10ème Congrès Français d'Acoustique Lyon, 12-16 Avril 2010 Envelope of signal and bandwidth : the key parameters for vertical seismic resolution Paul Cristini 1 , Grégoire Le Touzé 1 , Nathalie Favretto-Cristini 1 1 CNRS-Laboratoire de Mécanique et d’Acoustique, 31 chemin J. Aiguier, F-13402 Marseille Cedex 20, cristini@lma.cnrs-mrs.fr, le_touze@lma.cnrs- mrs.fr, favretto@lma.cnrs-mrs.fr At many stages of the interpretation and exploration life cycles, seismics is impacted by resolution. By definition, resolution is the ability to distinguish separate features. Improving resolution is the key problem to see thinner stratigraphic units, smaller details, lateral changes in rock properties... Whereas horizontal resolution is known to be linked to the size of the Interface Fresnel Zone, vertical resolution is usually considered as being enhanced by both high frequencies and broadband signals. This common belief comes from the fact that zero- phase signals, and particularly Ricker wavelets for which frequency and bandwidth are linearly linked, are used in seismic signal processing and modeling, as they provide easier interpretation of images, thanks to the direct link between the peaks/troughs and the reflection arrival times. Nevertheless, this belief is incorrect for mixed- phase signals (i.e., non zero-phase signals) or for signals with many oscillations. For this type of signals, we show by using the ambiguity function that, besides the bandwidth, the envelope of the signal is a fundamental tool to separate closed events and to provide reliable measurements of reflection arrival times. Bandwidth and envelope are therefore the key parameters for the analysis of seismic resolution. 1 Introduction At many stages of the interpretation and exploration life cycles, seismics is impacted by resolution. By definition, resolution is the ability to distinguish separate features [1]. Improving resolution to see thinner stratigraphic units, smaller details, or lateral changes in rock properties is still a topic of investigation and many papers are devoted to this problem. Horizontal resolution is characterized by the minimum distance between two features along a single interface such that these two features can be defined rather than one. It is well-known that Fresnel Zone considerations are the essence of horizontal resolution. Indeed, the size of the Interface Fresnel zone (IFZ) determines the spatial resolution with which important changes in the interface properties may be observed. Following Lindsey [2], “two events are visually independent at the same reflection level if they are separated laterally by approximately the Fresnel radius or more”. Depending on the shape of the interface, the diameter of the IFZ may be considerably great [3]. The process of migration however significantly improves the spatial (horizontal) resolution [2]. Vertical resolution is characterized by the minimum distance between two interfaces such that we can tell that there are two interfaces rather than a single. It is frequently stated that the seismic wavelength limits the resolving power. Following the Widess model, the resolution limit is about a quarter of the dominant wavelength of the signal. Following Sheriff [1], vertical resolution can be improved if higher frequencies and a broader band of frequencies can be recorded. We argue that this statement is correct for zero phase signals and incorrect for mixed phase signals. Conventional seismic analysis is based on a description of the real seismic trace. Event picks on the top and bottom of a thin layer and subsequent calculation of time shift and amplitude may be inaccurate due to interference. This resolution limit could lead to misinterpretation. As phase shifts affect the time resolution and side lobe effects affect the amplitude dramatically, we prefer to consider complex trace analysis because amplitude can be separated from phase in a natural way. Indeed, a single lobe is associated to a single wavelet, which avoids many problems usually encountered. 2 Seismic resolution and the ambiguity function In radar and sonar signal processing, the major tool for defining the resolution is the ambiguity function [4]. This function represents the time response of a filter matched to a given finite energy signal when the signal is received with a delay W and a Doppler shift Q relative to the nominal values (zeros) expected by the filter. It is given by :        ｳ f f    dt t j t u t u Q S W Q W F 2 exp * , , (1) where u is the complex envelope of the signal. Since Doppler shift is not always of interest to seismic applications, we only need to consider the cut along the delay axis. Setting Q = 0 then leads to the autocorrelation function of the envelope of the signal R(W ) :       W W W F R dt t u t u    ｳ f f  * 0 , . (2) In the same way as it is defined for real signals, Rayleigh’s criterion can be used for the quantification of the separation of resolved from unresolved domains for the signal envelopes. It is well-known in radar and sonar signal processing that the time resolution for the envelope is connected to the bandwidth B of the signal. We clearly show in Figure 1 that the key parameter for the separation of the envelopes of two seismic events is also the frequency