`Method of Effective Specular Points`
`or Reflector Geometry Reconstruction (RGR)`

Method of Effective Specular Points (MESP) [1] aims to reveal the geometry of unknown interface from reflected data, that is why another name of the approach is "Reflector Geometry Reconstruction", or RGR. It is supposed that the interface of interest is situated in 3D layered media, an interval velocity of the over-interface layer is under estimation as well, while velocities in all other upper layers are assumed  to be known precisely enough to define a reference kinematical model. The RGR deals with unstacked reflection data from a rather sparse and irregular -in general- source-receiver net {xs, xr}.
The RGR treats the data as caused by effective specular points, or e.s.p.. It means that seismogram is supposed to contain such a segment that can be interpreted as a result of reflection of a pulse  with a single elementary 'mirror', i.e. with an oriented element of the reflector surface,
which can be parametrized with a unit normal vector n and coordinates x* (5 parameters: 2 for and 3 for coordinates of location x*). After a proper preprocessing  the input data for the approach  are traveltimes estimated for all of source-receiver pairs,  {tobs}, which should be interpreted as corresponding to traveling along a "source -> e.s.p.-> receiver" ray tcal( xs; x*; xr).
No parametrization of the interface is used,  it is just supposed  a priori that the unknown interface can be represented with a rather smooth continuous function on the depth, which provides us with a way to "glue"  all effective specular points into one interface .
The mathematical problem is to find all effective specular points (i.e. to  find coordinates of the points and corresponding normal vectors), to define the velocity and  to reconstruct the interface, which  should be tangent to all elementary 'mirrors' (e.s.p.'s).   In this sense  the suggested approach allows us to reconstruct formally ( 1 +  5 x (number of source-receiver pairs) infinity of unknown parameters.
The strategy of MESP can be formalized as follows:
given measured travetimes of reflected by an interface pulses,
construct a 2D-surface which is tangent to all of one-parametrical families of isochrons.

```
MESP: testing with synthetic data.
```
```
Model # 1: 2D.```

`Fig. 1. FD-mimicking of seismic records along a line profile.`
`Three Common Shot Gathers are taken as examples.`
`Relevant acquisition is shown with Fig.5 beneath.`
```A 'true' location of the reflector
under reconstruction is given with```
`a stepwise  curve.`

`Fig. 2. Related traveltime curves.`
```Smooth ones represent times of travelling along
Fermat's extremals,```
```non-smooth ones are results of simulation of a -
`automatic picking of first arrivals.`

`Fig. 3. Traveltime gathers.`
```a) theoretical one, corresponding to traveltimes along
Fermat's extremals```
`b) automatically picked from  the records directly`

 a) b)

```Fig. 4.  Objective function for estimating
the layer velocity.```
```"True" velocity is 2000 m/s, the estimated-
after the 1st step- value is  2050 m/s.```

```Fig. 5.  True reflector (stepwise)
and  result (smooth) of reconstruction.```

```Model # 2: 3D.
```
```
Example of reconstructing the reflector
embedded in a 3D space.
```
` `

`Fig. 6a.  Reconstruction of a curved dipping reflector`
```Fragment of interest is illuminated by a set of
dimensions are taken in relative units: wavelengths)```
```Fig. 6b.  The same as Fig. 6a,
except dimension: [l] = m.```
```
```

``` References:
```
Ryzhikov, Gennady  and  Biryulina, Marina,  1996
Method of effective specular points
58th Mtg. Eur. Assoc. Expl Geophys.,
Extended Abstracts,  96, Session: P-134

Biryulina, Marina   and  Ryzhikov, Gennady,  2000
Imaging of reflectors under uncertainties in macromodel
62nd Mtg. Eur. Assoc. Expl Geophys.,
Extended Abstracts,  Session: B-55

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