The formalism of the dirty Image is developed from the fundamental Fourier Transform relationship between observed visibility and sky brightness. The convolutional gridding used to interpolate the irregularly sampled data onto a rectangular grid is examined, including aliasing of sources outside the primary field of view. Furthermore, how the visibility samples collected by an interferometric array can be used to produce a high quality image of the sky is described. These non-linear methods try to create estimates of the visibility function at positions in the Fourier plane where it is not measured. The various algorithms such as CLEAN, MEM, etc. are mentioned briefly.
Keywords: Dirty Image, Visibility, Sky Brightness, Convolutional gridding, Aliasing.
Data from a three-component broadband seismometer at ~270m deep show the lowest background noise levels recorded in the entire Global Seismographic Network (GSN) at frequencies above 2 Hz. However, with this reduction in noise, there is also an observed reduction in signal levels in some frequency bands. Modeling of transfer functions based on the velocity structure of the ice/firn near the surface suggest a spectral hole due to destructive interference from the free surface reflections will occur between 2 and 2.5 Hz. Analysis of body wave phases from actual earthquakes measured both at the surface and depth show that the transfer function between these two waveforms also show similar spectral scalloping. Results of this effort will be shared with the scientific community to ensure they are aware of the spectral bands were signal to noise ratios may be maximized for this quiet location.
Many ranging and detection applications where a pulse is sent out and echo received and processed can benefit for the technique of pulse compression. It allows for the simultaneous reduction of peak power and increase in range accuracy. Several generalized techniques for pulse compression will be explained. The technique of FM modulation or chirping will be examined in detail. Applications of pulse compression to radar will be discussed.
Non-uniform sample distribution presents a challenge to standard Fourier analysis. There are several work-a-rounds, e.g., employing interpolation to fill in the missing data points. For power spectral analysis however, Lomb developed a method for directly analyzing non-uniform samples. This method weights data on a per point basis by employing reverse interpolationrather than a per time interval basis and can thus return superior results to FFT methods. The resulting power peaks can be evaluated against a significance level that isbased on the chosen number of independent scanning frequencies. With adequate random sampling, this method can expose a cyclic pattern from data containing a single cycle, thus overcoming the Nyquist limitation found in uniform analysis. The Lomb method can be approximated with any desired precision using the fast processing FFT algorithms with an operation count of order N log N where N is the number of samples.
References:
Numerical Recipes in Fortran 77 http://www.library.cornell.edu/nr/bookfpdf/f13-8.pdf
PD Dr. Daniel Potts http://www.math.mu-luebeck.de/potts/nfft/
NASA Astrophysics Data System (ADS) http://adsbit.harvard.edu/cgi-bin/nph-iarticle_query?1989ApJ%2E%2E%2E338%2E%2E277P
The Wavelet Transform
Christian Lucero
The Fourier Transform is useful in giving the pectral content of a signal, but does not include any information about the time in which these spectral components exist. Typically, this information is not required when the signal is stationary (where the frequency content does not change in time). The wavelet Transform is a useful transform when the time-localization of the spctral components is needed (ususally in non-stationary signals), thus giving a time-frequency representation of the signal. This presentation will explore the properties of the Wavelet transform as well as a simple algorithm for computing the wavelet tranform for a time series.
Wavelet Shaping in Seismic Reflection Imaging
Garrett Kramer
Often in seismic reflection the stacked section has "ringy" and blurred characteristics that considerably limit resolution. Deconvolution allows us to image the source wavelet, which permits sharpening and brings out finer details in the seismic section. Several generalized techniques such as minimum phase and zero phase, will be explained and applied to a data set.