Reliable noise measure in time-gated NMR data

Summary Time gating is a commonly used approach in the pre-processing of nuclear magnetic resonance data before inversion. Gating suppresses spurious signals that can degrade recovered decay time distributions and therefore often stabilizes inversion. However, care must be taken in applying this technique when assumptions of uncorrelated Gaussianity break down and reduce stacking efficiency. If not properly accounted for, unreliable noise estimates introduce inversion artefacts through over- or underfitting of the data. Block-bootstrap resampling of noise realization proxies obtained through data phasing can be used to generate reliable noise estimates for the windows. Benefits of the approach are demonstrated through inversion of synthetic and borehole data. Analysis confirms that bootstrapped noise metrics are more reliable under variable noise conditions and result in more stable inversion results.

The Reliability of the Stochastic Active Rotator

The reliability of firing of excitable-oscillating systems is studied through the response of the active rotator, a neuronal model evolving on the unit circle, to white gaussian noise. A stochastic return map is introduced that captures the behavior of the model. This map has two fixed points: one stable and the other unstable. Iterates of all initial conditions except the unstable point tend to the stable fixed point for almost all input realizations. This means that to a given input realization, there corresponds a unique asymptotic response. In this way, repetitive stimulation with the same segment of noise realization evokes, after possibly a transient time, the same response in the active rotator. In other words, this model responds reliably to such inputs. It is argued that this results from the nonuniform motion of the active rotator around the unit circle and that similar results hold for other neuronal models whose dynamics can be approximated by phase dynamics similar to the active rotator.

On: “Homomorphic deconvolution” by D. J. Jin and J. R. Rogers (GEOPHYSICS, 48, 1014–1016).

The recent note by Jin and Rogers (1983) presented examples of the failure of the homomorphic transform to invert properly. Since this transform is not only of interest in geophysics, but has also found applications in other fields (Oppenheim and Schafer, 1975), these results are of concern. We consequently attempted to reproduce Jin and Rogers’ results. We failed to do so. In fact, in our experience, the transform has always inverted successfully. Our results using the first example of Jin and Rogers are shown in Figure 1. We used the algorithm of Tribolet (1977) with a modified Goertzel algorithm (Bonzanigo, 1978) for phase unwrapping. The figure is arranged as in Jin and Rogers’ paper. Figure 1a shows the input: impulses separated by 20 samples, of magnitude 2000 and 1999. Figure 1b shows its complex cepstrum. We have set the zero‐quefrency point to zero since this represents a scale factor and can dominate the plotting. Note the minimum delay cepstrum with a small amount of aliasing. The sequence returned by the inverse transform is shown in Figure 1c, demonstrating a successful inversion. The effect of noise is also shown. Noise with a standard deviation of 5 was added to the sequence of Figure 1a. This is shown in Figure 1d. Note that our noise realization is undoubtedly different from that of Jin and Rogers. The noise has changed the relative magnitude of the original spikes such that they are maximum delay. This is reflected in the cepstrum (Figure 1e). Figure 1f shows the returned sequence, again demonstrating the successful inversion.