 Solution

The volume alone does not determine a unique change of elevation. A local isostatic response to an added surface layer depends on its thickness and density, so the Deccan Traps' footprint must be specified or retained as a parameter.

Let the deposited basalt have uniform thickness $H=V/A$ and density ${\rho }_b$, where $V=10^{15}{\mathrm{m}}^3$ and $A$ is the loaded area. Let $w$ be downward displacement of the original land surface, and let ${\rho }_m$ be the density of displaced mantle. Assume a subaerial, laterally extensive deposit, negligible air density, unchanged thickness and density of the pre-existing crust, and complete local isostatic adjustment. Take the magma supply to be external to the local compensation column, and neglect cooling, erosion, magmatic underplating and any deep thermal or source-depletion anomaly.

Choose a compensation depth beneath all the displaced interfaces. The added column mass per unit area is ${\rho }_{b}H$. Downward translation of the pre-existing crust replaces a thickness $w$ of mantle within the column, reducing its mass by ${\rho }_{m}w$. Equality of hydrostatic pressure at the compensation depth therefore givesThe newly formed top of the basalt stands $H$ above the depressed original surface. Consequently the change of the region's top-surface elevation isThus the original substrate subsides, but the top of the deposit rises if ${\rho }_b<{\rho }_m$. Distinguishing these two surfaces prevents reporting the large downward displacement as the change in plateau elevation.

For an explicitly illustrative choice $A=10^{12}{\mathrm{m}}^2$, ${\rho }_b=2900\mathrm{kg}{\mathrm{m}}^{-3}$ and ${\rho }_m=3300\mathrm{kg}{\mathrm{m}}^{-3}$, the deposit is 1000 m thick, the substrate subsides about 879 m, and the final surface rises about 121 m. This is a conditional estimate, not a number fixed by the paper. Half that footprint would double all three heights under the same assumptions. For a nonuniform deposit, local isostasy gives the same relation pointwise with $H(x,y)$; the total volume still cannot specify a unique local elevation.