arXiv subjects

Chahana Nagesh

Publications and source records attributed to Chahana Nagesh.

3 recordsLinked to original sources

arXiv2026

Higher--order gradient modeling of velocity dispersion in tight sandstones with implications to agroseismology

Modeling frequency-dependent shear-wave propagation in fluid-saturated porous rocks remains a challenge today because existing poroelastic theories generally rely a large number of constitutive parameters that are difficult to constrain from seismic observations. We propose a dynamically consistent higher-gradient continuum model that reproduces shear-wave dispersion using only three effective material parameters: the macroscopic shear velocity and two characteristic length scales governing higher-order elasticity and gradient inertia. We validate the model against laboratory measurements of glycerin-saturated tight sandstone over a wide range of effective pressures. The inversion reproduces the measured shear-wave dispersion, including positive, weak and negative dispersion regimes. We finally discuss the relationship between the proposed formulation and micropolar theories, the computational challenges and the potential of reduced-order generalized continuum models for seismic exploration, reservoir characterization, and emerging agroseismology applications.

physics.geo-ph↗

arXiv2026

On the Fréchet interaction density of certain wave equations

We extend the adjoint method to complex-valued PDEs and introduce the \emph{Fréchet interaction density}, as the most fundamental interaction from which Fréchet sensitivity kernels can be derived. We apply this framework to four representative equations: two real-valued PDEs (the second-order wave equation and the Euler--Bernoulli beam equation) and two complex-valued PDEs (the complex transport equation and the Schrödinger equation with zero potential). We compute and analyze the Fréchet interaction densities for all four PDEs and show that the interaction shows consistent structure, with a waveform that depends on the initial conditions. For the Schrödinger equation, when the adjoint field is chosen as the complex conjugate of the forward wavefunction, the interaction density reduces algebraically to the Born probability density. Our results establish a unified approach to sensitivity analysis for real- and complex-valued PDEs.

physics.geo-ph↗

arXiv2026

From complex-step differentiation to a general reconstruction framework

The complex-step method is traditionally derived from the Taylor expansion of an analytic function and is widely used as a numerical technique for derivative approximation. We present an alternative formulation based on the Cauchy--Riemann equations and show that the classical complex-step relation arises naturally from the harmonic structure of holomorphic functions. In particular, the complex-step method admits two complementary harmonic interpretations: as a Cauchy problem, in which the derivative is identified with the normal datum of the imaginary component on the real axis, and as a reconstruction problem in a strip, in which the finite imaginary perturbation provides the upper-boundary data. The latter formulation leads explicitly to the strip Poisson and conjugate Poisson kernels and their derivatives. A related harmonic reconstruction framework in the upper half-plane leads to the Poisson, conjugate Poisson, and Cauchy kernels as elementary reconstruction operators for harmonic and holomorphic functions. Extending this reconstruction from ordinary boundary functions to finite measures yields the classical Stieltjes transform and its inversion formula. The same measure-theoretic structure appears in spectral theory, where scalar matrix elements of the resolvent are Stieltjes transforms of the associated spectral measures. These results establish a common complex-analytic structure connecting complex-step differentiation, harmonic reconstruction, Stieltjes inversion, and spectral reconstruction, while distinguishing the boundary-value problems through which the corresponding information is recovered.

math.NA↗