], [START_REF] Novikov | The inverse scattering problem on a fixed energy level for the twodimensional Schrödinger operator[END_REF] was implemented numerically in [BMRSVZ], [BBMRS], [BMR].

Reconstruction of [No8]

2.1. Preliminary presentation. In [START_REF] Novikov | The ∂-approach to approximate inverse scattering at fixed energy in three dimensions[END_REF] for v satisfying (1.7), n > d, for general nonlinearized case for d = 3 we succeeded, in particular, to give a stable reconstruction

f on M E stable reconstruction -------------→ v ± appr (•, τ, E) on R 3 (2.1) such that v(x) -v ± appr (x, τ, E) = O(E -(n-3)/2 ln E) in the uniform form as E → +∞, x ∈ R 3 , (2.2)
where ± and τ are fixed parameters. In addition, 0 < τ < δ(s, n, C) for appropriate δ (see formula (2.8) and conditions (2.6) discussed below), where C ≥ kvk n,1 s . That is in [START_REF] Novikov | The ∂-approach to approximate inverse scattering at fixed energy in three dimensions[END_REF] we succeded, in particular, to give a stable approximate solution of nonlinearized Problem 1.1 for d = 3 and v satisfying (1.7), n > d = 3 with the error term v(•)v ± appr (•, τ, E) decaying with almost the same rate for E → +∞ as in the linearized case (1.9), (1.11), (1.12b).

Reconstruction (2.1) (with estimate (2.2)) is based, in turn, on the following reconstruction of [START_REF] Novikov | The ∂-approach to approximate inverse scattering at fixed energy in three dimensions[END_REF] for v satisfying (1.7), n > d = 3,

f on M E stable reconstruction -------------→ v± appr (•, τ, E) on B 2τ √ E (2.3) such that |v(p) -v± appr (p, τ, E)| = O(E -(n-µ 0 )/2 ) (1 + |p|) µ 0 , as E → +∞, p ∈ B 2τ √ E , µ 0 ∈ [2, n[, (2.4)
where v is (the Fourier transform of v) defined by (1.6), O(E -(n-µ 0 )/2 ) is independent of p, and µ 0 , ±, τ are fixed parameters. In addition,

0 < τ < δ 1 (s, n, µ 0 , C), C ≥ kvk n,1 s , (2.5)
for appropriate δ 1 , where, in particular,

0 < δ 1 < 1, (2.6a) δ 1 → 0 as C → +∞.
(2.6b)

Using (2.3)-(2.6a) one can see that in (2.
3) we even do not try to reconstruct v on R 3 \B 2 √ E . In addition, we think that condition (2.6b) is not necessary and is related mainly with technical details of our proof of [START_REF] Novikov | The ∂-approach to approximate inverse scattering at fixed energy in three dimensions[END_REF].

In terms of the approximate reconstruction v± appr of (2.3), (2.4) the approximate reconstruction v ± appr of (2.1), (2.2) is given by

v ± appr (x, τ, E) = Z B 2 √ E e -ipx v± appr (p, τ, E)dp, x ∈ R 3 , (2.7)
where we use (2.3), (2.4) for µ 0 = 3. In addition,

δ(s, n, C) = δ 1 (s, n, 3, C), (2.8)
where δ was mentioned in connection with (2.1), (2.2). A stability estimate for (2.3) is given by formula (3.7) below. More detailed version of (2.4) is given by formula (3.6) below.

Note that before [START_REF] Novikov | Rapidly converging approximation in inverse quantum scattering in dimension 2[END_REF], [START_REF] Novikov | Approximate inverse quantum scattering at fixed energy in dimension 2[END_REF] for d = 2 and [START_REF] Novikov | The ∂-approach to approximate inverse scattering at fixed energy in three dimensions[END_REF] for d = 3, even for real v of the Schwartz class on R d , d ≥ 2, no result was given, in general, in the literature on finding v on R d from f on M E with the error term decaying more rapidly than O(E -1/2 ) in the uniform norm as E → +∞ (see related discussion given in [START_REF] Novikov | Rapidly converging approximation in inverse quantum scattering in dimension 2[END_REF]).

Note that inverse scattering at fixed energy for the Schrödinger equation (1.1) with spherically symmetric potential v in three dimensions was studied in many works, see [R], [Ne], [CS] and references therein. Nevertheless, the results (of [START_REF] Novikov | The ∂-approach to approximate inverse scattering at fixed energy in three dimensions[END_REF]) consisting in reconstruction (2.1)-(2.4) are new even for spherically symmetric potentials.

Note that the method of [START_REF] Novikov | The ∂-approach to approximate inverse scattering at fixed energy in three dimensions[END_REF] is obtained as a development of the ∂-approach to inverse scattering at fixed energy in dimension d ≥ 3 of [BC] and [HN] and involves, in particular, some results of [F1], [F2], [F3] and some ideas of the "two-dimensional" works mentioned in Remark 1.5. In addition, there is an interesting similarity between [R], on one hand, and [BC], [HN], [START_REF] Novikov | The ∂-approach to approximate inverse scattering at fixed energy in three dimensions[END_REF], on the other hand, consisting in an essential use of the complex analysis.

Background results. Reconstruction (2.3) is based on properties of the Faddeev function H(k, p),

where k ∈ C 3 \R 3 , p ∈ R 3 . For definitions of H see [F3], [HN]. Given v, to determine H one can use, in particular, the following integral equation

H(k, p) = v(p) - Z R 3 v(p + ξ)H(k, -ξ)dξ ξ 2 + 2kξ , k ∈ C 3 \R 3 , p ∈ R 3 , (2.9)
where v is (the Fourier transform of v) defined by (1.6). Actually, (2.9) is an equation for

H(k, •) on R 3 for each fixed k ∈ C 3 \R 3 . Reconstruction (2.3) is based, in particular, on properties of H on Ω E \Re Ω E ,
where

Ω E = {k ∈ C 3 , p ∈ R 3 : p 2 = 2kp, k 2 = E},
Re

Ω E = {k ∈ R 3 , p ∈ R 3 : p 2 = 2kp, k 2 = E}, (2.10)
and on properties of

H γ (k, p) = H(k + i0γ, p), (k, p) ∈ Re Ω E , γ ∈ S 2 ; ( 2 .11)
see formulas (2.20)-(2.29) given below.

To deal with f and H of (1.5) and (2.9) (in the framework of the aforementioned results) we consider, in particular, the functional spaces

C α,µ (R 3 ), H α,µ (R 3 ), C α (M E ): C α,µ (R 3 ) = {u ∈ C(R 3 ) : kuk α,µ < +∞}, α ∈]0, 1[, µ ∈ R, (2.12)
where kuk α,µ = kΛ µ uk α , (2.13a)

Λ µ u(p) = (1 + |p| 2 ) µ/2 u(p), p ∈ R 3 , (2.13b) kwk α = sup p,ξ∈R 3 , |ξ|≤1 (|w(p)| + |ξ| -α |w(p + ξ) -w(p)|); (2.13c) H α,µ is the closure of C ∞ 0 (R 3 ) in k • k α,µ , α ∈]0, 1[, µ ∈ R, (2.14)
where C ∞ 0 (R 3 ) is the space of infinitely smooth functions on R 3 with compact support;

C α (M E ) = {u ∈ C(M E ) : kuk C α (M E ),0 < +∞}, α ∈ [0, 1[, E > 0, (2.15) where kuk C 0 (M E ),µ = kuk C(M E ),µ = sup (k,l)∈M E (1 + |k -l| 2 ) µ/2 |u(k, l)|, µ ≥ 0, (2.16a) kuk C α (M E ),µ = max(kuk C(M E ),µ , kuk 0 C α (M E ),µ ), kuk 0 C α (M E ),µ = sup (k,l),(k 0 ,l 0 )∈M E |k-k 0 |≤1,|l-l 0 |≤1 (1 + |k -l| 2 ) µ/2 (|k -k 0 | α + |l -l 0 | α ) -1 × |u(k, l) -u(k 0 , l 0 )| for α ∈]0, 1[, µ ≥ 0. (2.16b) We use that if v satisfies (1.7), d = 3, then v ∈ H α,n (R 3 ) for α ∈ ½ ]0, s] for s < 1 ]0, 1[ for s ≥ 1 , kvk α,n ≤ Const 0 (s, α, n)kvk n,1 s ,
(2.17

)
where v is the Fourier transform of v. Further, for obtaining (2.3), (2.4) we use, in particular, that if v ∈ H α,n (R 3 ) for some α ∈]0, 1[ and some real µ > 1 ( 2 .18)

and

kvk α,µ ≤ D < E σ/2 Const 1 (α, µ, σ) for some σ ∈]0, min (1, µ -1)[ and some E ≥ 1, (2.19)
then (see [START_REF] Novikov | The ∂-approach to approximate inverse scattering at fixed energy in three dimensions[END_REF]):

I. For k 2 = E equations (1.5) for f and (2.9) for H are uniquely solvable in C α,µ (R 3 ) for each fixed k (considered as mentioned in these equations).

II. The function H γ of (2.11) at fixed E is well-defined and is related with the scattering amplitude f on M E by the following equation (of [F2], [F3])

h γ (k, l) = f (k, l) + πi √ E Z S 2 √ E h γ (k, m)χ((m -k)γ)f (m, l)dm, (k, l) ∈ M E , γ ∈ S 2 = S 2 1 ,
(2.20)

where

S 2 r = {m ∈ R 3 : |m| = r}, r > 0, (2.21) χ(s) = 0 for s ≤ 0, χ(s) = 1 for s > 0, (2.22)
and by the formula

H γ (k, p) = h γ (k, k -p), (k, p) ∈ Re Ω E , γ ∈ S 2 = S 2 1 , (2.23) where (k, p) ∈ Re Ω E ⇐⇒ (k, k -p) ∈ M E .
(2.24)

III. The following ∂-equation (of [BC], [HN]) holds:

∂kH(k, p) = -2π Z R 3 ( 3 X j=1 ξ j d kj )H(k, -ξ)H(K + ξ, p + ξ)× δ(ξ 2 + 2kξ)dξ for H on Ω E \Re Ω E , (2.25)
where δ is the Dirac function: Z

R 3 u(ξ)δ(ξ 2 + 2kξ)dξ = Z {ξ∈R 3 : ξ 2 +2kξ=0} u(ξ) |J(k, ξ)| |dξ 3 |, (2.26) where J(k, ξ) = 4[(ξ 1 + Re k 1 )Im k 2 -(ξ 2 + Re k 2 )Im k 1 ]
is the Jacobian of the map (ξ 1 , ξ 2 , ξ 3 ) → (ξ 2 + 2Re kξ, 2Im kξ, ξ 3 ) and u is a test function. In addition,

(k, p) ∈ Ω E =⇒ (k, -ξ) ∈ Ω E , (k + ξ, p + ξ) ∈ Ω E if ξ ∈ R 3 , ξ 2 + 2kξ = 0.
(2.27)

IV. The following formula (of [HN]) holds:

v(p) = lim (k,p)∈Ω E ,|k|→∞ H(k, p) for each p ∈ R 3 , (2.28)
where |k| = (|Re k| 2 + |Im k| 2 ) 1/2 . V. The following estimates hold:

|H(k, p)| ≤ D (1 -E -σ/2 Const 1 (α, µ, σ)D)(1 + |p| 2 ) µ/2 , (k, p) ∈ Ω E \Re Ω E , (2.29) kf k C(M E ),µ ≤ D 1 -E -σ/2 Const 1 (α, µ, σ)D , (2.30a) kf k C α (M E ),µ ≤ Const 2 D 1 -E -σ/2 Const 1 (α, µ, σ)D , (2.30b)
where D and Const 1 are the constants of (2.18), see [START_REF] Novikov | The ∂-approach to approximate inverse scattering at fixed energy in three dimensions[END_REF] (and [ER] in connection with (2.30)).

2.3. Reconstruction scheme. Reconstruction (2.3) is based on properties (2.11), (2.20), (2.23), (2.25), (2.28), (2.29), (2.30) of the functions H and f and consists of the following parts:

f on M E (2.20),(2.23),(2.30) ----------→ H γ (k, p), (k, p) ∈ Re Ω E , γ ∈ S 2 , γk = 0, ------→ H ± (k, p) = H γ ± (k,p) (k, p), γ ± (k, p) = ±p × (k -p/2) |p||k -p/2| , (k, p) ∈ Re Ω E , |p| 6 = 0, (2.31a)
where × denotes vector product;

H + and H -on Re Ω τ E (2.11),(2.25),(2.29) ----------→ H appr E,τ on Ω τ E \Re Ω τ E , Ω τ E = {(k, p) ∈ Ω E : p ∈ B 2τ √ E }, Re Ω τ E = {(k, p) ∈ Re Ω E : p ∈ B 2τ √ E }, (2.31b)
where τ is the parameter of (2. (2.32) where χ r denotes the multiplication operator by the function

∂kχ 2τ √ E H(k, p) = -2π Z R 3 ( 3 X j=1 ξ j d kj )χ 2τ √ E H(k, -ξ)χ 2τ √ E H(k + ξ, p + ξ)× δ(ξ 2 + 2kξ)dξ + R E,τ (k, p) for χ 2τ √ E H on Ω τ E \Re Ω τ E , τ ∈]0, 1[,
χ r (p) = 1 for |p| < r, χ r (p) = 0 for |p| ≥ r,
where p ∈ R 3 , r > 0, and, as a corollary,

χ 2τ √ E H = H on Ω τ E \Re Ω τ E , χ 2τ √ E H ≡ 0 on (Ω E \Re Ω E )\Ω τ E ,
(2.33) R E,τ is a remainder which can be written explicitly (see [START_REF] Novikov | The ∂-approach to approximate inverse scattering at fixed energy in three dimensions[END_REF]). Proceeding from (2.32), the definition of H ± of (2.11), (2.31a) and estimate (2.29), we obtain for

H E,τ = H ¯Ωτ E \Re Ω τ E
a nonlinear integral equation of the form (see [START_REF] Novikov | The ∂-approach to approximate inverse scattering at fixed energy in three dimensions[END_REF]): (2.34) where H 0 E,τ is expressed explicitly in terms of H ± , M E,τ is a nonlinear integral operator and Q E,τ is a remainder expressed in terms of R E,τ and admitting an estimate using (2.29). Proceeding from (2.34) we define H appr E,τ of (2.31b) by the nonlinear integral equation

H E,τ = H 0 E,τ + M E,τ (H E,τ ) + Q E,τ , τ ∈]0, 1[,
H appr E,τ = H 0 E,τ + M E,τ (H appr E,τ ), τ ∈]0, 1[. ( 2
(k, p) ∈ Ω E ⇐⇒ k ∈ Σ (3) E,p = {k ∈ C 3 : k 2 = E, p 2 = 2kp}
(2.36)

for fixed E > 0 and p ∈ R 3 . In addition,

Σ

(3)

E,p ≈ Σ (2) E-p 2 /4 , Σ (2)
E = {k ∈ C 2 : k 2 = E}, for 0 < |p| < 2 √ E.
(2.37)

It is interesting to note that the inverse scattering theory at fixed energy E > 0 in dimension d = 2 mentioned in Remark 1.5 of Introduction is, actually, based on some complex analysis on Σ

(2)

E .
In particular, it is very much used in these 2D considerations that the following formulas

λ = k 1 + ik 2 E 1/2 , k 1 = ¡ λ + 1 λ ¢ , k 2 = ¡ 1 λ -λ ¢ iE 1/2 2 (2.38)
give a diffeomorphism between Σ

(2)

E , E > 0, and C\0 (with the variable λ). Proceeding from these observations we introduce some convenient coordinates on Ω τ E , E > 0, τ ∈]0, 1], as follows [START_REF] Novikov | The ∂-approach to approximate inverse scattering at fixed energy in three dimensions[END_REF]).

Let (2.40)

L ν = {p ∈ R 3 : p = tν, t ∈ R}, (2.39a) B r,ν = {p ∈ B r : p 6 ∈ L ν }, (2.39b) Ω τ E,ν = {(k, p) ∈ Ω τ E,ν : p 6 ∈ L ν }, ReΩ τ E,ν = {(k, p) ∈ Re Ω τ E,ν : p 6 ∈ L ν }, (2.39c) where ν ∈ S 2 , r > 0, E > 0, τ ∈]0, 1]. For p ∈ R 3 \L ν we consider θ(p)
Conditions (2.40) imply that ω(p) = p × θ(p) |p| for p ∈ R 3 \L ν (2.41a) or ω(p) = - p × θ(p) |p| for p ∈ R 3 \L ν , (2.41b)
where × denotes vector product. To satisfy (2.40), (2.41a) one can take

θ(p) = ν × p |ν × p| , ω(p) = p × θ(p) |p| , p ∈ R 3 \L ν . (2.42) Let E > 0, τ ∈]0, 1], ν ∈ S 2
and the functions θ and ω of (2.40), (2.41a) be fixed. Then ( [START_REF] Novikov | The ∂-approach to approximate inverse scattering at fixed energy in three dimensions[END_REF]) the following formulas give a diffeomorphism between Ω τ E,ν and (C\0) (2.45)

× B 2τ √ E,ν : (k, p) → (λ, p), where λ = λ(k, p) = k(θ(p) + iω(p)) (E -p 2 /4) 1/2 , (2.43) (λ, p) → (k, p), where k = k(λ, p, E) = κ 1 (λ, p, E)θ(p) + κ 2 (λ, p, E)ω(p) + p/2, κ 1 (λ, p, E) = ¡ λ + 1 λ ¢ (E -p 2 /4) 1/2 2 , κ 2 (λ, p, E) = ¡ 1 λ -λ ¢ i(E -p 2 /4) 1/2 2 , (2.44) where (k, p) ∈ Ω τ E,ν , (λ, p) ∈ (C\0) × B 2τ √ E,ν ; in
We consider λ, p of (2.43), (2.44) as coordinates on Ω τ E,ν and on Ω τ E , E > 0, τ ∈]0, 1]. One can see a considerable similarity between formulas (2.43), (2.44) and formulas (2.38).

In the coordinates λ, p the restriction

H E,τ = H ¯Ωτ E \Re Ω τ E
can be written as

H E,τ = H(λ, p, E) = H(k(λ, p, E), p), λ ∈ C\(T ∪ 0), p ∈ B 2τ √ E,ν , (2.46a)
and H ± of (2.31a) can be written as

H ± (λ, p, E) = H ± (k(λ, p, E), p), λ ∈ T , p ∈ B 2τ √ E,ν . (2.46b)
In addition, (a) the following limit relation is valid:

H ± (λ, p, E) = H(λ(1 ∓ 0), p, E), λ ∈ T , p ∈ B 2τ √ E,ν ; ( 2 .47) (b) the ∂-equation (2.32) takes the form ∂ ∂ λ H(λ, p, E) = (H, H) E,τ (λ, p) + R E,τ (λ, p), λ ∈ C\(T ∪ 0), p ∈ B 2τ √ E,ν , (2.48)
where (H, H) is defined by (2.51) and R E,τ (λ, p) = R E,τ (k(λ, p, E), p); (c) the following formulas for H 0 E,τ and M E,τ of (2.35) hold:

H 0 E,τ (λ, p) = 1 2πi Z T H + (ζ, p, E) dζ ζ -λ , λ ∈ D + , p ∈ B 2τ √ E,ν , H 0 E,τ (λ, p) = - 1 2πi Z T H -(ζ, p, E) λdζ ζ(ζ -λ) , λ ∈ D -, p ∈ B 2τ √ E,ν , (2.49) M E,τ (U )(λ, p) = M + E,τ (U)(λ, p) = - 1 π Z Z D + (U, U) E,τ (ζ, p) dRe ζdIm ζ ζ -λ , λ ∈ D + , p ∈ B 2τ √ E,ν , (2.50a
)

M E,τ (U )(λ, p) = M - E,τ (U)(λ, p) = - 1 π Z Z D - (U, U ) E,τ (ζ, p) λdRe ζdIm ζ ζ(ζ -λ) , λ ∈ D -, p ∈ B 2τ √ E,ν , (2.50b) (U 1 , U 2 ) E,τ (λ, p) = - π 4 π Z -π µ (E -p 2 /4) 1/2 sgn(|λ| 2 -1)(|λ| 2 + 1) λ|λ| (cos ϕ -1) -|p| 1 λ sin ϕ ¶ × U 1 (z 1 (λ, p, E, ϕ), -ξ(λ, p, E, ϕ))U 2 (z 2 (λ, p, E, ϕ), p + ξ(λ, p, E, ϕ))× χ 2τ √ E (ξ(λ, p, E, ϕ))χ 2τ √ E (p + ξ(λ, p, E, ϕ))dϕ, λ ∈ D + ∪ D -, p ∈ B 2τ √ E,ν , (2.51)
where where k(λ, p, E) is defined by (2.44) and θ, ω are the functions of (2.40), (2.41a). Note also that in the coordinates λ, p formula (2.31c) takes the form

D + = {λ ∈ C : 0 < |λ| < 1}, D -= {λ ∈ C : |λ| > 1} (2.52) U 1 , U 2 , U 3 are test functions on (D + ∪ D -) × B 2τ √ E,ν , z 1 (λ, p, E, ϕ) = k(λ, p, E)(θ(-ξ(λ, p, E, ϕ)) + iω(-ξ(λ, p, E, ϕ))) (E -|ξ(λ, p, E, ϕ)| 2 /4) 1/2 , z 2 (λ, p, E, ϕ) = (k(λ, p, E) + ξ(λ, p, E, ϕ))(θ(p + ξ(λ, p, E, ϕ)) + iω(p + ξ(λ, p, E, ϕ))) (E -|p + ξ(λ, p, E, ϕ)| 2 /4) 1/2 , ( 2
v+ appr (p, τ, E) = lim λ→0 H appr E,τ (λ, p), λ ∈ D + , p ∈ B 2τ √ E,ν , v- appr (p, τ, E) = lim λ→∞ H appr E,τ (λ, p), λ ∈ D -, p ∈ B 2τ √ E,ν .
(2.55)

This completes a formal description (without estimates) of equations and formulas involved in (2.3), (2.31).

Error and stability estimates of [No8]

Note that it is more convenient (for estimates) to consider (2.3), (2.31) under condition (2.18), µ ≥ 2, than under our initial condition (1.7), n > d = 3. We have, in particular, the following results of [START_REF] Novikov | The ∂-approach to approximate inverse scattering at fixed energy in three dimensions[END_REF].

Let v satisfy (2.18), where µ ≥ 2 and kvk α,µ ≤ D for some D > 0. Let f be the scattering amplitude for equation (1.1). Let

0 < τ < τ 1 (α, β, µ, µ 0 , σ, D), E ≥ E 1 (α, β, µ, µ 0 , σ, D, g 1 , g 2 ) ( 3 .1)
for some special τ 1 and E 1 (which can be given explicitly, see [START_REF] Novikov | The ∂-approach to approximate inverse scattering at fixed energy in three dimensions[END_REF]), where β, µ, µ 0 , σ, D, g 1 , g 2 are some additional fixed numbers such that

2 ≤ µ 0 ≤ µ, 0 < σ < 1, 0 < β < min(α, σ, 1/2), g 1 > 1, g 2 > Const 2 (µ), g 2 > g 1 . (3.2)
In addition, in particular,

E 1 → +∞ as D → +∞, 0 < τ 1 < 1 and (in the framework of [No8]) τ 1 → 0, as D → +∞. (3.3)
Let also λ, p be the coordinates of (2.43), (2.44). Then ( [START_REF] Novikov | The ∂-approach to approximate inverse scattering at fixed energy in three dimensions[END_REF]): I. The following estimates hold:

kf k C(M E ),µ < g 1 D, kf k C α (M E ),µ < g 2 D, (3.4) |H 0 E,τ (λ, p)| ≤ R(α, β, µ, σ, D, E, |p|) def = µ 2 µ/2 D + Const 3 (α, β, µ, σ)D 2 E β/2 ¶ 1 (1 + |p|) µ , λ ∈ D + ∪ D -, p ∈ B 2τ √ E,ν , (3.5
(p, τ, E)| ≤ Const 4 (µ, µ 0 , τ)D 2 (1 + |p|) µ 0 (1 + 2τ √ E) µ-µ 0 , p ∈ B 2τ √ E,ν . (3.6)
III. In addition, if f is an approximation to f , where estimates (3.4) are valid also for f in place of f , then by means of (2.31), (2.35) f on M E determines ṽ± appr on B 2τ √ E and the following estimate holds:

|v ± appr (p, τ, E) - ṽ± appr (p, τ, E)| ≤ Const 5 (1 + ln E)kf -f k C(M E ),µ (1 + |p|) µ 0 + Const 6 (εβ)× (Const 7,1 (µ)g 2 D + Const 7,2 (β, µ)(g 1 D) 2 E -β/2 )(kf -f k C(M E ),µ ) 1-ε (1 + |p|) µ 0 , p ∈ B 2τ √ E,ν , ε ∈]0, 1[. ( 3
E (λ, p) = W E (|p|) ¡ H 0 E (λ, p) + M E ( Hj-1 E )(λ, p) ¢ , Hj=0 E (λ, p) = W E (|p|)H 0 E (λ, p), Hj E = (1 -ε) Hj-1 E + ε Hj E , 0 < ε ≤ 1, (4.4)
where ε is a relaxation parameter and W E (|p|) is a sufficiently regular filter such that W E (r) = 1 for 0 ≤ r ≤ 2τ 0 √ E, W E (r) = 0 for r ≥ 2τ 1 √ E for some τ 0 and τ 1 , 0 < τ 0 < τ 1 ≤ 1. In addition, the numerical implementation of [START_REF] Alexeenko | Solution of three-dimensional acoustical inverse scattering problem II. Modified Novikov algorithm[END_REF] is reduced to the algorithm of [START_REF] Alexeenko | Solution of three-dimensional acoustical inverse scattering problem based on Novikov-Henkin algorithm[END_REF] if (2.35) is "solved" by the zero approximation:

H appr E,1 ≈ H 0 E,1 . (4.5)
Note that sufficiently strong scatterers are successfully reconstructed in [START_REF] Alexeenko | Solution of three-dimensional acoustical inverse scattering problem II. Modified Novikov algorithm[END_REF] by the method of [START_REF] Novikov | The ∂-approach to approximate inverse scattering at fixed energy in three dimensions[END_REF] in the framework of numerical simulations for the acoustic equation (4.1). We emphasize that the Born approximation method does not give already a satisfactory reconstruction result for these scatterers. In [START_REF] Alexeenko | Solution of three-dimensional acoustical inverse scattering problem II. Modified Novikov algorithm[END_REF] it is also numerically shown that the method of [START_REF] Novikov | The ∂-approach to approximate inverse scattering at fixed energy in three dimensions[END_REF] is considerably more precise than its approximate version solving (2.35) by the zero approximation and actually used in [START_REF] Alexeenko | Solution of three-dimensional acoustical inverse scattering problem based on Novikov-Henkin algorithm[END_REF].

Note that in the numerical simulations on [START_REF] Alexeenko | Solution of three-dimensional acoustical inverse scattering problem II. Modified Novikov algorithm[END_REF] is is assumed that c and α of (4.1) are spherically symmetric functions of x. However, this condition is used in [START_REF] Alexeenko | Solution of three-dimensional acoustical inverse scattering problem II. Modified Novikov algorithm[END_REF] for reducing the volume of numerical operations only and is not an assumption of the method.

Note that some of results of [START_REF] Alexeenko | Solution of three-dimensional acoustical inverse scattering problem II. Modified Novikov algorithm[END_REF] are presented already in [START_REF] Alexeenko | Solution of three-dimensional inverse scattering problem by the modified Novikov algorithm[END_REF].

Open problem for the acoustic case

Note that the potential v = v(x, ω) of (4.3) depends on the frequecy ω in constrast with the energy independent potential v = v(x) of the Schrödinger equation (1.1).

For the acoustic equation (4.1) under condition (4.2) and even with α ≡ 0 on the whole space (in dimension d ≥ 2), analogs of (2.2), (2.4) are not obtained yet, in general, in the framework of (approximate but) sufficiently stable monochromatic inverse scattering (by methods of [START_REF] Novikov | Rapidly converging approximation in inverse quantum scattering in dimension 2[END_REF]- [START_REF] Novikov | The ∂-approach to approximate inverse scattering at fixed energy in three dimensions[END_REF] or by other methods). This is an important open problem for the acoustic case. In this connection, in addition to results of [START_REF] Novikov | Rapidly converging approximation in inverse quantum scattering in dimension 2[END_REF]- [START_REF] Novikov | The ∂-approach to approximate inverse scattering at fixed energy in three dimensions[END_REF] and [BBMRS], [BMR], [START_REF] Alexeenko | Solution of three-dimensional acoustical inverse scattering problem II. Modified Novikov algorithm[END_REF], we would like to mention also that:

In the particular case of the 3-dimensional acoustic equation in a half-space with horizontally-homogeneous velocity c (and zero absorption α) a stable monochromatic inverse scattering method with the error estimate (for ω → +∞) of the type (2.2) was developed in [MNPSF], [HeNo].

An interesting stability analysis for monochromatic acoustical inverse scattering with a decreasing error term as ω → +∞ is developed in [P] for d = 2.

  and ω(p) such that θ(p), ω(p) smoothly depend on p ∈ R 3 \L ν take their values in S 2 and θ(p)p = 0, ω(p)p = 0, θ(p)ω(p) = 0.

  addition, formulas (2.43), (2.44) give also diffeomorphisms between Re Ω τ E,ν and T × B 2τ √ E,ν and between Ω τ E,ν \Re Ω τ E,ν and (C\(0 ∪ T )) × B 2τ √ E,ν , where T = {λ ∈ C : |λ| = 1}.

  .53) ξ(λ, p, E, ϕ) = Re k(λ, p, E)(cos ϕ -1) + k ⊥ (λ, p, E) sin ϕ, k ⊥ (λ, p, E, ϕ) = Im k(λ, p, E) × Re k(λ, p, E) |Im k(λ, p, E)| , (2.54)

  -are the boundary values of H in the limits on Re Ω τ

	E from Ω τ E \Re Ω τ E (see E,τ is found from equation (2.35) (2.11), (2.31a)) and estimate (2.29), more precisely, H appr
	discussed below;
	v± appr (p, τ, E) Ω τ,± E = {(k, p) ∈ Ω τ (2.28) = (k,p)∈Ω τ,± lim E ,|k|→∞ E : Im k |Im k| = ±	H appr E,τ (k, p) for almost each p ∈ B 2τ |p × Re k| , Im k 6 = 0, p 6 = 0}, p × Re k	√ E ,	(2.31c)
	see also (2.55).
	To present (2.31b) in more detail we emphasize that (2.25) is only an approximate
	∂-equation for H on Ω τ E \Re Ω τ E , τ ∈]0, 1[:

3), H appr E,τ is an approximation to H on Ω τ E \Re Ω τ E , H appr E,τ is found using (2.25) (as an appropriate ∂equation for H on Ω τ E \Re Ω τ E ), the property that H + and H

  appr E,τ from H 0 E,τ via (2.35). One can see that (2.20) is a (standard) linear Fredholm integral equation of the second type, whereas (2.35) is a nonlinear integral equation. In [ABR3] equation (2.35) is solved for τ = 1 by iterations organized as follows:

	Hj
	.7)
	More precisely, (2.50), (2.55) with f , Hγ , H± , H0 ṽ± appr on B 2τ √ E is determined from f on M E via (2.31), (2.35), (2.49), E,τ , Happr E,τ , ṽ±

appr in place of f , H γ , H ± , H 0 E,τ , H appr E,τ ,

and finding H

v±

appr , where H0 E,τ arising in (2.49) is corrected also as follows: R(α, β, µ, σ, D, E, |p|),

where R is radius of (3.5).

In connection with (3.5), (3.8) note also that if f = f (that is the scattering amplitude is given with no errors) and H0 E,τ is calculated from f with no errors, then H0

and (due to (3.5)) correction (3.8) is not necessary. However, if some of these errors are present, then (3.8) seems to be necessary, in general, for solvability of (2.35) by iterations and for validity of (3.7).

The remark that (2.17) follows from (1.7), d = 3, and estimates (3.6), (3.7) imply the estimates of (2.3), (2.4).

Results of [ABR3]

The monochromatic inverse scattering method of [START_REF] Novikov | The ∂-approach to approximate inverse scattering at fixed energy in three dimensions[END_REF] was implemented for the first time numerically in [START_REF] Alexeenko | Solution of three-dimensional inverse scattering problem by the modified Novikov algorithm[END_REF], [START_REF] Alexeenko | Solution of three-dimensional acoustical inverse scattering problem II. Modified Novikov algorithm[END_REF] for the case of the inverse scattering problem for the acoustic equation

with velocity of sound c(x), amplitude absorption coefficient α(x, ω), at fixed frequency ω, under the assumption that

The possibility of applications of the method of [START_REF] Novikov | The ∂-approach to approximate inverse scattering at fixed energy in three dimensions[END_REF] (as well as of any method for solving Problem 1.1 for equation (1.1) under assumption (1.2)) to the inverse scattering problem for equation (4.1) under assumptions (4.2) (with known background c 0 ) is based on the fact that (4.1) can be written in the form (1.1), where

and that, as a corollary of (4.2), v = v(x, ω) ≡ 0 for |x| ≥ r.

(4.3b)

The main (or, at least, the most difficult for numerical implementation) points of monochromatic inverse scattering method of [START_REF] Novikov | The ∂-approach to approximate inverse scattering at fixed energy in three dimensions[END_REF] consists in finding h γ from f via (2.20)


Véronique Huth
email: veronique.huth@ifsttar.fr

Yann Sanchez

Corinne Brusque

Drivers' phone use at red traffic lights: a roadside observation study comparing calls and visual-manual interactions

Keywords: Mobile phone, distraction, observation, traffic light, use strategy, situation awareness, visualmanual interaction

Drivers' phone use at red traffic lights: A roadside observation study comparing calls and visual-manual interactions

Introduction

Communication is a central element of the digital age, and phone use has evolved into a habit for many people. The statistics of the International Telecommunication Union show that mobile phone subscriptions have dramatically increased over the last decade (ITU, 2014).

Inevitably, phone use is also taking place in road traffic and driver distraction by mobile phones has become one of the priority issues in road safety (WHO, 2011). In a survey conducted in 2012 with a representative sample of the French driver population, 23 % of the respondents indicated to use the phone with a hands-free kit while driving and 11 % admitted to use the phone hand-held. 13 % of the respondents said to read text (messages, mails or websites) and 11 % to write text while driving (ONISR, 2013). Accident analyses conducted in 2010 revealed that close to 10% of injury crashes in France could be attributed to mobile phone use while driving (OECD/ITF, 2014). On an international level, reviews of the effects of phone use on driving performance keep raising concerns on the cognitive, visual and manual distraction provoked by calls or other interactions with the phone and highlight the corresponding increase in crash risk [START_REF] Mccartt | Cell phones and driving: review of research[END_REF]Brace et al., 2007;[START_REF] Kircher | Mobile telephones and other communication devices and their impact on traffic safety[END_REF][START_REF] Bruyas | Impact of mobile phone use on driving performance: review of experimental literature[END_REF]. Given that they provoke interruptions of drivers' visual sampling of the driving environment, distracting activities with high visual demand are associated to higher crash risk than those that only imply cognitive distraction [START_REF] Young | Examining the relationship between driver distraction and driving errors: A discussion of theory, studies and methods[END_REF]. Simulated as well as naturalistic driving studies revealed a relationship between crashes and glances inside the vehicle that lasted more than 1.6 or 2 seconds respectively [START_REF] Horrey | In-vehicle glance duration: Distributions, tails and model of crash risk[END_REF][START_REF] Klauer | The Impact of Driver Inattention on Near-Crash/Crash Risk: An Analysis Using the 100-Car Naturalistic Driving Study Data[END_REF]. Related to phone use, the greatest increases in the odds for drivers to be involved in safety-critical events have been found for text messaging, followed by dialling a number and reaching for a mobile device [START_REF] Hickman | Distraction in commercial trucks and buses: assessing the prevalence and risk in conjunction with crashes and near-crashes[END_REF][START_REF] Olson | Driver distraction in commercial vehicle operations[END_REF].

While drivers used a mobile phone, failures to perceive and process traffic signs and hazards that are present on the road have been found [START_REF] Mcknight | The effect of cellular phone use upon driver attention[END_REF] and longer reaction times to traffic signals and events have been observed [START_REF] Strayer | Cell phone-induced failures of visual attention during simulated driving[END_REF][START_REF] Charlton | Driving while conversing: Cell phones that distract and passengers who react[END_REF]. Hence, phone use can lead to a general decrease in situation awareness, as defined by [START_REF] Endsley | Design and evaluation for situational awareness enhancement[END_REF], and to violations of traffic rules (Caird et al., 2008). These effects are amplified if the drivers' use of the mobile phone implies a diversion of the gaze towards the mobile phone. For example, while texting drivers make more in-vehicle glances and reduce their visual attention to the environment, which has proven to be associated to impaired lateral vehicle control [START_REF] Hosking | The effects of text messaging on young drivers[END_REF][START_REF] Drews | Text messaging during simulated driving[END_REF]. In addition to their decreased awareness of changing events in the road scene, drivers might not concentrate on the traffic situation for long enough between their off-road glances so as to get back 'in-the-loop' (Senders et al., 1967cited in Young & Salmon, 2012). As a consequence, drivers suffer from delayed detection of relevant information or complete hazard identification failures. Thus, a major issue of concern lies in the alteration of visual information uptake when using the phone.

The latest generations of mobile phones feature a wide range of use options, with a growing number of mobile phone applications that require visual-manual interactions. Smartphones are not only increasingly popular; they also tend to become an essential device in everyday life and imply strong temptations for visual-manual interactions. Their use can become an automated habit, driven by internal factors, such as the need to constantly stay informed and in touch with the social network, or by external triggers, i.e. a sound emitted by the phone.

These triggers can appear at any time and in any situation, and consequently also while driving (Bayer & Campbell, 2012).

In their deciding-to-be-distracted approach, [START_REF] Lerner | On-road study of willingness to engage in distracting tasks[END_REF] argue that the risk of secondary tasks while driving is not only determined by the distractive potential of the activity itself but also by the driver's exposure to the task, and that drivers can influence their risk by actively regulating exposure to secondary tasks. This regulation is based on higherlevel decisions, which depend on motivational and cognitive factors [START_REF] Summala | Hierarchical model of behavioral adaptation and traffic accidents[END_REF]. It can either be achieved by exposure limitation or by seeking for compatibility of the secondary task and driving, i.e. by choosing traffic situations in which the demands of the driving tasks leave enough attentional resources for the phone use [START_REF] Huth | Drivers' adaptation to mobile phone use: interaction strategies, consequences on driving behaviour and potential impact on road safety[END_REF]. In a simulator study, [START_REF] Schömig | Anticipatory and control processes in the interaction with secondary tasks while driving[END_REF] observed that drivers took the deliberate decision to engage in the secondary task based on an anticipation of the development of situational demands and the judgement of their compatibility with the execution of a concurrent task. Specifically, [START_REF] Stutts | Driver's exposure to distractions in their natural driving environment[END_REF] found that the most prominent factor influencing the decision to engage in a secondary task is whether the vehicle is moving or not. The compatibility of phone use with the driving task can vary dramatically, since the demands of driving can be relatively low or extremely high, according to the driving situation and the manoeuvre the driver is carrying out. The traffic context is dynamic and can evolve quickly, and drivers will have to apply phone use strategies that are adapted to these changing conditions.

Interactions with the phone that last longer than a few seconds are thus susceptible to take place in a context that does not correspond to the one initially chosen by the driver. In this regard, the possibility to interrupt the phone interaction plays a crucial role [START_REF] Huth | Drivers' adaptation to mobile phone use: interaction strategies, consequences on driving behaviour and potential impact on road safety[END_REF]. Interruptions of secondary tasks in order to execute controls of the primary task have been described as an interaction strategy with in-vehicle devices [START_REF] Rauch | User strategies for the interaction with in-vehicle devices while driving[END_REF].

However, phone interactions might not always be easily dividable into several chunks that allow the driver to pay attention to the traffic situations at regular and appropriate intervals.

Calls might be difficult to interrupt quickly, given that the conversation partner is not witnessing the traffic situation, whereas interruptions of visual-manual phone use can lead to interaction errors, which can also extend the interaction beyond the drivers' intentions.

In this regard, intersections represent a particularly interesting location due to the dynamic driving context that can considerably evolve within short periods of time [START_REF] Sandin | An analysis of common patterns in aggregated causation charts from intersection crashes[END_REF].

Driving through an intersection implies different phases that are associated to driving tasks of different difficulty levels [START_REF] Cooper | The impact of hands-free message reception/response on driving task performance[END_REF]. Furthermore, intersections are locations where the paths of several types of road users cross, which can lead to conflicts with potentially severe outcomes (Habibovic & Davidson, 2012). At the same time, intersections controlled by traffic lights can induce drivers to engage in phone use while safely stopped at a red traffic light. In this situation, drivers may take the chance to use the phone out of necessity or boredom, including highly demanding phone use modes such as texting or checking emails.

However, once the traffic light turns green, the driver enters a potentially complex situation with high attentional demands. Negative consequences of phone use could appear in this situation if the phone interaction was not interrupted when starting to move the vehicle or if the driver is still cognitively distracted by the recent phone use.

In sum, traffic light situations at urban intersections imply both a temptation to use the phone at the red traffic light and a potential threat due to the incompatibility of phone use and the driving task when the traffic signal turns green and the vehicle moves off. The aim of this study was to observe phone use at urban intersections during these two situations and to address the following two research questions:

1. The first research question focussed on the existence of a phone use strategy at the red traffic light. The aim was to determine if drivers took the chance to use their phone when waiting at a red traffic light by observing the initiation of phone use in this situation.

2. The second research question concerned the effectiveness of the phone use strategy at red traffic lights. It was addressed in two parts. Firstly, the cessation of phone use at traffic lights was observed in order to examine if drivers limit the phone use to the safe stopping situation.

Secondly, the relationship between phone use and delayed starts when the traffic signal turns green was tested.

Given that phone calls and visual-manual interactions affect driving to a different extent, the existence and effectiveness of a phone use strategy were tested for differences between these two phone use modes.

Method

Observations

The method applied in this study is based on traffic observation techniques, which have previously been used in studies related to the prevalence of distracting secondary activities while driving [START_REF] Sullman | An observational study of driver distraction in England[END_REF], and in particular regarding mobile phone use [START_REF] Walker | Unsafe driving behaviour and four wheel drive vehicles: observational study[END_REF][START_REF] Pickrell | Driver Electronic Device Use Observation Protocol[END_REF]. In these studies, trained observers collect information on driver characteristics and predefined target behaviours in the setting where they naturally occur.

Behaviour is measured directly, avoiding possible biases of self-reports, and the noninvasiveness of this research method makes for high construct and face validity [START_REF] Eby | Naturalistic Observational Field Techniques for Traffic Psychology Research[END_REF].

According to the research questions, the present study collected information on drivers who used their mobile phone while waiting at a red traffic light. With the aim to compare driver behaviour of mobile phone users and non-users, a control group of non-users was created in parallel.

Three urban intersections that are controlled by traffic lights were selected in Lyon, France.

At each intersection, the traffic light approach zone with best visibility for the observers and most convenient traffic flow was chosen. The observation area comprised the queue of the first four vehicles stopped at the red traffic light, on two parallel lanes. Drivers who passed while the traffic light was green were not included into the sample. The observations were carried out on-site by the observers rather than from video recordings, accounting for privacy concerns as well as for the difficulty to detect and precisely characterize phone uses from outside the vehicle. The observers were placed on the sidewalk opposite the observed lanes, in order to be less visible to the observed drivers and to have good visibility on the observation zone, including the traffic light signal. They were not assigned a fix point, in order to be able to adapt their position depending on the driver they were observing.

Procedure

In order to specify the observation variables and develop a corresponding observation grid, ad hoc observations were been carried out on the three selected observation sites. During this pre-test phase, two observers wrote down any noticeable detail that could be related to the research questions. Visible phone use modes and an observable indicator for the drivers' situation awareness were defined. During this process, the observers made sure that the variables were accurately observable from the roadside. At the same time, they were sensitised and trained for the observation of phone use in the chosen context.

Once the variables had been defined and integrated into a standardized observation grid, further training sessions were carried out until the observers converged towards common criteria for each observed variable. After about one hour at each observation site the observers had reached an interobserver reliability that was greater than 85% [START_REF] Eby | Naturalistic Observational Field Techniques for Traffic Psychology Research[END_REF]. During the adhoc observations and training sessions, the observers gained experience specifically related to the detection of different phone use modes, with the aim to increase sensitivity, i.e. their ability to track down phone use, and specificity, i.e. their ability to discriminate phone interactions from other secondary activities inside the car, such as the interaction with invehicle controls.

Standardized observations then took place in June and July 2013 on weekdays between 4 pm and 7 pm, ensuring a steady traffic flow that corresponded to evening rush hour traffic.

Observation sessions lasted 50 minutes, followed by a break of ten minutes. Data was collected during three sessions at each observation site, resulting in a total of nine sessions and 7.5 hours of observation. Weather conditions were good during all observation sessions. Data was collected paper-and-pencil, using the standardized observation grid that had been specifically created based on the ad hoc observations. The instruction given to the observers was to fill in the observation grid for each driver using a mobile phone while waiting at the red light. Exhaustiveness of this data collection could not be guaranteed, but the coding grid was designed so as to minimize the coding time and to maximize the number of observed phone users. For each driver who was observed using a mobile phone while waiting at the red light (target group), a driver who was not using the phone was observed (control group). The non-user was selected randomly, with the only requirement to occupy a similar position in the queue and to be observed in a narrow time frame after the user. These requirements were set to ensure that the traffic situation of the non-user resembled the one of the corresponding user. During each observation session, one observer registered the target group and the other observer took over data collection for the control group. The roles of the observers were counterbalanced among sessions so as to avoid observation artefacts.

Measures

All measures taken in this study were observation variables, which were included in the standardized grid that had been developed on the basis of the ad-hoc observations. In order to facilitate data collection, the variables were aligned in a row from the left to the right of the observation grid, and the observers simply had to tick boxes next to the variable categories that applied to the driver they were observing.

The first variable registered in the observation grid was phone use, distinguishing between users and non-users. All observed drivers were then assigned a vehicle type, gender and estimated age category. The vehicle type distinguished between passenger cars and minivans.

Drivers of other vehicles were not included into the sample. Three age categories were coded according to previous studies on mobile phone use (e.g., [START_REF] Sullman | An observational study of driver distraction in England[END_REF][START_REF] Young | Look who's talking! A roadside survey of drivers' cell phone use[END_REF].

The first category included those drivers who appeared to be under thirty years of age, the second one those between thirty and fifty, and the third category comprised those drivers who were judged as being over fifty.

The observed phone uses were assigned to three categories, which were based on the visible phone use modes collected in the pre-tests and which follow the categorization made in the NTHSA driver electronic device use observation protocol [START_REF] Pickrell | Driver Electronic Device Use Observation Protocol[END_REF]. Hands-free calls cover conversations held with the help of any hands-free device such as an integrated car-kit, a Bluetooth ear set or headphones; hand-held calls include conversations with the phone held to the ear or with the phone held in hand using the speaker of the phone; and visual-manual interactions are all phone uses except calls, which involve visual attention towards the phone and/or its manipulation and which can range from checking the screen of the phone to inserting a text. Drivers wearing headphones without speaking were not registered as phone users.

The observed phone use was furthermore characterized by its beginning with respect to the red traffic light situation, distinguishing between phone uses that were initiated while waiting at the red traffic light and those that were already ongoing when the driver entered the red traffic light situation. Similarly, the drivers' phone use behaviour after the end of the traffic light situation was coded, differentiating phone uses that were stopped when the traffic light turned green from those that were continued.

As an observable indicator for the drivers' situation awareness the drivers' response to the change of the traffic light signal was assessed, indicating if the drivers moved off as soon as it was correct and safe to do so. If the observer judged that it took the driver longer than other drivers to react to the green traffic light, this behaviour was coded as a delayed start. This variable was registered for users and non-users.

Data analysis

Data was cross-classified in RxC contingency tables and compared using chi-square tests of independence with a significance level of p <.05. Chi-squared tests were calculated to test for differences between phone use modes and between the behaviour of users and non-users. For the tests on differences between phone use modes, counts of hands-free and hand-held calls were collapsed into one variable (calls) and compared to the counts of visual-manual interactions, given that phone calls and visual-manual interactions affect driving to a different extent. Phi was calculated to measure effect sizes.

Results

Observed phone users

In total, 124 drivers who used the phone while waiting at the traffic light were observed across the three observation sites. Correspondingly, data on 124 non-users were collected. 54.0% of the observed phone uses were calls, consisting of 44 hands-free conversations (35.5%) and of 23 hand-held conversations (18.5%). Visual-manual interactions with the phone were observed in 57 phone use cases (46.0%).

Overall, in the sample there were 238 passenger cars (96.0%) and 10 minivans (4.0%). 108 observed drivers were women (43.5%) and 140 were men (56.5%). Target group and control group had the same gender distribution. Of the 54 female phone users 27 (50%) made a visual-manual interaction, and the 70 male phone users were calling in 40 cases (57.1%) and interacting visual-manually in 30 cases (42.9%). The chi-squared revealed no significant relationship between gender and phone use mode, X² (2, N=124)=0,626, n.s.. 131 drivers (52.8%) were estimated to be between thirty and fifty years of age, 86 drivers (34.7%) were judged as being over fifty and 31 drivers (12.5%) appeared to be under thirty.

The distribution of estimated age did not significantly differ between the control and target group, X² (2, N=248)=4,727, n.s. (table 1), neither did it between users who were calling and those who were using the phone in a visual-manual interaction, X² (2, N=124)=1,893, n.s.

(table 2).

Existence of phone use strategy

The first research question was addressed by analysing the distribution of the phone use modes that were observed in the red traffic light situation, and by distinguishing between uses that were already ongoing when the drivers entered the observation zone and those that were initiated during the waiting time at the traffic light. About half of the observed phone uses were initiated while waiting at the red traffic light and the other half were already ongoing when the driver arrived at the red traffic light. However, as illustrated in table 3 and confirmed by the chi-squared test, visual-manual interactions had significantly higher chances to be initiated at the red traffic light, whereas calls were more likely to be already ongoing, X²

(1, N=124)=68,144, p=.000, phi=.74. Among the 58 calls that were already ongoing, 42 were hands-free conversations and 16 hand-held conversations. Two hands-free conversations and seven hand-held conversations were initiated.

Effectiveness of phone use strategy

In order to determine whether the phone use was restricted to the red traffic light situation, the phone use observations were analysed regarding their interruption once the traffic light turns green. Table 4 shows the distribution of stopped and continued phone uses. Overall, the majority of phone uses extended beyond the red traffic light phase. Still, the phone use behaviour differs significantly between calls and visual-manual interactions. Statistical testing reveals that visual-manual interactions tend to be stopped before the vehicle moves off, while calls tend to be continued, X² (1, N=124)=44.358, p=.000, phi=.60. The seven interrupted calls that were observed were exclusively hand-held conversations. The indicator of impaired situation awareness, delayed start when the traffic light turns green, was observed more frequently for users than non-users (table 5). The chi-squared test confirms the relationship between delayed start and phone use, X² (1, N=248)=9.185, p=.002, phi=.19. Among phone users, visual-manual interactions are more likely to be associated with a delayed start than calls, as illustrated in table 6, X² (1, N=124)=16.729, p=.000, phi=.37.

However, no significant effect was detected when examining the relationship between interrupted and continued phone use and delayed start at the green traffic light, X² (1, N=124)=0.266, n.s. (table 7).

Discussion

The data collected in this study gives insights into the patterns of phone use at urban intersections. Although hand-held phone use and visual-manual interactions are not allowed behind the wheel in France, not even while stopped in traffic, these phone use modes represent more than half of the phone uses observed at the red traffic light. Gender and age were comparable among users and non-users in this study, contrary to previous findings suggesting that younger age and being male was a predictor for in-car phone use [START_REF] Pöysti | Factors influencing the use of cellular (mobile) phone during driving and hazards while using it[END_REF]Brusque & Alauzet, 2008;[START_REF] Sullman | An observational study of driver distraction in England[END_REF]. The distributions of phone use modes were not related to gender or age either.

In response to the first research question, a large effect was found when comparing the initiation of phone use at the red light between calls and visual-manual interactions. The results of this study show that drivers take the chance to initiate visual-manual interactions with the phone while waiting at the red traffic light. On one hand, the waiting time might incite them to interact with their phone, possibly out of boredom or due to the reflex to check the phone as soon as the opportunity arises. On the other hand, drivers might be aware of the interference visual-manual interactions with the phone represent and wait for a driving situation with low demands before engaging in this mode of phone use. The majority of visual-manual interactions do not require immediacy and can be strategically delayed, e.g. a driver can wait before answering a text message or before checking email. This is not necessarily the case for phone calls, which were more likely to be ongoing when the drivers arrived at the red traffic light in the present study. Firstly, the drivers might answer incoming calls immediately instead of calling back when the situation is more appropriate and secondly, the call might have started in a low demand situation (e.g., another traffic light) and last longer. The legality of hands-free phone use while driving could also play a role in this context. Avoiding fines could be a motivation to restrict phone use and the fact that handsfree phoning is allowed might even encourage drivers to consider that it does not increase crash risk [START_REF] White | Mobile phone use while driving: An investigation of the benefits influencing drivers' hands-free and hand-held mobile phone use[END_REF] and, hence, that there is no need to adapt their phone use behaviour.

Correspondingly, a large effect was found between calls and visual-manual interactions concerning the second research question. The results on the cessation of phone use when the traffic light turns green confirm that calls are more likely to be prolonged beyond the stopping situation than visual-manual interactions. This finding reflects the incompatibility of visualmanual interactions with the driving task within intersections due to their higher distractive potential. However, it cannot be determined whether this phone use strategy is the result of a conscious decision based on the drivers' perception of crash risk or whether it is only the consequence of the difficulties to comfortably deal with the concurrent tasks.

At the same time, the observation of a majority of phone uses extending beyond the red traffic light phase represents a potential safety concern, given that calls provoke cognitive distraction (e.g., [START_REF] Strayer | Multi-tasking in the automobile[END_REF]. This distraction has been found to narrow the perceptual field of view [START_REF] Recarte | Effects of verbal and spatial-imagery tasks on eye fixation while driving[END_REF], which is especially relevant for the driving performance in complex environments. The phone use behaviour observed in the present study concurs with previous findings on drivers' underestimation of cognitive distraction by a conversation and their judgement of visual-manual distractions as hazardous [START_REF] Mcevoy | Role of mobile phones in motor vehicle crashes resulting in hospital attendance: a case-crossover study[END_REF][START_REF] Huth | Drivers' adaptation to mobile phone use: interaction strategies, consequences on driving behaviour and potential impact on road safety[END_REF]. The results confirm that drivers particularly regulate their exposure to phone use when it includes visual-manual distraction, as concluded from a literature review on drivers' adaptation to mobile phone use while driving [START_REF] Huth | Drivers' adaptation to mobile phone use: interaction strategies, consequences on driving behaviour and potential impact on road safety[END_REF].

The results on the delayed start when the traffic light turns green support the claim that mobile phone use affects situation awareness. A small effect was found when comparing users and non-users, while the comparison of phone use mode and delayed start revealed a medium size effect. As it was expected, more demanding phone use (visual-manual interactions) is more often associated with impaired situation awareness. Interestingly, no distinction could be made between the effects of interrupted and persisting phone use.

Continued phone use was not associated to significantly more delayed starts than phone use that was interrupted. These data suggest that the distractive effects of phone use might last longer than actual phone use. Similarly, earlier findings have suggested that calls are significantly related to an increased crash risk for hazard intervals of up to 15 minutes [START_REF] Redelmeier | Association between cellular telephone calls and motor vehicle collisions[END_REF]. Furthermore, the interruption of the phone use and the moment at which the traffic light turns green might be too close in time as to permit an appropriate reaction of the driver.

In conclusion, choosing red traffic light situations to interact with the phone does not seem to interfere with the driving activity at a first glance, but the results show that drivers are limited in their ability to monitor the situation and this impaired situation awareness implies slower reactions to changing traffic situations. The threat of intersection scenarios lies in the contrast of a situation with very low demands while waiting at red traffic light and a highly complex situation when moving off, which requiring manoeuvres, interactions with other road users, gap estimations, etc. Phone interactions might not be flexible enough to adapt to the abrupt change of from one situation to the other. The findings of the present study suggest that red traffic light situations are too short as to allow effective application of phone use strategies.

This holds especially for calls, given that they tend to be longer and less easy to interrupt than visual-manual interactions. As a result, phone use extends beyond stopping situations and might lead to unintended concurrence of two tasks whose added demands exceed the maximum information processing capacity of the driver. Visual-manual distraction especially represent a hazard since they capture an essential share of the drivers' attentional resources, while the risk related to calls is constituted by a longer exposure to an underestimated cognitive distraction. At the same time, inner-urban traffic is characterized by relatively low speed and may often be part of a driver's familiar environment, which may induce drivers to underestimate the associated accident risk. The findings on lack of situation awareness hint towards a possible increase in crash risk due to driver distraction by phone use at traffic lights.

The types of crashes that could occur in this context range from low severity crashes, such as rear-end collisions that could be provoked by the following vehicle when a distracted driver does not move off at the green traffic light, to crashes with potentially severe outcome, such as those involving pedestrians or front-side collisions after red light running by a distracted driver or failure to give way at left turns with oncoming traffic during the green phase of the traffic light.

The limitations of this study are mostly inherent to the methodology applied. Its main advantage is the direct observation of behaviour in its natural environment. While relying on human observers provides certain flexibility during data collection and allows registering inferential variables that are based on expert judgement, some drawbacks have to be taken into account. First, manual data collection by the observers cannot be expected to be completely exhaustive in this study. The capacities of the observers could be limited due to impaired visibility to vehicles in the queue that were farther away from the observer and to vehicles on the far-side lane that were obstructed by vehicles on the near-side lane. Moreover, phone users could be missed when the observers were busy coding other drivers. Besides, determining phone use rates does not make sense in the scope of this study, and the total number of vehicles in the observation zone was not counted to that end.

Second, generalizability might be limited due to the relatively small sample. However, efforts were made to randomize as much of the design as possible so as to increase generalizability.

Three sites were included into the study and the three observation sessions at each site were conducted at different days of the week and at three different times (between 4 and 7 pm).

Still, it would be interesting to replicate this study on other French sites and in other countries.

Furthermore, the observers switched roles during data collection, registering the target group and the control group alternately. Yet, the use of inferential variables could affect the quality of the collected data. For increased reliability and validity of data collection, the researchers reached a high degree of familiarity with the contents of the observation during the ad hoc observations and the observer training. Moreover, the red traffic light situation represented a static traffic situation that was long enough for the observer to adequately assess the situation.

Lastly, the standardized observation grid and a strict observation protocol were implemented in order to minimize possible observer bias.

Nevertheless, the taxonomy of phone uses remains relatively unspecific. While hands-free and hand-held calls as well as visual-manual interactions could be distinguished, the exact phone activity could not be determined. For instance, no assumption can be made on the relatedness of the phone use to the driving activity. Visual-manual interactions could represent the use of mobile applications, such as navigation, parking information or other types of driving assistance, whereas calls could be made to receive directions or to inform about late arrival.

These issues need to be dealt with in self-report studies, and it should be tested in a controlled environment how different phone use content affects attention. The distinction between calls and visual-manual interactions in the present study is solely based on the underlying distraction mechanism, i.e. whether visual distraction is included or not.

Finally, the presence of observers could change the behaviour of drivers. In order to counteract this bias, the observers used inconspicuous clothing and were placed on the sidewalk of the opposite lane. In addition, the observation zone only comprised the first four vehicles at the traffic light, which gave the drivers less time to detect the observers.

Further research is needed so as to better understand the factors that underlie drivers' decisions to use the phone in different driving situations and to identify the factors that could prevent phone use while driving. For example, it should be analysed if phone use that is not compatible with the driving situation is due to a judgement error, or if it is the neglect of a known risk due to the priority of a competing need. Phone use behaviour is often driven by internal motives concerning the management of time and resources, the enjoyment of technology use, socializing and general lifestyle attitudes [START_REF] Lerner | On-road study of willingness to engage in distracting tasks[END_REF][START_REF] White | Mobile phone use while driving: An investigation of the benefits influencing drivers' hands-free and hand-held mobile phone use[END_REF].

Moreover, a differentiation between active decisions or impulse-driven habits would help defining appropriate countermeasures. This aspect is particularly relevant since smartphones start shaping the behaviour of many people in a way that is not always socially appropriate but might be driven by an addiction-like urge to stay connected and to stay informed [START_REF] Chóliz | Mobile-phone addiction in adolescence: The Test of Mobile Phone Dependence (TMD)[END_REF]. In this context, an analysis of the social component of phone use while driving is also of interest. Heavy mobile phone use from an early age is common in today's society, phone use can take place anywhere and at anytime, and distracted driving has become a normative behaviour. In a recent study, drivers who were confronted with crash scenarios recognized the greater responsibility of a distracted driver for an accident, but they did not suggest greater punishments (fines and jail time) for this driver than for an undistracted driver (Atchley et al., 2012). The social pressure to communicate in combination with the missing social reprobation of mobile phone use while driving could incite drivers to engage in this distracting activity. It could therefore be beneficial to change social norms regarding mobile phone use in traffic, such as it has been achieved regarding drink driving (Atchley et al., 2012).

Further countermeasures could be based on technical solutions that block the phone or limit its functions when the car is moving. However, their technical feasibility and user acceptance need to be assessed. Alternatively, more driving task compatible human-machine interfaces which shift visual-manual behaviour to a speech user interface could help to better manage distraction and mitigate the effects of intersection distraction. A message system interface giving the driver control over the timing of speech-based e-mail presentation, e.g., proved to have a positive effect on driving performance under most circumstances, whereas it was detrimental in highly demanding situations [START_REF] Jamson | Speech-based e-mail and driver behaviour: Effect of an in-vehicle message system interface[END_REF]. The cognitive costs associated with the decision-making process imposed by the system use need to be considered.

Technical solutions based on a real-time diagnosis of the driving situation have been designed

to assist the driver in deciding when to carry out secondary tasks. Depending on the situational demands, these so-called workload managers regulate the information input that is potentially distractive, for instance by diverting an incoming call automatically to the mailbox if the driver is performing a manoeuvre at an intersection, and give advice to the driver when not to perform a self-paced distractive task, e.g. outgoing calls. Simulator tests on such a system revealed that drivers positively rate the support in combining the secondary task with driving, but that they often overrule the system messages [START_REF] Mühlbacher | Workload management while driving: Do subjects' judgements reflect driving safety[END_REF]. It has to be carefully evaluated whether such measures can resolve the issue of phone use that extends over changing driving situations and if they are adapted to intersections, where contrasting situational demands alternate with one another.

Acknowledgements

The research leading to these results has received funding from the European Community's Seventh Framework Programme (FP7/2007-2013) under grant agreement no.

238833/ADAPTATION project (www.adaptation-itn.eu).

Bayer

  , J.B. & Campbell, S.W., 2012. Texting while driving on automatic: Considering the frequency-independent side of habit. Computers in Human Behavior, 28, 2083-2090. Brace C.L., Young K.L., Regan M.A. Analysis of the literature: the use of mobile phones while driving. Monash University Accident Research Centre, 2007 & Vagverket, Swedish Road Administration, 2007. Brusque, C. & Alauzet, A. (2008). Analysis of the individual factors affecting mobile phone use while driving in France: Socio-demographic characteristics, car and phone use in professional and private contexts. Accident Analysis and Prevention, 40, 35-44.

Table 1 :

 1 Cross tabulation of phone use and estimated age: frequency counts and contingent percentages.

	<30	30-50	>50	∑
	Non-user	15	58	51	124
		12.1%	46.8%	41.1%	100%
	User	16	73	35	124
		12.9%	58.9%	28.2%	100%
	∑	44	23	57	248

Table 2 :

 2 Cross tabulation of phone use mode and estimated age: frequency counts and contingent percentages.

	<30	30-50	>50	∑
	call	7	38	22	67
		10.5%	56.7%	32.8%	100%
	visual-manual	9	35	13	57
	interaction	15.8%	61.4%	22.8%	100%
	∑	16	73	35	124

Table 3 :

 3 Cross tabulation of phone use mode and beginning of phone use in relation to red traffic light situation:

	frequency counts and contingent percentages.
	ongoing	initiated		∑
	call	58	9	67
		86.6%	13.4%	100%
	visual-manual	7	50	57
	interaction	12.3%	87.7%	100%
	∑	65	59	124

Table 4 :

 4 Cross tabulation of phone use mode and interruption of phone use after end of red traffic light situation: frequency counts and contingent percentages.

	continued	interrupted		∑
	call	60	7	67
		89.6%	10.4%	100%
	visual-manual	18	39	57
	interaction	31.6%	68.4%	100%
	∑	78	46	124

Table 5 :

 5 Cross tabulation of phone use and delayed start when traffic light turns green: frequency counts and contingent percentages.

		no delayed start	delayed start	∑
	non-user	116	8	124
		93.5%	6.5%	100%
	user	100	24	124
		80.6%	19.4%	100%
	∑	216	32	248

Table 6 :

 6 Cross tabulation of phone use mode and delayed start when traffic light turns green: frequency counts and contingent percentages.

		no delayed start	delayed start	∑
	call	63	4	67
		94.0%	6.0%	100%
	visual-manual	37	20	57
	interaction	64.9%	35.1%	100%
	∑	100	24	124

Table 7 :

 7 Cross tabulation of interruption of phone use and delayed start when traffic light turns green: frequency counts and contingent percentages.

		no delayed start	delayed start	∑
	interrupted	36	10	46
		78.3%	21.7%	100%
	continued	64	14	78
		82.1%	17.9%	100%
	∑	100	24	124