In the TF90 approach the overall tilt term can be retrieved if a suitable NGS is available in the field. In the present approach we show in the following that the knowledge of the absolute tilt of only one of the various LGS allow to obtain the absolute tilt free from the conical anisokinetism effect.
Let us consider a wavefront distortion W(x,y)
introduced by a turbulence layer and defined in some domain of the
(x,y) plane at the height of the layer itself
(see Fig. 1 (click here) for an unidimensional section). Let us also consider two
sub-domains of this plane denoted by the indexes A and B.
While the absolute tilt of W can be unknown, the knowledge of the
higher order shape of the wavefront enables us to define uniquely the
relative tilt measured considering the wavefront portions A and B.
Such a quantity is denoted in the following by 
and does not
depends upon the absolute tilt of W. For each of the
layers in the TF90 technique one can figure out any quantity of this
type between single portions of the same wavefront.
As a particular case (see Fig. 2 (click here)) TF90 allows for the determination of
the detailed wavefront shape for each layer as seen on the cylindrical
shape beam going from the telescope entrance to the astronomical target.
For any of the projected LGSs forming the constellation required
to the proper turbulence tomography one can define as 
the
relative tilt between the tilt generated by the i-th layer for the
astronomical target and the tilt originated in the sub-domain defined by
that LGS cone.
Figure 1: In this unidimensional section of the wavefront W(x,y) the
relative tilt difference experienced by the wavefront in the domains A
and B is denoted by . This quantity does not depends upon the
absoulte tilt knowledge of W in the whole domain
Figure 2: For any i-th layer the quantity is defined as the
differential tilt produced by that layer into the two domains A
related to an infinitely distant astronomical target and B related to
a particular LGS used for the TF90 tomographyc sensing
We assume that (for istance using any of the cited techniques) the absolute tilt of an LGS is sensed.
Let us define 
as the
angular tilt introduced by the sub-domain of
the i-th layer intersected by the LGS beam.
Because of the conical shape of the LGS beam,
the optical path disturbance propagates from the layer to the
telescope aperture with the same magnitude, but it is distributed
on a larger surface. This effect reduces the angular LGS tilt contribution
by a factor 
, where Hi and 
are the heights
of the layer and of the LGS respectively. This factor is given by the
ratio between the diameter of
the cone at the height Hi and at the telescope level.
The measured angular tilt of the LGS is given by:
Following Ragazzoni et al. (1997),
with the use of Eq. (4 (click here)) we define an effective tilting layer
height 
such that:
On the other hand one can write the similar subdivision over the whole set of k layers for the cylindrical shaped beam of an hypotethical Natural Guide Star (NGS):
The complete set of , that are estimated through the TF90
technique, are given by:
Summing over the whole k layers the latter relationship, and using the Eq. (5 (click here)), one can easily derive the following:
where the terms
and 
on the right side of Eq. (9 (click here)) are experimentally measured, and the
effective layer height can be estimated as decribed in
Ragazzoni et al. (1997).
Care is to be given in a practical implementation to the effects of the
error propagation from the estimation of the angles and
the parameter (this last figure, morevoer, can change with
time).
The proposed approach can be seen as an extension of the constraints
imposed in the TF90 scheme. Their Eq. (13), in fact, imposes a dummy
zero-tilt condition on the system to be solved to have the tomography
of the turbulence layers. Subsituting such a constraint by the full set
of Eq. (8) is an equivalent way to obtain the absolute tilt free from the
conical anisokinetism error.
As a further remark one should recall that for practical reasons one
should limits the number of perturbing layer sensed by the TF90
technique and it is to be evaluated the impact of this approach to the
overall tilt error budget. A detailed calculation and an estimation of
the achievable Sthrel ratio is well beyond the limits of this paper and
is not treated here.
