========================================================================== ZOMBIE VOLCANO -- CLUB MOGI-SOURCE DEFORMATION MODEL Anchor: Liu, Kendall, Zhang, Blundy, Pritchard, Hudson & MacQueen (2025) PNAS 122(18), e2420996122 doi:10.1073/pnas.2420996122 This is the club's own simplified model on SYNTHETIC data, NOT the paper's analysis. The paper is a seismology paper. ========================================================================== PART A0 -- THE THREE PROPERTIES OF A MOGI SOURCE ------------------------------------------------ uz(r) = (1-nu) dV d / (pi (r^2+d^2)^1.5) [vertical] ur(r) = (1-nu) dV r / (pi (r^2+d^2)^1.5) [radial] nu = 0.25 P1 sign(uz) = sign(dV) at EVERY radius. A single point source cannot change the sign of the vertical displacement anywhere. Demonstration, one inflating source at d = 20 km, dV = +1e7 m3: r (km) : 0 10 30 60 100 200 500 uz (mm): 5.9683 4.2706 1.0187 0.1887 0.0450 0.0059 0.0004 minimum over that range = 0.000381 mm (sign never flips) P2 ur is maximal at r = d/sqrt(2) = 0.7071 d. For d = 20 km that is r = 14.14 km. Numerical check by brute-force scan: 14.14 km. P3 uz half-width r_half = d * sqrt(2^(2/3) - 1) = 0.7664 d. An uplift bump 70 km across (r_half = 35 km) needs d = 35 / 0.7664 = 45.7 km. Deep. Ratio uz/ur = d/r at every radius, so a single pair of vertical and horizontal measurements already fixes the depth. PART B -- THE SYNTHETIC SOMBRERO (our fabricated 'observations') ---------------------------------------------------------------- radial profile: 61 samples, 0 to 120 km, 2 km spacing seeded noise : Gaussian, sd = 0.20 mm/yr, seed = 20250428 TRUE (hidden) model used to manufacture the data: source 1 depth 20.0 km dV = +2.3457e+07 m3/yr (inflating) source 2 depth 80.0 km dV = -1.0723e+08 m3/yr (deflating) |dV2|/dV1 = 4.57 net dV = -8.3776e+07 m3/yr Shape of the resulting 'observed' profile: peak uplift at r = 0 +9.71 mm/yr zero crossing 24.7 km deepest subsidence -1.97 mm/yr at r = 42 km moat-to-peak amplitude ratio 0.203 still subsiding at r = 120 km -0.38 mm/yr For comparison, the published Uturuncu pattern: roughly 10 mm/yr of central uplift in the 1990s over a dome tens of km wide, ringed by 1 to 4 mm/yr of subsidence reaching past 70 km radius (Pritchard & Simons 2002; Fialko & Pearse 2012; Henderson & Pritchard 2017). Our synthetic is a caricature of that, built so we know the answer. Profile, every 10 km (mm/yr): r(km) : 0 10 20 30 40 50 60 70 80 90 100 110 120 uz : 9.71 6.20 0.84 -0.90 -1.57 -1.62 -1.57 -1.53 -1.54 -0.74 -0.73 -1.04 -0.38 PART C1 -- BEST SINGLE MOGI SOURCE ---------------------------------- grid: 793 depths from 2 to 200 km, volume rate solved exactly at each best-fit depth 12.25 km best-fit volume rate +6.6246e+06 m3/yr (inflating) RMS misfit 1.291 mm/yr variance reduction 81.77 % noise floor (the best any model could do) 0.20 mm/yr misfit / noise floor 6.5x predicted uz at r = 0 +10.54 mm/yr (observed +9.71) predicted uz minimum +0.0110 mm/yr (observed -1.97) sign changes in prediction: 0 sign changes in data : 1 THE MOAT TEST. Take the annulus 40 to 100 km, where the data subside, and compare: observed mean uz there -1.366 mm/yr single-source prediction +0.080 mm/yr mean signed error +1.447 mm/yr RMS error in the moat 1.491 mm/yr Note the single source cannot get the sign right in the moat AND the sign right at the centre. It is not being out-fitted by a better model; it is barred by P1 from making the shape at all. PART C1b -- WHAT THE SINGLE-SOURCE MISFIT LOOKS LIKE ACROSS DEPTH ----------------------------------------------------------------- There is no depth that rescues a single source. RMS misfit versus assumed depth, sampled every 10 km: depth(km) : 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 RMS(mm/yr): 1.343 1.577 2.055 2.376 2.581 2.712 2.799 2.857 2.898 2.926 2.947 2.962 2.973 2.981 2.988 2.993 2.997 3.001 3.003 3.005 flattest value anywhere on the grid: 1.291 mm/yr worst value anywhere on the grid : 3.005 mm/yr Even the best depth leaves a misfit several times the noise. PART C2 -- BEST PAIR OF MOGI SOURCES ------------------------------------ coarse grid 56 x 86 depth pairs, then refined at 50 m shallow source depth 20.80 km dV = +2.5441e+07 m3/yr (inflating) deep source depth 77.40 km dV = -1.0750e+08 m3/yr (deflating) |dV_deep| / dV_shallow 4.23 net volume rate -8.2063e+07 m3/yr RMS misfit 0.171 mm/yr variance reduction 99.68 % misfit / noise floor 0.86x RECOVERY OF THE HIDDEN TRUTH: depth 1 true 20.00 km recovered 20.80 km error +4.0 % depth 2 true 80.00 km recovered 77.40 km error -3.2 % dV 1 true +2.346e+07 recovered +2.544e+07 error +8.5 % dV 2 true -1.072e+08 recovered -1.075e+08 error -0.3 % PART C3 -- HEAD TO HEAD, AND WHY THE MOAT IS THE DIAGNOSTIC ----------------------------------------------------------- model params RMS mm/yr VR % sign flips ---------------------------------------------------------------- one Mogi source 2 1.291 81.77 0 two Mogi sources 4 0.171 99.68 1 the data themselves - - - 1 misfit ratio, one source / two sources : 7.53x residual variance removed by the 2nd source: 98.24 % The sombrero is diagnostic for a reason that has nothing to do with statistics. Adding a second source does not merely fit better; a single source is mathematically incapable of producing the shape. Two numbers make that concrete: moat depth in the data -1.97 mm/yr deepest a single source can go, given it fits the centre: +0.0110 mm/yr The second of those can never be negative while the first is. The gap is not a fitting residual. It is a structural impossibility. PART C4 -- DEPTH READING FROM THE RADIAL DISPLACEMENT ----------------------------------------------------- Horizontal motion is the other half of the measurement, and it carries the depth in a way the vertical does not. shallow source: ur peaks at r = d/sqrt(2) = 14.71 km deep source : ur peaks at r = d/sqrt(2) = 54.73 km combined radial field peaks at r = 14 km, +4.66 mm/yr Vertical and radial, every 20 km (mm/yr): r(km) : 0 20 40 60 80 100 120 uz : 9.75 1.37 -1.63 -1.62 -1.22 -0.86 -0.61 ur : 0.00 4.05 1.10 -0.22 -0.63 -0.70 -0.65 For a single source uz/ur = d/r exactly. Measuring both at one radius fixes the depth with no inversion at all: r = 10 km : uz/ur = 2.0800 -> d = 20.80 km r = 20 km : uz/ur = 1.0400 -> d = 20.80 km r = 40 km : uz/ur = 0.5200 -> d = 20.80 km PART D -- WHAT THE INFLATION RATE WOULD HAVE TO BUILD ----------------------------------------------------- fitted shallow inflation rate dV = 2.5441e+07 m3/yr = 0.02544 km3/yr = 0.806 m3/s published estimate for Uturuncu: about 1 m3/s, 1e-2 km3/yr (Sparks et al. 2008, from 1992-2006 InSAR). Same ballpark, which is the only claim we make for our number. Uturuncu's own long-run magma output rate: edifice 85 km3 over 640 kyr = 1.328e+05 m3/yr RATIO, inflation rate / long-run magma output rate: 192x That ratio is the first thing that should make you suspicious. If the volume change were magma arriving, Uturuncu would currently be supplied 192 times faster than it was during the 640,000 years when it was actually building a mountain. PART D2 -- THE REDUCTIO ----------------------- If the present rate had run steadily since the last eruption (250 ka), the accumulated volume would be: 6.3604e+12 m3 = 6,360 km3 = 75x the entire volcano = 2.5x the Atana ignimbrite (2,500 km3), the largest eruption the Altiplano-Puna has ever produced. Nothing like that is underground. So the rate is not steady, and whatever the ground is doing it has not been doing it for long. Since the uplift was first measured (33 years): accumulated volume change 8.3957e+08 m3 = 0.840 km3 that is 8.4x a VEI 4, 0.84x a VEI 5, 0.084x a VEI 6 PART D2b -- THE NET BUDGET, WHICH POINTS THE WRONG WAY ------------------------------------------------------ The fit does not just find a shallow source filling up. It finds a deep source emptying out, and the deep one is the larger of the two. shallow source +2.5441e+07 m3/yr deep source -1.0750e+08 m3/yr NET -8.2063e+07 m3/yr = -0.0821 km3/yr Over the 33 years of measured unrest the net volume change is -2.71 km3. Negative. Taken at face value the system as a whole is losing volume while its top inflates, which is exactly what you expect if something mobile is moving UP out of a deep reservoir rather than new material arriving from below. No eruptible body is being assembled on that budget. It is being disassembled and redistributed upward. PART D3 -- CENTURIES NEEDED, THE LADDER --------------------------------------- target km3 yr, naive yr, mush yr, long-run ---------------------------------------------------------------------------- VEI 4, a modest dome or flow 0.1 4 16 753 VEI 5, 1991 Pinatubo scale 1.0 39 157 7,529 VEI 6, 1883 Krakatau scale 10.0 393 1,572 75,294 Uturuncu's whole edifice 85.0 3,341 13,364 640,000 Pastos Grandes ignimbrite 500.0 19,653 78,612 3,764,706 Atana ignimbrite 2500.0 98,265 393,060 18,823,529 'naive' : every cubic metre of volume change is eruptible magma. 'mush' : the magma body is 25% melt, so 4 m3 of mush must be reorganised per m3 of eruptible melt extracted. 'long-run' : magma arrives at the rate that actually built the volcano over the last 890,000 years. Headline: assembling one cubic kilometre of eruptible magma at the observed inflation rate, allowing for the 25% melt fraction the paper images, takes 1.6 centuries. At the rate that actually built the mountain it takes 75 centuries. PART D4 -- MASS, NOT VOLUME: WHY FLUID IS THE CHEAP EXPLANATION --------------------------------------------------------------- The same 2.544e+07 m3/yr of volume change costs: as dacitic magma at 2400 kg/m3 6.106e+10 kg/yr = 61.06 Mt/yr as supercritical fluid at 200 kg/m3 5.088e+09 kg/yr = 5.09 Mt/yr as supercritical fluid at 400 kg/m3 1.018e+10 kg/yr = 10.18 Mt/yr mass ratio magma / fluid : 6x to 12x And that is still an overstatement of the fluid's cost, because a compressible fluid can change the volume of a pore network by expanding in place, with no new mass arriving at all. Volume change is not mass change. The Mogi model cannot tell the two apart, and that single blind spot is what forty years of deformation-only interpretation at Uturuncu kept walking into. PART E -- HEADLINE NUMBERS FOR THE ARTICLE AND THE INTERACTIVE PAGE ------------------------------------------------------------------- synthetic peak uplift +9.71 mm/yr synthetic moat minimum -1.97 mm/yr at 42 km zero crossing 24.7 km moat/peak ratio 0.203 ONE SOURCE depth 12.2 km ONE SOURCE dV +6.625e+06 m3/yr ONE SOURCE RMS 1.29 mm/yr ONE SOURCE variance reduction 81.8 % ONE SOURCE min predicted uz +0.0110 mm/yr TWO SOURCE shallow depth 20.8 km TWO SOURCE shallow dV +2.544e+07 m3/yr TWO SOURCE deep depth 77.4 km TWO SOURCE deep dV -1.075e+08 m3/yr TWO SOURCE RMS 0.17 mm/yr TWO SOURCE variance reduction 99.68 % TWO SOURCE |deep|/shallow 4.23 TWO SOURCE net dV -8.206e+07 m3/yr net over 33 yr -2.71 km3 misfit ratio 1-source : 2-source 7.5x inflation / long-run magma supply 192x volume if steady since 250 ka 6,360 km3 = 75x the edifice 33 years of inflation 0.84 km3 centuries for 1 km3 eruptible 1.6 magma:fluid mass ratio 6x to 12x ========================================================================== APPENDIX -- PROFILE TABLE FOR THE FIGURES (mm/yr) ------------------------------------------------- r(km) observed 1-source 2-source radial 0 9.710 10.539 9.755 0.000 4 8.923 9.053 9.027 2.336 8 7.336 6.186 7.198 3.954 12 5.127 3.842 4.989 4.623 16 2.948 2.368 2.967 4.546 20 0.838 1.502 1.370 4.051 24 0.219 0.990 0.211 3.393 28 -0.735 0.679 -0.585 2.719 32 -0.902 0.482 -1.109 2.098 36 -1.462 0.352 -1.436 1.557 40 -1.571 0.265 -1.625 1.099 44 -1.751 0.203 -1.719 0.718 48 -1.595 0.159 -1.747 0.406 52 -1.503 0.127 -1.731 0.152 56 -1.614 0.103 -1.686 -0.053 60 -1.575 0.084 -1.622 -0.216 64 -1.377 0.070 -1.546 -0.346 68 -1.702 0.059 -1.465 -0.447 72 -1.579 0.050 -1.381 -0.525 76 -1.187 0.042 -1.298 -0.585 80 -1.543 0.037 -1.217 -0.628 84 -1.226 0.032 -1.138 -0.659 88 -1.062 0.028 -1.063 -0.680 92 -1.020 0.024 -0.992 -0.693 96 -1.014 0.021 -0.926 -0.699 100 -0.732 0.019 -0.864 -0.699 104 -0.818 0.017 -0.806 -0.696 108 -0.631 0.015 -0.752 -0.689 112 -0.753 0.014 -0.702 -0.679 116 -0.594 0.012 -0.655 -0.667 120 -0.378 0.011 -0.612 -0.654 APPENDIX -- SINGLE-SOURCE RMS VERSUS ASSUMED DEPTH (mm/yr) ---------------------------------------------------------- 2.0 km 2.484 7.0 km 1.594 12.0 km 1.292 17.0 km 1.425 22.0 km 1.683 27.0 km 1.927 32.0 km 2.131 37.0 km 2.295 42.0 km 2.425 47.0 km 2.529 52.0 km 2.612 57.0 km 2.678 62.0 km 2.732 67.0 km 2.776 72.0 km 2.812 77.0 km 2.842 82.0 km 2.867 87.0 km 2.887 92.0 km 2.904 97.0 km 2.919 102.0 km 2.931 107.0 km 2.941 112.0 km 2.950 117.0 km 2.958 122.0 km 2.964 127.0 km 2.970 132.0 km 2.975 137.0 km 2.979 142.0 km 2.983 147.0 km 2.986 152.0 km 2.989 157.0 km 2.992 162.0 km 2.994 167.0 km 2.996 172.0 km 2.998 177.0 km 3.000 182.0 km 3.001 187.0 km 3.003 192.0 km 3.004 197.0 km 3.005 ==========================================================================