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Figure 1 – (a) Seismic amplitude, and its corresponding instantaneous envelope and phase. The envelope is linearly interpolated and exhibits no strong intersample errors. In contrast, the linearly interpolated phase generates inaccurate estimates between samples (yellow arrows) when the phase “wraps” around the circle. The spline-interpolated phase (used in many commercial software implementations) exhibits the same artifacts, but also overshoots (blue arrows) and undershoots (red arrows) values beyond ±180 degrees. (b) A cartoon showing the relationship between the original amplitude, u, its quadrature component, v, instantaneous envelope, e, and instantaneous phase, φ. Then first define u=ecos φ and v= ecos φ before interpolation. After interpolation of u and v, compute φ=ATAN2(v,u). We will use this relationship to improve our interpolation.
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Figure 1 – (a) Seismic amplitude, and its corresponding instantaneous envelope and phase. The envelope is linearly interpolated and exhibits no strong intersample errors. In contrast, the linearly interpolated phase generates inaccurate estimates between samples (yellow arrows) when the phase “wraps” around the circle. The spline-interpolated phase (used in many commercial software implementations) exhibits the same artifacts, but also overshoots (blue arrows) and undershoots (red arrows) values beyond ±180 degrees. (b) A cartoon showing the relationship between the original amplitude, u, its quadrature component, v, instantaneous envelope, e, and instantaneous phase, φ. Then first define u=ecos φ and v= ecos φ before interpolation. After interpolation of u and v, compute φ=ATAN2(v,u). We will use this relationship to improve our interpolation.