Applications of detection and estimation theory to large array seismology

Continuous recordings of ambient seismic noise across large seismic arrays allows a new type of processing using the cross‐correlation technique on broadband data. We propose to apply double beamforming (DBF) to cross correlations to extract a particular wave component of the reconstructed signals. We focus here on the extraction of the surface waves to measure phase velocity variations with great accuracy. DBF acts as a spatial filter between two distant subarrays after cross correlation of the wavefield between each single receiver pair. During the DBF process, horizontal slowness and azimuth are used to select the wavefront on both subarray sides. DBF increases the signal‐to‐noise ratio, which improves the extraction of the dispersive wave packets. This combination of cross correlation and DBF is used on the Transportable Array (USArray), for the central U.S. region. A standard model of surface wave propagation is constructed from a combination of the DBF and cross correlations at different offsets and for different frequency bands. The perturbation (phase shift) between each beam and the standard model is inverted. High‐resolution maps of the phase velocity of Rayleigh and Love waves are then constructed. Finally, the addition of azimuthal information provided by DBF is discussed, to construct curved rays that replace the classical great‐circle path assumption.

The use of information theory to design waveforms for the measurement of extended radar targets exhibiting resonance phenomena is investigated. The target impulse response is introduced to model target scattering behavior. Two radar waveform design problems with constraints on waveform energy and duration are then solved. In the first, a deterministic target impulse response is used to design waveform/receiver-filter pairs for the optimal detection of extended targets in additive noise. In the second, a random target impulse response is used to design waveforms that maximize the mutual information between a target ensemble and the received signal in additive Gaussian noise. The two solutions are contrasted to show the difference between the characteristics of waveforms for extended target detection and information extraction. The optimal target detection solution places as much energy as possible in the largest target scattering mode under the imposed constraints on waveform duration and energy. The optimal information extraction solution distributes the energy among the target scattering modes in order to maximize the mutual information between the target ensemble and the received radar waveform. >

Borehole seismology is the highest resolution geophysical imaging technique available today to the oil and gas industry for characterization and monitoring of oil and gas reservoirs. However, the industry's ability to perform high resolution 3D imaging of deep and complex gas reservoirs using borehole seismology has been hampered by the lack of acquisition technology necessary to record large volumes of high frequency, high signal-to-noise-ratio borehole seismic data. This project took aim at this shortcoming by developing a 400 level 3C clamped downhole seismic receiver array, and accompanying software, for borehole seismic 3D imaging. This large borehole seismic array has removed the technical acquisition barrier for recording the data volumes necessary to do high resolution 3D VSP and 3D cross-well seismic imaging. Massive 3D VSP{reg_sign} and long range Cross-Well Seismology (CWS) are two of the borehole seismic techniques that promise to take the gas industry to the next level in their quest for higher resolution images of deep and complex oil and gas reservoirs. Today only a fraction of the oil or gas in place is produced when reservoirs are considered depleted. This is primarily due to our lack of understanding of detailed compartmentalization of oil and gas reservoirs. In this project, we developed a 400 level 3C borehole seismic receiver array that allows for economic use of 3D borehole seismic imaging for reservoir characterization and monitoring. This new array has significantly increased the efficiency of recording large data volumes at sufficiently dense spatial sampling to resolve reservoir complexities. The receiver pods have been fabricated and tested to withstand high temperature (200 C/400 F) and high pressure (25,000 psi), so that they can operate in wells up to 7,620 meters (25,000 feet) deep. The receiver array is deployed on standard production or drill tubing. In combination with 3C surface seismic or 3C borehole seismic sources, the 400 level receiver array can be used to obtain 3D 9C data. These 9C borehole seismic data provide both compressional wave and shear wave information that can be used for quantitative prediction of rock and pore fluid types. The 400-level borehole receiver array has been deployed successfully in a number of oil and gas wells during the course of this project, and each survey has resulted in marked improvements in imaging of geologic features that are critical for oil or gas production but were previously considered to be below the limits of seismic resolution. This added level of reservoir detail has resulted in improved well placement in the oil and gas fields that have been drilled using the Massive 3D VSP{reg_sign} images. In the future, the 400-level downhole seismic receiver array is expected to continue to improve reservoir characterization and drilling success in deep and complex oil and gas reservoirs.

The optimum detection of binary antipodal signals in additive white Gaussian noise (AWGN) channels with Gaussian channel estimation error has been studied in prior work. In this paper, we present the optimum detector based on the maximum- likelihood criterion for binary orthogonal signals in the presence of Gaussian distributed channel estimation error and AWGN. It is shown that the optimum detector is a linear combination of the optimum coherent and optimum noncoherent detectors. We derive the exact closed-form expression of the average bit error probability of the proposed optimum detector in Rayleigh fading channels with AWGN. It is found that if the variance of channel estimation error for a given average SNR is greater than a threshold, then orthogonal signalling outperforms antipodal modulation, and the analytical expression of this threshold is derived.

Borehole seismology is the highest resolution geophysical imaging technique available today to the oil and gas industry for characterization and monitoring of oil and gas reservoirs. However, the industry's ability to economically do high resolution 3D imaging of deep and complex gas reservoirs using borehole seismology is currently frustrated by the lack of the acquisition technology necessary to record the large volumes of the high frequency, high signal-to-noise-ratio borehole seismic data needed to do 3D imaging. This project takes direct aim at this shortcoming by developing a 400 level 3C clamped downhole seismic receiver array, and accompanying software, for borehole seismic 3D imaging. This large borehole seismic array will remove the technical acquisition barrier for recording the necessary volumes of data to do high resolution 3D VSP or 3D cross well seismic imaging. 3D VSP and long range Cross-Well Seismology (CWS) are two of the borehole seismic techniques that will allow the Gas industry to take the next step in their quest for higher resolution images of the gas reservoirs for the purpose of improving the recovery of the natural gas resources. Today only a fraction of the original Oil or Gas in place is produced when reservoirs are considered depleted. This is primarily due to our lack of understanding of the detailed compartmentalization of the oil and gas reservoirs. The 400 level 3C borehole seismic receiver array will allow for the economic use of 3D borehole seismic imaging for reservoir characterization and monitoring by allowing the economic recording of the required large data volumes that have a sufficiently dense spatial sampling. By using 3C surface seismic or 3C borehole seismic sources the 400 level receiver array will furthermore allow 3D reservoir imaging using 9C data. The 9C borehole seismic data will provide P, SH and SV information for imaging of the complex deep gas reservoirs and allow quantitative prediction of the rock and the fluid types. The data quality and the data volumes from a 400 level 3C array will allow us to develop the data processing technology necessary for high resolution reservoir imaging.

Beamforming of ambient noise recorded by regional arrays of seismometers is presented as an alternative imaging approach to cross-correlations between pairs of sensors. The method is used to obtain phase velocities and propagation directions of Rayleigh surface waves around the first and second microseism peaks in southern California. The derived velocity maps and propagation directions correlate with major geological structures and changes of the coastal shape in the region. The results are consistent with and complementary to those obtained using cross-correlations of long-duration data between pairs of sensors. Significant advantages of the presented high-resolution adaptive beamforming method over point-to-point noise cross-correlations are the short time interval of required data (hours to days compared to a year) and robust performance with directive (rather than omnidirectional) noise propagation. Given the recent trend toward dense and large seismic arrays at various scales, the combination of beamforming and noise-correlation processing may provide an optimal strategy for performing noise-based tomography.

We show that the concept of correlation detection of deterministic signals in additive Gaussian noise can be extended in a natural manner to the detection of signals that are transmitted through a "Gaussian" random channel besides being corrupted by additive Gaussian noise. Such situations are typical in communication over scatter-multipath channels (with or without a specular component). In the deterministic case, the receiver essentially crosscorrelates the received signal with the signal before the additive noise was introduced. When a random channel is present, however, this latter signal, i.e., the output of the random channel, is not known at the receiver, However, knowing the statistics of the channel and the noise, the receiver can make an estimate of it from the received signal on the hypothesis that a particular signal was transmitted. The optimum receiver then crosscorrelates this estimate with the received signal.

The Wigner distribution (WD) is applied to the detection and localization in time, of narrow-band transient signals of unknown waveform in an additive background comprised of quasiharmonic and random components. A traditional method of processing such signals is the spectrogram or short-time power spectrum. Based on an important relationship between the spectrogram and the WD, the application of the WD to the problem is investigated. The smoothed-pseudo-WD is shown to provide superior time and frequency resolutions of the received signal. Based on monitoring the received power in localized regions of the WD spectrum, a detector that achieves good time localization of the transient waveform and is relatively insensitive to changes in signal duration and frequency is obtained. The signal-to-noise ratio of the proposed detection statistic is derived for a transient signal in additive Gaussian noise and its dependence on various signal and detector parameters discussed.

: The report proposes a design of an adaptive receiver for the detection and estimation of nearly periodic signals in additive Gaussian noise. A nearly periodic signal is defined to be a sample function of a Gaussian random process which can be divided into equal length intervals, called periods, in such a manner that the correlation between periods decreases exponentially with their separation. The receiver computes a low signal-to-noise ratio conditional likelihood ratio from which the observer must make decisions. The likelihood ratio is conditional because the receiver estimates any unknown parameters necessary for the computation of the true likelihood ratio. Thus the receiver can only compute a likelihood ratio conditioned upon these estimates being the true values of the unknown parameters. The receiver consists of pre- emphasis filters followed by a comb filter, energy detector, and weighted summation. A theoretical evaluation of the receiver, in terms of ROC curves, is made for the special case of nearly periodic signals with statistically independent equal-strength harmonics in white noise of known power.

In this paper, we put forward a computationally efficient algorithm to estimate the frequency and complex amplitude of a real sinusoidal signal in additive Gaussian noise. The novel method extends an iterative frequency estimator for single complex exponentials that is based on interpolation on Fourier coefficients to the real case by incorporating an iterative leakage subtraction strategy. Simulation results are presented to verify that the proposed algorithm can obtain more accurate estimation than both time and frequency domain parameter estimators in the literature, and the estimation variance of the method sits on the Cramer-Rao lower bound with only a few iterations required.

The linear prewhitening matched filter (PWMF) is the optimal decision function for discriminating exactly-specified signals in additive Gaussian noise. When the signals are less well specified the optimal decision function contains higher than linear terms in the data. Several examples of detection and discrimination tasks are presented for which only the linear term is required and other examples for which higher order terms are necessary. Even in the nonlinear case decision functions can often be approximated by linear operations on the data followed by logical operations. This combination of linear weights plus logic operations is typical of neural network models, which are thought to be elementary models of human processing mechanisms. A number of experiments suggest that there is no decrease in performance of humans for some complex tasks that ideally require such nonlinear operations. However, there are other such tasks where human performance is degraded and this appears to be due more to the complexity of the task and the nature of the correlations in the image than the order of the task. This suggests applications in diagnostic imaging where it might be advantageous for a machine viewer to substitute or work in conjunction with the human observer.

: Recent results on detection of random nonGaussian signals in additive Gaussian noise are discussed. A discrete-time approximation to a likelihood ratio detection algorithm is discussed, and preliminary results are presented for a computational evaluation of the performance for this approximation.

This paper derives the performance of the joint radar and communication system (JRCS). Firstly, for the radar application of the JRCS, we derive the mutual information (MI) between target and received signal in additive Gaussian noise. For communication, we derive the communication data rate that the JRCS can achieve. Then, under partial coexistence between radar and communication bands, the MI and communication data rate for the JRCS are derived. In terms of the overlap between the received radar and communication signals, the successive interference cancellation (SIC) scheme is adopted. Finally, the provided numerical results demonstrate the performances for the JRCS.

One approach to the analysis of the high signal-to-noise ratio (SNR) capacity of noncoherent wireless communication systems is to ignore the noise component of the received signal in the computation of its differential entropy. In this paper, we consider the error incurred by this approximation when the transmitter and the receiver have one antenna each and when the noise has a Gaussian distribution. We consider the complex and real cases, and we show that when the probability density function (pdf) of the signal component of the received signal is piecewise differentiable, the approximation error decays as 1/SNR, which tightens the available result that the error decays as o(1). In addition, we consider the special instance in which the signal component of the received signal corresponds to a signal transmitted over a channel with a Gaussian fading coefficient. For that case, we provide explicit expressions for the first nonconstant term of the Taylor expansion of the differential entropy, and we invoke Schwartz's inequality to obtain an efficiently computable bound on it. Our results are supported by numerical examples.

Data collected via seismic and infrasound array deployments are leveraged in the geosciences to detect and characterize a myriad of natural and anthropogenic sources. These deployments consist of numerous sensors placed in a predetermined configuration to amplify signal strength and improve the efficacy of array processing techniques used to measure signal directionality and waveform coherence. High-fidelity feature extraction is often predicated on interstation distance as well as the frequency content and wavelength of an incident signal. Numerous array processing softwares analyze data in sequential frequency bands to obtain a more detailed characterization of a signal. However, current algorithms are limited in their ability to determine optimal array configuration for each band. We introduce an open-source Python code, called Cardinal, to process seismic and infrasound array data in discretized time–frequency space with the option of applying an adaptive array design to determine optimal subarray configuration for each frequency band. To reduce computational time, the array processing step can be run in parallel using multithreading. Moreover, the software has the capability to aggregate array processing results from different time–frequency pixels to produce separate sets of detections, or families, with added utility via the application of an adaptive semblance threshold, which aids in isolating signals-of-interest from coherent background noise. Upon appropriate configuration, Cardinal exhibits the potential to combine distinct seismic and infrasound phases into separate families.