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Time–Frequency Analysis of Seismic Reflection Data
Published in Antonia Papandreou-Suppappola, Applications in Time-Frequency Signal Processing, 2018
Steeghs Philippe, Richard G. Baraniuk, Jan Erik Odegard
Figure 8.2 illustrates the composition of a generic seismic TF sequence. We have constructed an impedance model of the subsurface that consists of three types of components. The first component is a rectangular boxcar function — Figure 8.2(a) — of unit impedance. This homogeneous impedance is perturbed with a purely harmonic impedance variation (a cosine, Figure 8.2[b]). We have introduced a frequency change at t = 0.5 sec. The third component is a perturbation with a linearly increasing frequency (a linear chirp, Figure 8.2[c]). The overall, superposition impedance model is shown in Figure 8.2(d). The simplest model for the seismic response u(t) to this impedance variation is the reflectivity of the impedance function r(t) convolved with a seismic source signal s(t): () u(t)=∫−∞+∞r(t′)s(t−t′)dt′
Sampling and Reconstruction of Functions—Part II
Published in Eleanor Chu, Discrete and Continuous Fourier Transforms, 2008
Example 5.3 (Figure 5.4 Time-Limited Pair) We consider the rectangular pulse function (also known as the square wave or boxcar function), which is assumed to have a pulse width of 2t0 and it is de ned as xrect(t)={12t0,fort∈[−t0,t0];0,for|t|>t0.
Spectroscopy and Spectroscopic Imaging
Published in Christakis Constantinides, Magnetic Resonance Imaging, 2016
Loss of data during tD leads to significant baseline artifacts. In effect, if the first few points in the acquisition of data are missed, this can be equivalent to neglecting the DC term in the Fourier series representation of either the FID or the spatially encoded data array. Setting the missing data to zero is equivalent to multiplying the true complete time domain signal by a boxcar function g(t): () strunc=s(t)⋅g(t)
Perturbation approach for damage localization in beam-type structures: analytical, experimental and numerical exposition
Published in Journal of Structural Integrity and Maintenance, 2023
Md. Arif Faridi, Koushik Roy, Vaibhav Singhal
In this paper, a novel VBDD mathematical formulation is proposed for damage localization in beam-type structures. The first-order perturbation in stiffness properties of a beam is ideated and its effect on the eigen properties is used for damage localization. The present study aims in formulating the mathematical basis behind using difference in mode shape curvature (DMSC), accomplished utilizing the analytical mode shape of the beam. Here, a mathematical formulation of open crack termed as normalized boxcar function (NBF) is developed. NBF and analytical mode shape formulae are substituted in the expression of DMSC, to find a novel mathematical formulation. This study put forward an effective boxcar filtration technique to supplement the DMSC in damage localization. An experiment is further executed on a miniature model of a beam to assess the analytical results under the real experimental conditions with limited sensors (using two-point roving techniques). Finally, for the completeness of the study, the numerical validation is conducted in SAP2000 on the miniature model of beam to support the analytical findings. The simulation studies are performed with various boundary conditions to check the robustness of the analytical findings.
An improved inversion method for determining two-dimensional mass distributions of non-refractory materials on refractory black carbon
Published in Aerosol Science and Technology, 2021
A. Naseri, T. A. Sipkens, S. N. Rogak, J. S. Olfert
This expression now forms the basis for a set of linear equations, where the linear operator is defined as and which results in the system b = Ax + ε, where b ∈ ℝni×1, A ∈ ℝni×nj, x ∈ ℝnj×1, ε ∈ ℝni×1, and ni and nj are the number of data and reconstruction points, respectively. Broda et al. (2018) used a simplified version of the problem in which (mrBChigh)j = mihigh, and (mrBClow)j=milow, such that one reconstructs using an identical set of bins or grid points as was used to bin the original SP2 data. In this case, the boxcar function in Aij is also uniform over each element and can be removed from the expression, allowing for simplification of Equation (13) to
Determination of intervention programs for multiple municipal infrastructure networks: considering network operator and service costs
Published in Sustainable and Resilient Infrastructure, 2020
Clemens Kielhauser, Bryan T. Adey
where is the binary variable indicating whether an intervention is executed on object at time phase ; is the Heaviside function, defined as ; and is the Boxcar function, defined as With this equation, the conductivity of the object and hence the object state can be described formally.