Foundations

A tectonic earthquake involves sudden slip on a fault and release of part of the surrounding rock's stored elastic energy, including energy radiated as seismic waves.

In a compression-positive Coulomb model for a closed fault, effective normal traction is σ′_n=σ_n−p.

For a planar shear-fault source with uniform shear modulus μ and consistently directed slip magnitude D, scalar moment is M₀=μAD̄, equivalently μ∫_A D dA.

For a homogeneous isotropic linearly elastic solid of densityρ, P waves have vP²=(λ+2μ)/ρ=(K+4μ/3)/ρ and S waves have vS²=μ/ρ.

In a high-frequency ray model, travel time along a pathγ is t=∫_γ ds/v.

For source and receiver on a flat surface a horizontal distance x apart above a uniform layer of depth h and speed v₁, direct travel time is x/v₁ and reflection from its horizontal base has time 2√(h²+x²/4)/v₁.

A gravity anomaly is an observed/modelled field difference relative to a stated reference and correction convention.

For an infinite line parallel to y at horizontal coordinate x₀ and depth b>0 with signed excess mass per lengthm_l, its downward surface anomaly is Δg=2Gm_lb/[(x−x₀)²+b²].

Different subsurface density structures can yield the same external gravitational field.

For stationary homogeneous material with constant positiveρ,c_p andk, no internal heat production and one-dimensional conduction, ρc_pT_t=kT_zz, with thermal diffusivityκ=k/(ρc_p).

With depthz positive downward, constant conductivityk and uniform volumetric heat productionH, a stationary conductive slab obeys kT_zz+H=0.

For z>0 initially at uniform T_i and a surface held at T_s for t>0, constant-κ conduction gives T(z,t)=T_s+(T_i−T_s)erf[z/(2√(κt))].

Passive sensors receive externally supplied or naturally emitted radiation; active radar/lidar supplies illumination and measures its return.

For the specified Landsat Level-1 product, metadata supplies band-specific rescaling L_λ=M_LQ_cal+A_L and uncorrected planetary reflectanceρ′=M_ρQ_cal+A_ρ.

A Landsat thermal-band TOA radianceL_λ>0 maps to product brightness temperature T_B=K₂/ln(K₁/L_λ+1), using its band-specific metadata constants.

For compatible near-infrared and red surface-reflectance bands, NDVI=(ρ_NIR−ρ_R)/(ρ_NIR+ρ_R).

For a monostatic pulse with corrected round-trip delayτ and constant propagation group speedv_g, one-way slant range is R=v_gτ/2.

Ocean-colour band-ratio models estimate near-surface chlorophyll-a from water-leaving remote-sensing reflectances.

LetΣ be the symmetric compression-positive stress tensor, the negative of tensile-positive Cauchy stress.

In an Andersonian idealization, one principal stress axis is vertical and two are horizontal.

In a quasistatic elastic fracture model, crack extension releases mechanical potential energy at rateG=−∂Π/∂A per projected crack area.

A weak-field linear isotropic rock model separates magnetizationM=M_r+χH into retained remanence and an induced response to localH.

An inverse solution is a posterior distribution conditioned on observations, a forward operator, likelihood and prior.

Ensemble assimilation estimates forecast-error covariances from model realizations and updates the state using observations and their error model.

Fingerprinting compares observed patterns with simulated responses to specified forcing and internal-variability covariance.

TEOS-10 seawater calculations distinguish Absolute Salinity, Conservative Temperature and sea pressure.

Mixing waters can change density through nonlinear temperature/salinity dependence even when each source has the same density at the reference pressure.

A mineral assemblage at specified pressure, temperature and bulk composition minimizes Gibbs energy subject to component balance.

Combining mineral phase fractions with elastic properties can predict seismic velocities, but equilibrium databases and shear-modulus data have separate calibration domains.

Ice thermal models combine advection, diffusion and deformation heating; enthalpy tracks sensible heat and liquid fraction at the pressure-melting boundary.