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Magnetometry is used to detect ferrous objects at various scales, but detecting small-size, compact sources that produce small-amplitude anomalies in the shallow subsurface remains challenging. Magnetic anomalies are often approximated as dipoles or volumes of dipoles that can be located, and their source parameters (burial depth, magnetization direction, magnetic susceptibility, etc.) are characterized using scalar or vector magnetometers. Both types of magnetometers are affected by space weather and cultural noise sources that map temporal variations into spatial variations across a survey area. Vector magnetometers provide more information about detected bodies at the cost of extreme sensitivity to orientation, which cannot be reliably measured in the field. Magnetic gradiometry addresses the problem of temporal-to-spatial mapping and reduces distant noise sources, but the heading error challenges remain, motivating the need for magnetic gradient tensor (MGT) invariants that are relatively insensitive to rotation. Here, we show that the finite size of magnetic gradiometers compared to the lengthscales of magnetic anomalies due to small buried objects affects the properties of the gradient tensor, including its symmetry and invariants. This renders traditional assumptions of magnetic gradiometry largely inappropriate for detecting and characterizing small-size anomalies. We then show how the properties of the finite-difference MGT and its invariants can be leveraged to map these small sources in the shallow critical zone, such as unexploded ordnance (UXO), landmines, and explosive remnants of war (ERW), using both synthetic and field data obtained with a triaxial magnetic gradiometer (TetraMag). 

Abstract The critical zone (CZ) is the region of the Earth’s surface that extends from the bottom of the weathered bedrock to the tree canopy and is important because of its ability to store water and support ecosystems. A growing number of studies use active source shallow seismic refraction to explore and define the size and structure of the CZ across landscapes. However, measurement uncertainty and model resolution at depth are generally not evaluated, which makes the identification and interpretation of CZ features inconclusive. To reliably resolve seismic velocity with depth, we implement a Transdimensional Hierarchical Bayesian (THB) framework with reversible‐jump Markov Chain Monte Carlo to generate samples from the posterior distribution of velocity structures. We also perform 2D synthetic tests to explore how well THB traveltime inversion can resolve different subsurface velocity structures. We find that THB recovers both sharp changes in velocity as well as gradual velocity increases with depth. Furthermore, we explore the velocity structure in a series of ridge‐valley systems in northern California. The posterior velocity model shows an increasing thickness of low velocity material from channels to ridgetops along a transect parallel to bedding strike, implying a deeper weathering zone below ridgetops and hillslopes than below channels. The THB method enhances the ability to reliably image CZ structure, and the model uncertainty estimates it yields provides an objective way to interpret deep CZ structure. The method can be applied across other near‐surface studies, especially in the presence of significant surface topography.