The Mogi (1958) model treats an inflating magma reservoir as a small pressurized sphere of radius a at depth d inside an elastic half-space — the same closed-form equations real observatories fit to GPS and InSAR ground-deformation data. This 2D view renders the two instruments observatories actually read: a plan-view interferogram (concentric colour fringes, each one a fixed 2.8 cm of line-of-sight motion, wrapping like real InSAR phase) and a radial cross-section profile through the summit.
ΔV = a³·ΔP / μ (effective volume change)
u_z(r) = (3ΔV / 4π) · d / (d² + r²)^(3/2) [uplift]
u_r(r) = (3ΔV / 4π) · r / (d² + r²)^(3/2) [radial, horizontal]
- Depth d — shallower sources produce tighter, more closely-spaced fringes and a sharper cross-section bulge; deeper sources spread fringes wider and flatten the profile.
- Reservoir radius a — sets the source's elastic stiffness and the critical volume at which surrounding rock is assumed to fail and erupt.
- Influx rate — magma supply rate from depth; ΔV integrates this rate over time until an eruption releases it.
- r of max u_r — the horizontal (radial) displacement is not largest at the summit; it peaks at exactly r = d/√2, a classical analytic result of this model (verified numerically for this build — see report). The live readout tracks that ring on the plan view.
- Trigger Eruption — ΔP collapses rapidly; fringes race outward as the dome deflates and the profile subsides, exactly as GPS/InSAR record post-eruptive subsidence at real volcanoes (e.g. Yellowstone, Campi Flegrei, Kīlauea).
Unlike a 3D terrain mesh, both panels here are computed directly from the analytic displacement field — the fringe map is coloured by wrapping u_z into a fixed-wavelength cycle (the real InSAR technique), and the profile plots u_z(r) and u_r(r) as literal curves rather than mesh height.