**Authors**

**Abstract**

The application of NMO (normal moveout) has been recognized as an effective method of generating quasi-zero-offset traces in traditional common-midpoint processing. Artifacts of the NMO method relate to the NMO-stretch effects. Conventional application of normal-moveout correction to a common-midpoint (CMP) reflection generates a stretch that increases with offset and decreases with zero-offset time. Shatilo and Aminzadeh (2000) introduced the technique which implies constant normal moveout (CNMO) for a finite time interval of a seismic trace. Perroud and Tygel (2004) introduced the implementation, called nonstretch NMO, automatically, which avoids the undesirable stretch effects that are present in the conventional NMO. They applied their new method (Nonstretch NMO) to shallow seismic data including high resolution (HR) seismic data and ground-penetrating radar (GPR) measurements.

In this paper, nonstretch NMO (Perroud and Tygel, 2004) is applied to seismic reflection data. NMO correction is usually considered in a hyperbolic equation where , is travel time, related to the x, offset between the source and the receiver, , two-way zero offset travel time, and , is NMO velocity which estimates the root-mean-square (RMS) velocity in a case of horizontal stratified earth. Hyperbolic equation represents a hyperbola whose asymptote passes through the origin and has slope equal to . Ideally, the entire pulse width must be shifted to the horizontal line without any distortion. Traditional NMO correction moves the samples in the vicinity of traveltime onto and with substituting these quantities in hyperbolic equation, then by comparing these equations the stretch ratio can be extracted. For avoiding stretching we try to parallel the hyperbolae traveltimes. The traveltimes in conventional NMO converge to each other whereas in nonstretch NMO the traveltimes are almost parallel to each other. It can be if , in this case, we obtain the adjusted velocity ( ) relation. It can be seen from the adjusted velocity relation that for the set of recorded events on a given trace, stronger effects on the stacking velocity are observed at shorter zero offset times. Also, we know that always decreases when the time shift increases. Therefore, even setting NMO velocity constant is not sufficient to avoid stretching, as it is done in the constant-velocity-stack (CVS) approach. In addition, the conventional increase of NMO velocity with time that results from interpolation, the time-velocity distribution is going the wrong way and further increases the stretching effect of the NMO. With adjusted velocity equation the original time-velocity point picked from conventional velocity analysis is replaced by adjusted velocity in the curve segment that was obtained by it. The quantity time shift ( ) is obtained by inversing the bandwidth of the propagating signal. In this method, the time-velocity distribution is dependent on a trace in the range as for each sample in the range about the the modified velocity calculated for all of traces. Because in the range the velocity decreases with time, the interpolated NMO velocity between events will increase faster, and thus the NMO stretch effect will be increased between events. The problem arises when events cross each other and it can be solved by processing the reflection events one at a time. At first, we obtained the traveltime corresponding to each reflection by the traditional velocity analysis. Then, for each event, we mute all samples above the corresponding hyperbola and below those for the next events and apply the nonstretch NMO. The process completed by summing all the events.

The no stretch NMO technique has been tested on synthetic and real data. The synthetic data include CMP gathers of a flat layer, two layers with crossing reflectors and four flat layers with crossing reflectors which contain multiples and noise. The real CMP data belong to one of the 2-D seismic reflection operations in Iran. The method improves the resolution of CMP stack. Nonstretch NMO correction reduces the stretch effects of conventional NMO. This results in higher spectral frequencies and smaller spectral distortion of shallow far offset reflected events. Following the nonstretch NMO correction, muting may be less when compared with conventional NMO.

In this paper, nonstretch NMO (Perroud and Tygel, 2004) is applied to seismic reflection data. NMO correction is usually considered in a hyperbolic equation where , is travel time, related to the x, offset between the source and the receiver, , two-way zero offset travel time, and , is NMO velocity which estimates the root-mean-square (RMS) velocity in a case of horizontal stratified earth. Hyperbolic equation represents a hyperbola whose asymptote passes through the origin and has slope equal to . Ideally, the entire pulse width must be shifted to the horizontal line without any distortion. Traditional NMO correction moves the samples in the vicinity of traveltime onto and with substituting these quantities in hyperbolic equation, then by comparing these equations the stretch ratio can be extracted. For avoiding stretching we try to parallel the hyperbolae traveltimes. The traveltimes in conventional NMO converge to each other whereas in nonstretch NMO the traveltimes are almost parallel to each other. It can be if , in this case, we obtain the adjusted velocity ( ) relation. It can be seen from the adjusted velocity relation that for the set of recorded events on a given trace, stronger effects on the stacking velocity are observed at shorter zero offset times. Also, we know that always decreases when the time shift increases. Therefore, even setting NMO velocity constant is not sufficient to avoid stretching, as it is done in the constant-velocity-stack (CVS) approach. In addition, the conventional increase of NMO velocity with time that results from interpolation, the time-velocity distribution is going the wrong way and further increases the stretching effect of the NMO. With adjusted velocity equation the original time-velocity point picked from conventional velocity analysis is replaced by adjusted velocity in the curve segment that was obtained by it. The quantity time shift ( ) is obtained by inversing the bandwidth of the propagating signal. In this method, the time-velocity distribution is dependent on a trace in the range as for each sample in the range about the the modified velocity calculated for all of traces. Because in the range the velocity decreases with time, the interpolated NMO velocity between events will increase faster, and thus the NMO stretch effect will be increased between events. The problem arises when events cross each other and it can be solved by processing the reflection events one at a time. At first, we obtained the traveltime corresponding to each reflection by the traditional velocity analysis. Then, for each event, we mute all samples above the corresponding hyperbola and below those for the next events and apply the nonstretch NMO. The process completed by summing all the events.

The no stretch NMO technique has been tested on synthetic and real data. The synthetic data include CMP gathers of a flat layer, two layers with crossing reflectors and four flat layers with crossing reflectors which contain multiples and noise. The real CMP data belong to one of the 2-D seismic reflection operations in Iran. The method improves the resolution of CMP stack. Nonstretch NMO correction reduces the stretch effects of conventional NMO. This results in higher spectral frequencies and smaller spectral distortion of shallow far offset reflected events. Following the nonstretch NMO correction, muting may be less when compared with conventional NMO.

**Keywords**