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", orRGR. 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 {x_{s},x_{r}}.The RGR treats the data as caused by, oreffective specular pointse.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 vectornand coordinatesx* (5 parameters: 2 fornand 3 for coordinates of locationx*). After a proper preprocessing the input data for the approach are traveltimes estimated for all of source-receiver pairs, {t_{obs}}, which should be interpreted as corresponding to traveling alonga "source -> e.s.p.-> receiver" rayt_{cal}(x_{s};x*;x_{r}).No parametrization of the interface is used, it is just supposeda priorithat the unknown interface can be represented with a rathersmooth continuousfunction on the depth, which provides us with a wayto "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, whichshould 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.ThreeCommon Shot Gathersare taken as examples.Relevant acquisition is shown with Fig.5 beneath.A 'true' location of the reflector under reconstruction is given withastepwisecurve.

Fig. 2. Related traveltime curves.Smooth ones represent times of travelling along Fermat's extremals,non-smooth ones are results of simulation of a - not far advanced -automatic picking of first arrivals.

Fig. 3. Traveltime gathers.a) theoretical one, corresponding to traveltimes along Fermat's extremalsb) 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 reflectorFragment of interest is illuminated by a set of of sources-receivers ( 4x4 grid, 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 points58th Mtg. Eur. Assoc. Expl Geophys.,Extended Abstracts, 96, Session: P-134Biryulina, Marina and Ryzhikov, Gennady, 2000

Imaging of reflectors under uncertainties in macromodel62nd Mtg. Eur. Assoc. Expl Geophys.,Extended Abstracts, Session: B-55

Геннадий Рыжиков