============================================================================== PART A -- THE SLIP PULSE ============================================================================== ------------------------------------------------------------------------------ A1. The shape factor, and why the classical pulse cannot fit ------------------------------------------------------------------------------ Published central values: D = 2.5 m, T = 1.3 s, v_peak = 3.2 m/s Shape factor k = D / (v_peak * T) = 0.6010 Reference shapes: boxcar k = 1.000 half-sine k = 0.637 triangle k = 0.500 sharp crack-tip spike k < 0.5 k = 0.601 sits between a half-sine and a boxcar. It is FLATTER-TOPPED than a triangle, and every regularized-Yoffe pulse has k <= 0.5. ------------------------------------------------------------------------------ A2. Regularized Yoffe, duration pinned at 1.3 s: how peaked does it stay? ------------------------------------------------------------------------------ tau_S/T tau_R (s) tau_S (s) v_peak (m/s) t_peak (s) shape k 0.10 1.1700 0.1300 8.6578 0.0876 0.2221 0.25 0.9750 0.3250 5.9405 0.2148 0.3237 0.50 0.6500 0.6500 4.8667 0.4106 0.3951 0.75 0.3250 0.9750 4.4534 0.5424 0.4318 0.90 0.1300 1.1700 4.0858 0.6074 0.4707 0.98 0.0260 1.2740 3.8929 0.6418 0.4940 Even at the flattest end of the family the peak stays near 3.85 m/s, which is +20% above the reported 3.2 m/s. The three published central values are jointly inconsistent with a Yoffe-type shape. Two honest ways out, both inside the published error bars: (i) keep D = 2.5 m and v_peak = 3.2 m/s -> duration must be >= 1.562 s (quoted: 1.3 +/- 0.2 s) (ii) keep T = 1.3 s and v_peak = 3.2 m/s -> slip must be <= 2.080 m (quoted: 2.5 +/- 0.5 m) Neither is a contradiction. It just means the real pulse has a flat shoulder that a crack-tip formula does not produce. ------------------------------------------------------------------------------ A3. Model 2 -- asymmetric smoothed ramp fitted to all three numbers ------------------------------------------------------------------------------ v(t) = C * u^a * (1-u)^b with u = t/T, T = 1.3 s required mode u_m = 0.3846 (peak at 0.5 s) required shape k = 0.6010 solved exponents a = 1.01612 b = 1.62579 check: shape factor of solution = 0.60096 Resulting pulse: total slip = 2.5000 m (target 2.5) duration = 1.3000 s (target 1.3) peak slip velocity = 3.2000 m/s (target 3.2) time of peak = 0.5000 s (target 0.5) mean slip velocity = 1.9231 m/s PEAK / MEAN ratio = 1.6640 rise time (5% -> 95%) = 0.8871 s velocity FWHM = 0.8109 s acceleration limb = 0.5000 s (paper 0.5) deceleration limb = 0.8000 s (paper 0.8) asymmetry decel/accel = 1.6000 slip before velocity peak = 1.0861 m (43.4%) slip after velocity peak = 1.4139 m (56.6%) peak slip acceleration = 12.68 m/s^2 = 1.29 g (at t = 0.030 s) NB: this acceleration is a LOWER bound. A 24 fps camera plus a 0.2 s smoother cannot resolve how fast the onset really was. ------------------------------------------------------------------------------ A4. Implied strain rate ------------------------------------------------------------------------------ moletrack, 1.5 m wide peak gamma_dot = 2.133e+00 /s mean = 1.282e+00 /s total strain = 1.667e+00 deformation zone, 20 m peak gamma_dot = 1.600e-01 /s mean = 9.615e-02 /s total strain = 1.250e-01 seismogenic width, 15 km peak gamma_dot = 2.133e-04 /s mean = 1.282e-04 /s total strain = 1.667e-04 interseismic shear strain rate (18 mm/yr over 100 km) = 5.704e-15 /s coseismic peak across the moletrack = 2.133e+00 /s ratio = 3.740e+14 as orders of magnitude (log10) = 14.57 2.5 m of slip at 18 mm/yr of loading = 139 years of stored strain released in 1.3 s. Time compression factor = 3.372e+09 (10^9.5) ------------------------------------------------------------------------------ A5. Slip function sampled at 30 fps (article filmstrip) ------------------------------------------------------------------------------ frame t (s) slip (m) v (m/s) % of total 0 0.0000 0.0000 0.0000 0.0% 1 0.0333 0.0072 0.4307 0.3% 2 0.0667 0.0285 0.8353 1.1% 3 0.1000 0.0626 1.2056 2.5% 4 0.1333 0.1084 1.5423 4.3% 5 0.1667 0.1651 1.8464 6.6% 6 0.2000 0.2312 2.1166 9.2% 7 0.2333 0.3058 2.3545 12.2% 8 0.2667 0.3880 2.5615 15.5% 9 0.3000 0.4763 2.7370 19.1% 10 0.3333 0.5699 2.8828 22.8% 11 0.3667 0.6683 3.0001 26.7% 12 0.4000 0.7697 3.0892 30.8% 13 0.4333 0.8737 3.1514 34.9% 14 0.4667 0.9797 3.1881 39.2% 15 0.5000 1.0861 3.2000 43.4% 16 0.5333 1.1925 3.1885 47.7% 17 0.5667 1.2985 3.1546 51.9% 18 0.6000 1.4027 3.0997 56.1% 19 0.6333 1.5047 3.0252 60.2% 20 0.6667 1.6043 2.9318 64.2% 21 0.7000 1.7001 2.8217 68.0% 22 0.7333 1.7920 2.6959 71.7% 23 0.7667 1.8797 2.5554 75.2% 24 0.8000 1.9623 2.4027 78.5% 25 0.8333 2.0396 2.2389 81.6% 26 0.8667 2.1115 2.0651 84.5% 27 0.9000 2.1773 1.8842 87.1% 28 0.9333 2.2369 1.6974 89.5% 29 0.9667 2.2904 1.5062 91.6% 30 1.0000 2.3373 1.3137 93.5% 31 1.0333 2.3779 1.1217 95.1% 32 1.0667 2.4122 0.9321 96.5% 33 1.1000 2.4401 0.7486 97.6% 34 1.1333 2.4621 0.5739 98.5% 35 1.1667 2.4785 0.4109 99.1% 36 1.2000 2.4897 0.2650 99.6% 37 1.2333 2.4964 0.1411 99.9% 38 1.2667 2.4994 0.0469 100.0% 39 1.3000 2.5000 0.0000 100.0% ============================================================================== PART B -- THE CURVED SLIP PATH (rotating slip vector) ============================================================================== Fitted rake law (degrees): theta(t) = 35*exp(-(t/0.4600)^5.5600) - 14.9167*(t/0.1842)*exp(1 - t/0.1842) normalised misfit against 4 published targets = 0.0167 initial rake = 35.000 deg (paper: max rake 35) most negative rake = -4.646 deg rake at arrest = -0.246 deg (paper: 'nearly pure strike slip') total rake swept = 39.646 deg peak dip slip = 0.2813 m (target 0.3) time of peak dip slip = 0.4985 s (target 0.5) final dip slip = 0.2162 m (target 0.2) final strike slip = 2.4533 m net slip magnitude = 2.4628 m net rake = 5.037 deg mean |rake|, first 1.00 m of slip = 18.79 deg mean |rake|, remaining 1.46 m = 1.85 deg (paper: first metre oblique to 35 deg, subsequent 1.5 m nearly pure strike slip) ------------------------------------------------------------------------------ B1. Curved versus straight: what does the curve actually cost? ------------------------------------------------------------------------------ arc length of curved path = 2.50000 m straight chord (net slip) = 2.46281 m excess path length = 3.719 cm (1.5102%) sagitta (max bow off chord) = 19.535 cm at t = 0.456 s sagitta in camera pixels = 2.579 px max curvature |kappa| = 1.12631 /m tightest radius of curvature= 0.88786 m (at 0.050 m of slip) mean curvature = 0.23001 /m -> mean radius 4.348 m A tape measure laid along the ground would read the CHORD. The extra 37.2 mm of path that the curve costs is well inside the +/- 0.5 m uncertainty on the offset. That is exactly why nobody could settle this from field measurements: the curve is a SHAPE, not a LENGTH. rake swept during acceleration (0 -> 0.50 s) = 35.15 deg over 1.086 m of slip rake swept during deceleration (0.50 -> 1.30 s) = 0.10 deg over 1.414 m of slip -> 99.7% of the turning happens before peak velocity ------------------------------------------------------------------------------ B2. Sweep: rotation amplitude versus detectability ------------------------------------------------------------------------------ A (deg) arc (m) chord (m) excess (cm) excess % sagitta cm sagitta px R_min (m) 3.0 2.50000 2.49973 0.0274 0.0110 1.688 0.223 10.3583 5.0 2.50000 2.49924 0.0762 0.0305 2.812 0.371 6.2150 10.0 2.50000 2.49695 0.3049 0.1221 5.622 0.742 3.1075 15.0 2.50000 2.49314 0.6856 0.2750 8.426 1.112 2.0717 20.0 2.50000 2.48782 1.2181 0.4896 11.222 1.481 1.5538 25.0 2.50000 2.48098 1.9017 0.7665 14.008 1.849 1.2430 35.0 2.50000 2.46281 3.7193 1.5102 19.535 2.579 0.8879 45.0 2.50000 2.43870 6.1302 2.5137 24.988 3.298 0.6906 The column that decides everything is sagitta in PIXELS. Sub-pixel cross correlation is good to roughly 0.1-0.3 px, so a curve is only visible if its sagitta clears that floor. Smallest rotation here clearing 0.3 px: 5 deg (0.37 px, 2.8 cm on the ground) ------------------------------------------------------------------------------ B3. Curved slip trajectory sampled every 1/30 s (article figure 3) ------------------------------------------------------------------------------ frame t (s) strike (m) dip (m) rake (deg) |v| (m/s) 0 0.0000 0.0000 0.0000 35.000 0.0000 1 0.0333 0.0062 0.0037 28.881 0.4307 2 0.0667 0.0252 0.0132 24.776 0.8353 3 0.1000 0.0565 0.0266 22.201 1.2056 4 0.1333 0.0993 0.0433 20.733 1.5423 5 0.1667 0.1524 0.0630 20.031 1.8464 6 0.2000 0.2146 0.0855 19.796 2.1166 7 0.2333 0.2847 0.1107 19.735 2.3545 8 0.2667 0.3621 0.1384 19.551 2.5615 9 0.3000 0.4455 0.1676 18.941 2.7370 10 0.3333 0.5344 0.1970 17.613 2.8828 11 0.3667 0.6286 0.2250 15.335 3.0001 12 0.4000 0.7272 0.2492 12.065 3.0892 13 0.4333 0.8295 0.2674 8.012 3.1514 14 0.4667 0.9349 0.2781 3.690 3.1881 15 0.5000 1.0413 0.2813 -0.150 3.2000 16 0.5333 1.1476 0.2782 -2.891 3.1885 17 0.5667 1.2534 0.2714 -4.309 3.1546 18 0.6000 1.3573 0.2631 -4.644 3.0997 19 0.6333 1.4590 0.2550 -4.382 3.0252 20 0.6667 1.5582 0.2478 -3.918 2.9318 21 0.7000 1.6539 0.2417 -3.443 2.8217 22 0.7333 1.7456 0.2365 -3.012 2.6959 23 0.7667 1.8333 0.2322 -2.627 2.5554 24 0.8000 1.9158 0.2287 -2.287 2.4027 25 0.8333 1.9930 0.2258 -1.988 2.2389 26 0.8667 2.0649 0.2234 -1.725 2.0651 27 0.9000 2.1306 0.2216 -1.495 1.8842 28 0.9333 2.1902 0.2201 -1.294 1.6974 29 0.9667 2.2437 0.2190 -1.118 1.5062 30 1.0000 2.2907 0.2182 -0.965 1.3137 31 1.0333 2.3312 0.2175 -0.832 1.1217 32 1.0667 2.3655 0.2171 -0.717 0.9321 33 1.1000 2.3934 0.2167 -0.617 0.7486 34 1.1333 2.4154 0.2165 -0.530 0.5739 35 1.1667 2.4318 0.2164 -0.455 0.4109 36 1.2000 2.4430 0.2163 -0.391 0.2650 37 1.2333 2.4497 0.2163 -0.335 0.1411 38 1.2667 2.4527 0.2162 -0.287 0.0469 39 1.3000 2.4533 0.2162 -0.246 0.0000 ============================================================================== PART C -- PIXEL-TRACKING DEMO (what camera noise costs you) ============================================================================== Ground sampling at the target: 13.200 px/m = 7.58 cm per pixel (from the paper's +33 px of tracked motion for 2.5 m of slip) RNG seed = 20250328; Monte Carlo trials = 4000 Model peak velocity being recovered: 3.2000 m/s ------------------------------------------------------------------------------ C1. Raw central differences, NO smoothing, 30 fps ------------------------------------------------------------------------------ sigma px sigma cm predicted RMS v err peak found peak err err % 0.05 0.379 0.0804 0.0800 3.2778 +0.0778 +2.4% 0.10 0.758 0.1607 0.1595 3.3867 +0.1867 +5.8% 0.25 1.894 0.4018 0.3983 3.7624 +0.5624 +17.6% 0.50 3.788 0.8035 0.7971 4.4658 +1.2658 +39.6% 1.00 7.576 1.6071 1.5918 5.9688 +2.7688 +86.5% Note that the peak is biased HIGH, and the bias grows with noise. Taking the maximum of a noisy series is a rigged game: you are selecting for whichever frame the noise happened to push furthest up. ------------------------------------------------------------------------------ C2. With the paper's 0.2 s moving average, 30 fps ------------------------------------------------------------------------------ sigma px window predicted RMS v err peak found peak err err % 0.00 6 0.0000 0.0894 3.1599 -0.0401 -1.3% 0.05 6 0.0328 0.0914 3.1679 -0.0321 -1.0% 0.10 6 0.0656 0.0967 3.1812 -0.0188 -0.6% 0.25 6 0.1640 0.1289 3.2401 +0.0401 +1.3% 0.50 6 0.3280 0.2058 3.3612 +0.1612 +5.0% 1.00 6 0.6561 0.3838 3.6433 +0.4433 +13.9% The sigma = 0 row isolates the SYSTEMATIC cost: even with a perfect camera, a 6-frame moving average shaves 0.040 m/s (1.3%) off the true peak. Smoothing trades random error for a guaranteed underestimate. ------------------------------------------------------------------------------ C3. Frame rate: same 0.25 px noise, 0.2 s smoothing ------------------------------------------------------------------------------ fps frames/pulse raw sigma_v smoothed RMS v err peak found err % 12 15 0.1607 0.1136 0.2523 3.2074 +0.2% 15 19 0.2009 0.1160 0.0996 3.1920 +0.1% 24 31 0.3214 0.1437 0.0963 3.2204 +0.6% 30 39 0.4018 0.1640 0.1285 3.2387 +1.2% 60 78 0.8035 0.2320 0.1078 3.2786 +2.5% 120 156 1.6071 0.3280 0.1018 3.3138 +3.6% 240 312 3.2141 0.4639 0.1000 3.3448 +4.5% Raw differencing noise gets WORSE at high frame rate (you divide by a smaller dt), but a fixed-DURATION smoother averages more frames, so the two effects cancel almost exactly. Frame rate buys you time resolution, not velocity precision. ------------------------------------------------------------------------------ C4. The headline numbers ------------------------------------------------------------------------------ 30 fps, 0.25 px tracking noise (= 1.89 cm per frame): predicted raw velocity noise = 0.4018 m/s = 12.6% of the peak measured RMS velocity error = 0.3983 m/s recovered peak (raw) = 3.772 +/- 0.210 m/s (true 3.200) upward bias from taking a max = +0.572 m/s (+17.9%) Same data with a 0.2 s moving average (6 frames): predicted velocity noise = 0.1640 m/s measured RMS velocity error = 0.1284 m/s recovered peak = 3.240 +/- 0.069 m/s net peak error = +0.040 m/s (+1.2%) random error cut by factor = 3.10x (sqrt(6) = 2.45 predicted) At the real video's 24 fps, same recipe: recovered peak = 3.221 +/- 0.067 m/s (+0.6%) which is inside the paper's quoted +/- 1.0 m/s: yes ------------------------------------------------------------------------------ C5. How many frames do you actually get? ------------------------------------------------------------------------------ 24 fps: the whole 1.3 s pulse is 31 frames; the 0.5 s acceleration limb -- where ALL the curvature lives -- is 12 frames. 30 fps: the whole 1.3 s pulse is 39 frames; the 0.5 s acceleration limb -- where ALL the curvature lives -- is 15 frames. ------------------------------------------------------------------------------ C6. Is the curvature signal above the noise? ------------------------------------------------------------------------------ Fitted 35 deg trajectory bows 19.54 cm off the chord = 2.58 px at this camera's scale. Against 0.25 px of per-frame tracking noise: SNR = 10.3 : 1 and the sagitta is spread over ~12 frames, so the effective SNR on the SHAPE is nearer 36 : 1. A 5 deg rotation would bow only 0.37 px: indistinguishable from noise. Real, but not lavish. This is a one-camera, one-site result. ============================================================================== SUMMARY -- NUMBERS USED IN THE ARTICLE ============================================================================== shape factor implied by paper k = 0.601 (boxcar 1, triangle 0.5) Yoffe-family ceiling k = 0.500 -> classical pulse cannot fit fitted smoothed-ramp exponents a = 1.016, b = 1.626 slip pulse D = 2.50 m over T = 1.30 s peak slip velocity v_max = 3.20 m/s mean slip velocity v_avg = 1.92 m/s peak / mean = 1.66 rise time (5-95% of slip) = 0.89 s velocity FWHM = 0.81 s peak slip acceleration = 12.7 m/s^2 = 1.29 g shear strain rate, 1.5 m zone = 2.13 /s vs interseismic = 10^14.6 times faster strain stored over = 139 years, spent in 1.3 s rake swept = 39.6 deg rake at arrest = -0.25 deg arc length = 2.5000 m chord (what a tape measures) = 2.4628 m excess path length = 37.2 mm = 1.510% sagitta = 19.5 cm = 2.58 px tightest radius of curvature = 0.89 m turning done before peak velocity = 100% 5 deg rotation would bow = 2.8 cm = 0.37 px (invisible) 30 fps + 0.25 px, raw v-noise = 0.40 m/s (13% of peak) raw peak overshoot = +18% after 0.2 s smoothing = 0.13 m/s RMS, peak error +1% smoothing-only bias (sigma = 0) = -1.3% frames in the pulse at 30 fps = 39 frames in the pulse at 24 fps = 31 (the real video) frames in the curved section = 12 at 24 fps All of the above is the club's forward model, not the authors' inversion. The published measurements are 2.5 +/- 0.5 m, 1.3 +/- 0.2 s, 3.2 +/- 1.0 m/s, maximum rake ~35 deg, transient dip slip 0.3 -> 0.2 m.