Fractional cable models for spiny neuronal dendrites

Cable equations with fractional order temporal operators are introduced to model electrotonic properties of spiny neuronal dendrites. These equations are derived from Nernst-Planck equations with fractional order operators to model the anomalous subdiffusion that arises from trapping properties of dendritic spines. The fractional cable models predict that postsynaptic potentials propagating along dendrites with larger spine densities can arrive at the soma faster and be sustained at higher levels over longer times. Calibration and validation of the models should provide new insight into the functional implications of altered neuronal spine densities, a hallmark of normal aging and many neurodegenerative disorders

Free-surface Multiples and full-waveform inversion spectral resolution

Low frequencies play a crucial role in the convergence of full-waveform inversion to the correct model in most of its current implementations. However, the lower the frequencies, the bigger are the amplitudes of the surface waves, causing the inversion to be driven by the latter. If they are not blanked out or removed, this may lead to convergence problems. To analyze this situation, we consider the simplest case where surface waves are present: an acoustic layer over a halfspace. We earlier analyzed the contributions of various wave types to the wavenumber spectrum of a velocity perturbation above a reflecting halfspace, without a free surface. Here, we extend this spectral sensitivity analysis to the case with a free surface, which generates multiples and ghosts. In this setting, the surface guided P-waves can be considered as a superposition of free-surface multiples. Our analysis shows that the conditioning of the linearized inverse problem, which is solved at each iteration of full-waveform inversion, becomes worse when multiples are taken into account. At the same time the inclusion of multiples increases the sensitivity to some low wavenumbers in the model spectrum, which should be beneficial for full-waveform inversion once a suitable preconditioner has been found.Geoscience & EngineeringCivil Engineering and Geoscience

FWI sensitivity analysis in the presence of free-surface multiples

It is generally believed that full waveform inversion needs very low frequencies in the data to avoid convergence to a local minimum, which would lead to an incorrect velocity model. Often, low frequencies are not present in the data. Then, a kinematically accurate initial model is required that contains the long-wavelength structures that cannot be reconstructed from the data. Mora (1989), however, showed that reflector below the target area may help in recovering some of the longwavelength information. Here, we extend his analysis to the case with a free surface, generating multiples. At first sight, multiples should improve the sensitivity and resolution. Indeed, our study shows that sensitivity becomes higher in some parts of the model spectrum when multiples are included. If we consider thin layers and only first-order multiples, then the sensitivity is improved. However, when the layer is thick enough to allow for normal modes of higher orders, areas of high sensitivity appear and this leads to a poorer conditioning of the inverse problem in the wavenumber domain.Geoscience & EngineeringCivil Engineering and Geoscience

Estimating a continuous p-wave velocity profile with constant squared-slowness gradient models from seismic field data

We inverted seismic field data for a continuous, laterally invariant P-wave velocity profile. Instead of the usual approach that involves horizontal layers with piecewise constant densities and velocities, we consider models of one or two layers with a constant gradient of the squared slowness above a homogeneous halfspace. With a single layer above a halfspace, there are three parameters. With two layers, there are five. We solve the inverse problem by a direct grid search over a wide range of parameters. The results were compared to that of a piecewise-constant multi-layer inversion result. In the single-layer case, either the shallow or the deeper part of the model would match the multi-layer case, depending on which modes of the surface waves were selected. With two layers, a considerably better agreement is obtained over a larger depth range. Our method is limited to cases with a small Vs/Vp-ratio but has only 5 parameters. It could be a useful alternative to piecewise-constant multi-layer inversion, in particular if continuous P-velocity profiles are sought. These are sometimes better suited as a starting model for full waveform inversion than models with many discontinuities.Geoscience and EngineeringCivil Engineering and Geoscience

Coupling of Elastic Isotropic Medium Parameters in Iterative Linearized Inversion

An elastic isotropic medium is described with three parameters. In seismic migration the perturbation of one elastic parameter affects the images of all the three, which means that these parameters are coupled. For an effective quantitative reconstruction of the true elastic medium reflectivity one can apply an iterative linearized migration/inversion, where minimization of the misfit functional is done by the conjugate gradient method. The final result of the iterative approach can be obtained directly by Newton’s method, using the pseudo-inverse Hessian matrix. Calculation of this matrix for a realistic model is an extremely resource-intensive problem, but for a model of a scatterer in a homogeneous elastic medium it is quite feasible. This paper presents the numerical results of elastic linearized inversion for this simple model, calculated both with iterative approach and Newton’s method. Experiments show that in the both cases the elastic parameters have coupled weaker than in the case of migration. The iterative approach allows achieving acceptable quality of the inversion, but requires a large number of iterations. For faster convergence it is necessary to use the preconditioned conjugate gradient method. The optimal preconditioning will improve the convergence of the method as well as the quality of inversion.Geoscience & EngineeringCivil Engineering and Geoscience

Estimation of the P-wave Velocity Profile of Elastic Real Data Based on Surface Wave Inversion

Recently, we proposed an analytical approach to invert for a smoothly varying near-surface P-wave velocity profile that has a squared slowness linearly decreasing with depth. The exact solution for such a velocity profile in the acoustic approximation can be expressed in terms of Airy functions and leads to a dispersion equation. The method was successfully applied to synthetic elastic data with small Vs/Vp-ratio. Here, we apply the method to land data. The result agrees with that of multi-layered inversion, confirming its potential to provide an initial P-wave velocity model for acoustic full waveform inversion. Compared to multi-layered inversion, the method is simpler to use and produces a smooth model characterized by three parameters. In some cases, having a smooth rather than a blocky initial model for full waveform inversion is more appropriate.Geoscience & EngineeringCivil Engineering and Geoscience

Surface wave inversion for a p-wave velocity profile: Estimation of the squared slowness gradient

Surface waves can be used to obtain a near-surface shear wave profile. The inverse problem is usually solved for the locally 1-D problem of a set of homogeneous horizontal elastic layers. The output is a set of shear velocity values for each layer in the profile. P-wave velocity profile can be estimated if higher modes and P-guided waves are used in the inversion scheme. Here, we use an exact acoustic solution to invert for the P-velocity profile in an elastic model with a decreasing constant vertical gradient of the squared P-wave slowness, bounded by a free surface on the top and a homogeneous halfspace at the bottom. The exact acoustic solution can be expressed in Airy functions and leads to a dispersion equation. We can invert several modes of the dispersion equation for the single gradient parameter of the squared P-wave slowness from elastic data. As a first test case, we invert for the P-wave velocity profile of synthetic 2-D isotropic elastic data with a small Vs/Vp-ratio, using the first two dispersive P-wave modes. The method does not require any picking and should be able to provide an initial model for full waveform inversion when applied to real data.Geoscience & EngineeringCivil Engineering and Geoscience

On the Contribution of Head Waves to Full Waveform Inversion

Full waveform inversion suffers from local minima, due to a lack of low frequencies in the data. A reflector below the zone of interest may, however, help in recovering the long-wavelength components of a velocity perturbation, as demonstrated in a paper by Mora. With the Born approximation for the perturbation in a reference model consisting of two homogeneous isotropic acoustic halfspaces, analytic expressions can be found that describe the spatial spectrum of the recorded seismic signal as a function of the spatial spectrum of the inhomogeneity. We study this spectrum in more detail by separately considering direct, reflected and head waves. Taking the reflection coefficient of the deeper reflector into account, we obtain sensitivity estimates for each of these types of waves. Although the head waves have a relatively small contribution to the reconstruction of the velocity perturbation, compared to the other waves, they contain reliable long-wavelength information that can be beneficial for full waveform inversion.Geoscience & EngineeringCivil Engineering and Geoscience