Layered Column Density Averager Calculator

The thickness-weighted mean density of a stratified column is simply its mass per unit area divided by its total height. That single number is the one that matters for three separate calculations: the Bouguer slab correction in gravity reduction, the mass balance that determines whether a column is in isostatic equilibrium, and the lithostatic pressure at the base of the section. Thicker layers dominate the result, which is physically correct because they hold more mass per unit area. The harmonic mean reported alongside it answers a different question. Because vertical travel time through a stack is the sum of thickness divided by velocity, quantities that combine reciprocally through layers, such as slowness or series thermal resistance, require harmonic rather than arithmetic averaging. The harmonic mean is always less than or equal to the arithmetic mean and the two coincide only when all layers share one density. Realistic inputs span a narrow but consequential range. Sedimentary cover runs about 2200 to 2700 kg/m3 depending on compaction, upper continental crust of granite and gneiss sits near 2700, lower continental crust of granulite and mafic rock reaches 2900 to 3100, oceanic crust of basalt and gabbro is about 2900 to 3000, and uppermost mantle peridotite is close to 3300. The pressure figures here assume a constant gravitational acceleration of 9.81 m/s2, which is accurate to better than one percent through the crust and uppermost mantle but degrades deeper, since g rises to roughly 10.7 m/s2 in the lower mantle. The model also treats each layer as internally homogeneous and laterally infinite, ignoring self-compression, porosity loss with burial, and lateral density variation.

Layered density models of the Earth emerged from seismology rather than from direct sampling. Emil Wiechert argued in 1897 that the mean density of the planet, roughly 5.5 g/cm3 against about 2.7 for surface rock, required a dense iron interior beneath a silicate shell. Beno Gutenberg fixed the depth of the core-mantle boundary near 2900 km in 1913 by tracing the shadow zone in P-wave arrivals. Converting travel-time curves into a density profile required a physical bridge, and the Adams-Williamson equation, published in 1923, supplied it by relating the density gradient in a self-compressed homogeneous region to the seismic velocities. Harold Jeffreys and Keith Edward Bullen built the first comprehensive layered models through the 1930s and 1940s. Bullen divided the Earth into regions labeled A through G in 1942, each assigned a density-depth relation, and the scheme dominated for a generation. The modern reference is the Preliminary Reference Earth Model, published by Adam Dziewonski and Don Anderson in 1981, which specifies density and velocity as functions of radius and remains the standard against which regional columns are compared. For shallow work, empirical velocity-density relations do the practical job of populating layers: the Nafe-Drake curve from the late 1950s for marine sediments and crust, Birch law relating compressional velocity to density published in 1961, and the Gardner relation of 1974 widely used in petroleum exploration.