How to find MH370?

How to f ind MH 370? Martin Kristensen ( Department of Engineering, Aarhus University, Finlandsgade 22, DK-8200 Aarhus N, Denmark ) (Email: m k@eng.au.dk) The disappearance of flight MH370 is possibly the greatest mystery in aviation history. A large zone in the Southern Indian Oc ean was searched unsuccessf ully leaving an open case and an unacceptable si tuation for the family m embers. We discuss the sc ientific diff iculties with locating the plane through satellite data and develop an im proved analysis using least square curve fitting of analytical non - Euclidean route equations providing robust topology - optimization with perturbation theory handling satellite movement. We find four independent solutions with the fin al part of the flight follow ing a great c ircle. Two are located in st able minim a for the error - f unction, and two unstable ones agree poorly with most data. One stable solution coincides with the I nmarsat - result , but fails to explain additional data. Our b est solution leads to an entirely diff erent loc ation agreeing with other data from debris, acou stics, an eyewitness report, t he receive d microwave power, s everal contrail s, two seismic detectors at Christmas Island , and with a mushroom impact cl oud providing the final proof where to find the rest of the ai r plane and the black boxes. KEY WORDS 1. MH370. 2. Satellite Comm unications. 3. Entanglement. 4. Topolog y - Optimization. Originally submitted: 2 March 2018 . 1. INTRODUCTION. After contact was lost with flight MH370 on the route from Kuala Lumpur to Beijing on March 8, 2014 the only available da ta concerning its route came from a military r adar facility in Malaysi a (News, 2017) and a number of handshakes with a satel lite from I n marsat (Ashton et al., 2014). R adar data tells us that after shutting down transponders and communication systems it made a sharp turn and flew along the border to Thailand, past Penang to the w aypoint MEKAR in the Malacca Strait wh ere military radar lost contact ( News, 2017) . Then the satellite handshakes r estarted after interruptio n of the satellite un it (SDU), possibly due to a power failure. The handshakes provide accurate tim e differences and thereby distances to the satellite 3F1 at a pproximately geostation ary location west of the Maldives , represented by so - called Burst Time Offset (BTO) (Ashton et al., 2014). The sum of the Doppler shifts for the communication loop was also measured and the local Doppler shift from the airplane movement deduced through the Burst Frequency Offset (BFO) . Since these represent frequency shifts due to the radial pro jection of the movement of the plane, measured simultaneously with BTO at roughly one -hour intervals aft er the reboot , and two data points due to attempted phone call s , it is possible to deduce some route information. However, the accuracy of the Doppler s hifts (1% of t he maximum value) was be lieved insufficient to det ermine a precise route. If the satellite were perfectly geost ationary we would have been practic ally stuck here, since we could only conclude that it reached a certain distance from the last BTO, giving roughly a circle called the seventh arc after numbering the handshakes. This corresponds to 2662 nmi ( nautical m ile ( nmi ) = 1.852 km) along a spherical earth s urface from the ground projection of 3F1. This arc is cut around 5 5% shorter due to th e maxi mum f uel range of the plane but still represents an unsurmountable difficulty to search , particularly when allowing for moderate movement from the last handshake to the end. In other words, the plane effectively disappeared. Physically this is easy to understand, s ince the satellite measurements only provide radial information, but no direction. Fortunately the satellite is not completely ideal and wanders, predominantly in north -south direction (Ashton et al., 2014), and the maximum speed and minimum turning r a dius for the plane also give some guidance. During most of the flight the satellite moved south after reaching its northern extremum roughly an hour after the SDU restarted. This opens the possibility to distinguish between northern and southern flight routes since the perturbation to the relat ive airplane velocity (fo r SDU - satellite Dop pler shift) has opposite sign for northern and southern routes. For routes going straight north or south , this gives a 10% difference. This is ten times the measurement uncertainty for the BFO leaving no doubt routes going straight north are inconsistent with the d ata, as shown by A shton ( 2014) . Routes towards northeast are even worse. However, in north- western direction there is a c hance of a route with poor fit and range near the acceptan ce limit because the east - w est satellite m ovement m ixes with the north - south movement relaxing fitting condition s. We initially ign ore this type o f solution since it is unlike ly that any airplane could penet rate radar s urveillance by several countries including India and China on a north- western route. In addition , it would have to fly at relatively low speed along a curved route to fit the d ata, and thereby be forced to, but at the same time practically unable t o, pass the tall mountains in Himalay a. In addition , some of the phones belonging to the passengers would likely have been on and left electronic footprints by handshakes with the Chinese network. Finally , all debris would have to be planted. Later we develop a sim ple test to find all remaining solutions and settle the issue rigorously. This leads us to conclude that only one of the solutio ns is correct, and define a new and dramatically sm aller search zone near Christm as Island . This also explain s why all se archin g until now was unsuccessful. 2. ANALYSIS OF SATELLITE DATA. Soon a fte r militar y radar contact was lost, the SDU restarted unexpectedly . After some time without power, it was cold, so it r ebooted and reheated simultaneously leading to some highly unreliable handshakes (BTO/BFO) due to temperature drift. These points must be discarded. Around 18:28 UTC, the measur ement quality was improving in agreement with results from previous SDU cold- starts (Holland, 2017). At 18:40 , Malaysia A i rlin e s called the plane via sate llite phone. Nobody answer ed, but the call provided BFO valu e s and defined the star ting point fo r the hourly handshakes. Unfortunately , telephone calls give no BTO (Ashton et al., 2014) . Sinc e the airplane had moved from MEKAR so its height, speed and course were no longer known i ndependently from radar data, it is im possible to ex tract precis e knowledge from these two points without making assumptions or having additional k nowledge. It is therefore not until the first regular handshake at 19:41 that complete and precise data is available for system atic analysis. Figure 1. Equation (1) com pared to t he radi al dista nces  =   on a spherical earth. At 18:28, there a re two qua litative options: A norther n or a southern route. The handshake at 18:28 allows for these two solutions to a quadratic equation. Concerning the route after 18:28 the BFO at 18:40 provides some selection among the p ossibilities but unfortunately the problem is underdetermined and the data t oo inaccurate to distinguish between north and south. Importantly the rest of the points look like they are located on a straight line with constant spee d when considering the BTO v alues. If one converts the BTO values to distances, R , along the earth surface f rom the sa tellite g round projec tion using equations (3 ) and ( 4), th ese are modelled well by a simple equation describing a straight line on a flat earth with an exclusively transverse speed of v = 800 km/h, starting at t 1 (19:41)  = 1758   1 +  0. 2457 (     )   (1) where the front f actor is R at t 1 = 19:41 and the value 0.2457 is the ratio of v and R ( t 1 ). This is a strong l ead, in particular sinc e the agreement with m easured values is good as shown in figure 1. Furthermore, the BFO values also agree with expectations from such a strai ght flight, but with less precision, and with the exception of 19:41, which has movement towards the satellite rather than away from it as the rest of the p oints, comp ared to zero movement as equation (1 ) predicts. In fact , it is impossible to reconcile the BFO and BTO information at 19:41 without assuming one of the following: 1) The point at 19:41 is not at all on the same straight line as the rest. 2) There is roughly 10-degree heading change between 19:41 and 20:41. 3) The minimum distance to the satellite lies between 19:41 and 20:41. 4) There is a U -turn between 19:41 and 20:41. We combined BTO and BFO information in a detailed mathematical analysis to gain deeper insight and found the sign -change for the aircraft Doppler - shift of key importance to select the correct option . We elimina te two of the four possibilities by initially conside ring a flat earth, and improve b y a Taylor expansion of the spherical solu tion. For a flat ea rth the surfa ce angle toward s the satelli te ground projection ,  , is linked to the speed, v , and radius along the earth surface from the ground projection, R , by 0, 0 0, 5 1,0 1, 5 2, 0 2,5 3, 0 3, 5 4, 0 4,5 1800 1950 2100 2250 2400 2550 2700 R ( n mi ) Ti me (h) R exp Eq. ( 1) 1.1        cos  = 1.1  󰇧    󰇨   (     ) (2) where R J is the earth radius and v the airplane speed. Using th at v ·cos(  ) is proport ional to the Doppler shift we plot this relatio nship in figure 2 against tim e, t, measured from 19:41 using the BFO values converted to Doppler shifts (Ashton et al., 2014) and the R - values. We find a relatively good linear fit for all the lat er points but a poor one for 19:41, and moderate deviation for 20:41. By replacing the simpli f ied flat earth version ( 2) (only valid for short distances) with a spherical solution derived from equations ( 7 ) and ( 8 ) and Taylor expanded to second order in the point at 22:41 (the best approxim ation which is still a straight line , and labelled equation 2A ) one finds a nice fit from 20:41 as shown in the same figure. We multiplie d e quation ( 2) by 1.1 to match the simple equation with the op timally Taylor- expanded Doppler shifts at short di stances. A linear f it gives v = 784 km/h using the Taylor formula. As illustrated, t he point at 19: 41 deviates by a Doppler sign change (indicating U - turn) and is therefore not included in the fit. In stead , we fi nd the natural zero - point, t 0 , at - 0.61 hours (before 19:41) eliminating option 3, s ince the minim um satellite distance, R min , and corresponding sign change must occur at t 0 . A greement with equation (1) and sign change at t = 0 makes option 1 impossible. This leaves us only with options 2 or 4 (most li kely 4) . These lead to solution s in the Southern I ndian Ocean and the Eastern Indi an Ocean resp ectively, while the Iannello solution ( Iannello and Godfrey, 2016 ) can connect to both options. Figure 2. Data points from e quation ( 2 ) using a flat ear th description (red circles) , and a second - order T aylor e xpansion (2A) of the spherical equation at 22:41 ( black squares) provi ding the most accurate line ar description for all distances , plotted against time after 19:41 and fitted with a straight line. It illustrates tha t the point at 19:41 would fit much better if the Doppler - ef fect changes sign due to U- turn soon after 19:41. After establishing this overview of possible solutions , we p erform the detailed an alysis of the rest of the dat a . The analytical con nection between th e satellite position, the BTO value and the radius, R , along the earth surface from the satellite ground projection to t he airplane is given by Ashton (2014) and Steel (2016) 0, 0 0,5 1, 0 1, 5 2, 0 2, 5 3,0 3, 5 4, 0 4, 5 -0, 01 0, 00 0, 01 0, 02 0, 03 0, 04 0, 05 0, 06 0, 07 0, 08 0, 09 Equati on 2A (h -1 ) Ti me (h) Spheri cal Planar ^ | Sign change   =    +    +   +    2   +    +  cos 󰇧    󰇨 (3)   =  2 (    )       (4) where R sat is the lin e -of- sight distan ce from the satellite to the airpl ane, R J = 6378 km is the earth radius, H is the satellite height above earth , h is the flying height, c is the speed of light, bias is an internal Inmarsat parameter and R sat -Perth the distance from the satellite to the relay station. W e calculate R for a spherical earth model to sim plify the non - Eucli de an equations. Finally , we make obla te projection s (Wikipedia-1) to place them on the true earth. Please n ot e that the R -values do not agree with a table originally published in the blog of (Steel, 2016) and later deleted . According to our calculations those values had up to one percent error . All Doppler- shifts are class ical , since it is sa fe to ignore rela t ivistic contributions, atmospheric influence and gravitational shift s , which are all at least 30 tim es below the measurement uncertainty . We calculate the Doppler - shift from the change in the line -of- sight distance between the ai rplane and the satellite b y differentiation of equation ( 3), initially ign oring the sate llite earth -projection m ovement     =   + 󰇧 1 +     +  cos 󰇧    󰇨󰇨  sin 󰇧    󰇨   cos  (5) where  is the angle between R a nd the flight direction. We use h = 11 km, ignore he ight changes until the seventh arc, and assume H >> R J >> h . This flight height was chosen af ter optimizing the entire problem (Plougmann and Kristensen , 2004) with h f ixed at 10, 11 and 12 km respectively, and finding slightly better fit for both stabl e solutions at 11 km (most pronounced south- e asterly ). In addition to the expected Doppler - shift due to the airplane velocity with respect to the satellite there is a he ight contrib ution from th e satellite. It i s unclear how Inmarsat handled this contribution. They split up the Doppler shift into two contributions: One entirely due to the satellite and one entirely due to the plane. Strictly speaking, this is w rong since Doppler- shift is a relativ e effect as pointed out by ( Einstein , 1905) , since the Doppler effect depends only on the relative motion of source and receiver. One can circumvent this by choosing a suitable inertial - system as reference. Howev er, Inmarsat has not m ade a well - defined choice of inertial - system . If we calculate the contribution from the satel lite ground - proje ction movem ent relative to its most northern point, (not an iner tial system because of e arth rotation ) we get agreem ent with the ir satellite contribution except for last decimal round - off and the point at 21 :41 where there is a somewhat larger deviation. This issue cannot be ignored since our model is so accura te that even the last decimal matters, and the deviation at 21:41 is more than 2 σ for the BFO for other routes than Inmarsat’s rou te where eve rything fits. W e tried to add t he satellite h eight change, but this makes agreement worse. Therefore , we have chosen to ignor e the height - change and add its m aximum size lin early to the error - bar lead ing to a doubling of σ for the BFO. This only moves the predicted crash site a few km, so w e can live with it. Under these conditions and using equation (5 ) the airp lane Doppler sh ift,  , i s  =     󰇧 1 +     +  cos 󰇧      󰇨󰇨 sin 󰇧      󰇨   cos     (6) where ν 0 = 1646652500 Hz is the communication frequency (Ashton et al., 2014), and experimenta l numbers includin g perturbatio n effects from satellite movem ent are symbolized with a tilde over the variables (  ,    ). We think u sing a de tailed recip e from Inmarsat the σ enlargement can be avoided and th e agreement will be better for 21:41. The published Inmarsat procedure is perfect for their specific route but potent ially problem atic for other routes. However, an alternative possibility is that the plane passed a thunderstorm at 21:41. Table 1. First or der pert urbati ons from the sat ellite movement o n R , v (along R ) and  for the t wo solut ions towards 13. 3˚S and 34.6 ˚S using val ues from (A shton et al., 2014) and o ur own recal culations. Satellite perturbation Solution towards 13.3˚S Solution towards 34.6˚S Time (UTC) - ΔR (nmi) -v R (km/h)  ( ° ) - ΔR (nmi) -v R (km/h)  ( ° ) 20:41:05 -1.54 -3.94 0.113 0.6 3 0.02 21:41:27 -1.72 -2.09 0.410 6 12 0.19 22:41:22 2.48 3.85 0.785 18 21 0.47 00:11:00 18.71 14.48 1.304 48 34 1.02 After calculating the values o f R and Δν we take the perturb ations from the satellite ground-projection movement into account. We use classical first order perturbation theory (Stewart, 19 90) to find the projection on the individual R - values and satellite D oppler effect for two solutions going south and southeast in table 1. By subtracting the pert urbations from the measured value s , we get the unperturbed result for a spherical earth in c ase the direction is co rrect. After a cou ple of tries , we found that we only arrive at self - consistent routes with good fits in two directions. In addition we found indication of a third solution between them, but its error ( χ 2 ) minimum is too shallow f or a stable fit as shown in figu re 3 . Starting at any other direction l eads to an itera tive convergence towards one of t he two stable solutions. South of the shallow minimum converging t o the southern solution and north of it to t he south - eastern solution , both after 3 - 4 itera tions where th e satellite perturbations are reca lculated for each ite ration. Table 2. F itting results with equations (6 - 8) for th e solutions to wards 13.3˚S a nd 34.6˚S with   .  .  = 31.7 n mi 2 and σ Doppler = 15 Hz, gi ving     = 1.25 nm i 2 . Route t 0 (h) v (km/h) R min ( nmi )    ( nmi 2 )   ( nmi 2 )    ( ° ) 13.3˚S -0.4709±0.0035 796.87±0.50 1661.6±1.2 0.475 0.845 43.294±0.030 34.6˚S -0.3800±0.0035 822.74±0.50 1675.8±1.6 2.64 2.86 42.963±0.035 To extract precise end - points for the solutions w e used simultaneous least square curve fitting of exact analyt ical expressions for R and  ( t ) deriv ed from spherical Non - Eucli dean algebra . We fitted R- and Doppler -values simultaneously to expressions for a rig ht -angled triang le wit h minimum satellite distan ce R min and flight length v ( t-t 0 ) on a spherical earth for the angle    i n the lower end of the t riangle with 1000 times less weight on the Doppler - part of χ 2 than on the pr ecise R - part with free param eters R min , v and t 0 . The analytical expressions in the spherical approximation without satellite movement are given by (Wikipedia - 2): cos 󰇧  (   )   󰇨 = cos 󰇧     󰇨 cos 󰇧  (      )   󰇨 (7) cos (   ) = cos        cos   (      )    cos   (   )    sin   (      )    sin   (   )    (8) For continuous calculations, t n is replaced by t and   by  . Complete expressions including pertu rbation f rom satellite m ovement are sym bolized with a tilde over the variables (  ,    ). The fitting parameters for the southern and south- eastern routes are listed in table 2 tog ether with    for the R - contribution and χ 2 for the entire fit. Exclusively 21:41, 22:41, 23:14 and 00:11 were used for the fits, since only for these point s we are sure of a straight flight with constant speed. χ 2 shows the best fit for the south- eastern route. This value is 3.4 times lower than for the southern route (Ashton et al., 2014) . T he statistica l χ 2 due to measurement uncertainty is roughly 1.5 times our best val ue , meaning both are within the expected uncertainty range, with the southern around the upper statistical lim it. The fitted values for   and  are listed in tabl e three where the perturbations are added back facilitating comparison with raw data. Table 3. Fitted values for   and  where t he pertur bations a re adde d back, a nd the D oppler - shift calculated for all relevan t points using a southern route for  sim plifying com parison wi th raw data. Solution towards 13.3˚S Solution towards 34.6˚S Time (UTC) Δν (Hz) Deviation (Hz)   (nmi) Dev . (nmi) Δν (Hz) Deviation (Hz)   (nmi) Dev. (nmi) 21:41:27 -385 -18 (±15) 1) 1952.96 0.46 -377 -10 (±7) 1953.5 1.0 22:41:22 -522 -1 (±15) 2192.77 -0.43 -521 0 (±7) 2192.2 -1.0 23:14:00 -594 6 (±15) 2342.3 - -589 11 (±7) 2341.6 - 00:11:00 -705 -3 (±15) 2620.58 0.28 -703 -1 (±7) 2621.1 0.8 00:19:29 -721 70 (±15) * ) 2664.55 2.55 * ) -718 73 (±7) * ) 2664.5 2.5 * ) 1) Larger deviati on perha ps due to passage of a thu nderstorm as the plane ent ered the intertr opical converge nce zone (Schnei der et al ., 2014) * ) Due to flame out (irrelevan t for fit quality) The last task is projecting the results on earth, which is slig h tly oblate due to rotation. This is done using formulas for the radii of curvature in t he relevant directions and for the relevant lati tudes (Wikiped ia - 1) to convert distances to oblate geom etry. Table 4 lists relevant m odified R - val ues. After corre ction , we manually placed the solutions on the earth surface using Google Maps (GM) by demanding that all d istances and angles should fit as shown in figure 4 for the south - eastern solution. The southern solution becomes practically identical to the Inm arsat sol ution with a deviation on the 6 th and 7 th arcs of only 21 km and an end point at (34.591˚ South, 93.161˚ East). Thi s means that there is no reason to re fit t he initial part of this solution and fine - adjust the rest , since everything will be pra ctically identical to the findings of Ashton (2014). Figure 3. Sketch of the an gle    (full line) and   (dashed line) as a function of φ (angle from ea st to sout h from satellite ground proje ction at 00:11 to 6 th arc crossi ng) with the search - zo nes highli ghted by col oured lines wit h thic kness ro ughly pr oportional to the zone wi dth. Thre e routes to the south - eastern quarter of the 6 th arc are labelled and indicated with red circles, The tw o stable solutions are con nected with a dashed red line to illustrate how the angular dependence determines the local length of t he search zone in the same way as dispersion determines th e width of a n optical filter , where lar ge dispers ion gi ves a nar row filt er (Plougm ann and Krist ensen, 2004). The dash - do tted blue line is t he measurement uncertainty. The search zone (ATSB , 2017) ha d small angul ar depen dence and hi gh   with small chance of finding the airplane. Table 4. R - values corrected to oblate earth. Oblate corre ction of R for placing on earth     (nmi)     (nmi)     (nmi) * ) Solution towards 13.3˚S 1658.03 2618.98 2661.6 Solution towards 34.6˚S 1675.8 2612.6 2654.6 * ) Using aver age of mode l and measur ed value as end - position Figure 4. GM i llustration of the entire south - eastern solution includ ing the initial part from MEKAR via the U- turn to the mergi ng point near B andar Ac eh with t he airpl ane coo rdinat es when ha ndshakes took p lace. MEKAR (6.50 o , 96.49 o ) P -2 = ( 6.98 o , 95.81 o ) P -1 = ( 7.86 o , 94.73 o ) P 2 = ( 8.14 o , 93.81 o ) P 1 = ( 14.26 o , 91.21 o ) P 3 = ( 2.73 o , 96.98 o ) P 4 = (- 3.36 o , 100.72 o ) P 5 = ( - 6.68 o , 102.77 o ) P 6 = ( - 12.44 o , 106.42 o ) P 7 = ( - 13.28 o , 106.96 o ) Crash Coral Bay 6.3 o - Bend U-turn Perth Kate T. Delay 8 min. MH370 flight route For the south - eastern solution we co mbined the radar da ta at MEKAR with the h andshakes at 18:28, 19:41 and 20:41, and the BFO value at 18:40 with a U - turn soon after 19:41 and a final m erger with th e straight rou te soon afte r 20:41 (with a small delay t o match ) to construct th e initial part o f the route. A fter several att empts , we are convinced ther e is only little room for different solutions, but minor deviations (up to ±10 km) are possible. In f act , many solutions i n south - easterly d irections can be made to follow straight l ines by introducing a few minutes delay near Bandar Aceh in contrast to the conclusions by Inmarsat , but with sig nificantly w orse fitting quality than o ur particular so lution . In addition, we extrapolated the fit to 00:19:29 in the other end and found almo st perfect agreement with the R - value, while the Doppler - shift deviated significantly downwards, as expected for an engine flame out (Holland, 2017). We choose a middle point as our best guess for the crash - position (13.279˚ South, 106.964˚ East). 3. MATHEMATICAL TESTING OF THE VALIDITY AND P LACING OF THE STABLE SOLUTIONS USING ELLIPTICAL TRIANGLES. We derived a nalytical solutions to com plete an elliptical tri angle includ ing satellite perturbation s by adding the small triang le formed by the net satellite movement in the upper corner as illustrated in figure 5 . The first and last positions are at R min and    (6 th ar c), wh ere R min i s linked up to the most northern satellite position. The small triangle is handled with simple Euclidean geometry. We use the non - Euclidean value for the angular sum in the combined triangle with area, A , extracted from GM by back - extrapolation of the R min and    lines to their crossing point O as sketched in figure 5 . The resulting equati ons are 90° +    +  = 180° 󰇧 1 +      󰇨 (9)   =   sin  sin ( 92 . 679°   ) (10)   =   sin  sin ( 87 . 321°   +  ) (11) where r s = 115.748 km is the length of the linear satellite projection m ovement from 19:41 to 00:11, γ is t he top angle at O for both t riangles, φ is the angle from east at the satellite projection point at 00:11 in clockwise direction t o the airplane position at the 6 th ar c, a 0 i s the length of the back-extension of R min to O and a 6 is the extension di stance of    back to O . We again use the tilde to illustrate that the satellit e ground p rojection mo vement is ad ded back on, since we are here placing the solution on eart h using the true satellite projections as fix -points . Equation ( 9 ) is based on (Wikipedia -1) , while we have derived ( 10 ) a nd ( 11 ) u sing the information above and the data on satellite projection movement. (    , A ) assume the values (43.294˚, 6.678·10 6 km 2 ) and (42.963˚, 6.729·10 6 km 2 ) for the south- eastern and southern solutions at the 6 th arc respectively. Figure 5 . Sketch of the c omposit ion o f the small (satellite movement) triangle used to co mplete a spheri cal triangle for P ythagoras - testing of the placing and v alidity of the stable solutions. The dash - dotted part of the large right - angled tr iangle is deliberately drawn too small, and the satellite pr ojection is added as a curved arrow (in movement dir ection ) with the relevant star t - and end - point s m arked by filled blue circles. Equations ( 10 ) and ( 11 ) are used as independent checks of the solution (assuming negligible change in A as a functi on of φ near optimum), and the ac curacy of the GM placing by comparing the values of φ , a 0 and a 6 from GM and ( 10 ) and ( 11 ). The first is do ne by insertion into equation ( 7 ) which is used as a non - Euclid ean analogue of Pythagoras (Wikipedia-2) . For the two stable solutions we find φ = 19.3˚ and φ = 51.6˚ respectively b y demanding exact validity of ( 7 ) for the large tri angle. The south- eastern solution agrees within 20 km with GM , while the southern solution deviates almost 200 km. The reason is that this region has large non - Euclidean modification due to rapid changes in A and relativ ely large oblate corre ction factors (see table 4). All valu es of a 0 and a 6 agree within 0.8 % with ( 10 ) and ( 11 ), except a 0 in the southern solution which deviates almost 6 % (for the same non -Euclidean and o blate reasons). W e have hereby performed tests of the validity and placing of our stable solutions. 4. OTHER POSSIBLE SOLUTIONS SATISFYING THE SATELLITE DATA. In order to be completely sure we find all possible solutions (candidate rou tes) for the model we derive formulas for identifying additional (unsta ble) solution s. Here larg er approxim ations are made, so the result only allows crudely estimating sixth arc points within 250 km along it. The radi al precision is still conserved. First order perturbation treatmen t of the average movement of the satellite during the flight tell s us t hat flights directed 3˚ south (from sat ellite ground projection) have minimum ne t sensiti vity to satel lite movement. All other flight routes will be affected predominantly by t he satellite movem ent perpendicular to this direction. The velocity, v 0 , going into the Doppler- effect is therefore given by                            (  ) =  +   sin (   3° ) (12) This veloci ty will contr ibute to the Doppler effect through it s cosine projection along R given by   (  ) cos 󰇡  (  )  󰇢 =   (13) v sat ≈ - 50 km/h near 6 th arc, meaning that the Doppler effect is reduced for flights in southerly direction, so they need higher v to give the same Doppler effect . The ind ex ( p ) labels three diff erent solution types: p =1: Normal speed routes: The flight joins up with the model - line after the R min -point at t 0 (so t 0 is negative) p =2: High - speed routes: Near airp lane perform ance limit, join ing up before R min (giving positive or z ero t 0 ). p =3: Low - speed routes: Unrealist ic curved routes with small v and   (  )  , typical speeds around 650 km/h. The optimum values for this simp lified model a re determined f rom the three kno wn solutions: v sat = - 56.3 km/h, v 0 (1) = 780.5 km/h, v Doppler = 571.2 km/h, v 0 (2) = 845.5 km/h As a test we independently find (due to surplus information) v V.I. = 879.9 km/h , which is c lose to the value used by ( Iann ello and Godfrey, 2016). Because the exact starting point is unknown, the largest uncertainty associated w ith using this simple model is the determinatio n of   (  )  . For eventual extra southern routes, we chose a virtual starting point at a strategi c position in th e middle of th e southern mou th of the Malacca Strai t (west of Bandar Aceh) which all routes must (roughly) pass through. U sing this we find no additional solutions within the fuel range, confi rm ing that the so lution is complete in southerly directions. For norther n direction s, we use MEKAR as starting point and find four optional solutions. Two are unphysical (strongly cu rved and/or zig - zag ) routes ending in the Yunnan province of China and o ne leading to western Kazakhstan is outside the fuel range. The last one towards a 6 th arc crossing at ( 43.87˚ North, 70.06˚ East) in south- easter n Kazakhstan is almost possible, but is ruled out by impossible timing and/or flame - out before 6 th arc without unrea listically s trong tail wind against the global trade winds (Schneider et a l. , 2014) during the first half of the route in addition to the difficulties mentioned before. The solution by Iannello and Godfrey (2016) is timely even though it includes loiter or a U- tu rn near Perka. However , a large    speaks against it. The total χ 2 is 37.5 times higher than our best solution and 25 times above the measurement uncert ainty. It also suffers from relatively po or agreement with some of the additional data. Most importantly , it b alances on a mathematical knife - edge. Norm ally there are two such solutions as in northerly direction but this one lies at the bifurcation poin t where there is e xactly one (double root) . A l ocation near the b ifurcation leaves little room f or improvement. The reason is that it uses maximum speed to match the margina l angle in this area. In conclusion, we f ind this type of solution an unlikely candidate for the flight route. After effectively comp leti ng this manu script ( Iannello , 2017) published a more elaborate analy sis which fit s better. However, it displays   making the deviation look smaller than    a nd includes all points from 19:41, which lifts the bottom level for   so the curve looks very flat , surprisin g some of th e bloggers . This illustra tes how e ssential it is to exclude 19:41 to find t he south- easterly solution with the U - turn. If one includes 20:41 , it is possible to find this solution but the fit is sys tematically poor without delay and a small dire ction change between 20:41 and 21:41. 5 . INCLUSION OF ADDITI ONAL DATA A ND INFORMATION I N THE ANALYSIS. We use all other pub lically available data to choose between the two stab le solutions. There are mixed opinions on the debris beaching (Iannello, 2017) , but several reports conclude it fits better th e further north o ne gets the crash as long as it is not near the Indonesian coast. This goes for back - tracing of the flaperon from Reunion performed by Geomar in (2015) and (2016), a report from a group of oceanographers (Theguardian, 2016) and b ack - tracing of temporarily b eached deb ris performed by ( Chilli t , 2018) . In addition nothing was fo und during the aerial search or the seabed sea rch in the official search zone (including i ts later extension) , which ( in combination ) effectively covered most high - probability area near the Inmarsat and Iannello solutions . Together t he se issues ma ke southern route s more than an order of magnitude less pr obable than the south- eastern route. Analysis of the flapero n biofouling also delivers important results (Wise , 2018 ) . There was only one species p resent (tropica l Goose barn acles). Furth ermore Ca/M g analysis of a large barnacle shell ( Dailymail, 2016) and (ATSB, 2017 ) shows that it experienced an unusual thermal history with initially very high tem peratures th en dropp ing to values near its growth l im it of 18˚C and then gradually rising to values typical for Reunion. No one has been able to come up with a good explanation for this peculiar result. However, looki ng at sea currents and temperature w orld maps (Hunter, 2013) this kind of behaviour is possible when s tarting out n ear our south- eastern solution in the fall . Here a weak current of hot, nutrient- poor tropica l water from north carrie s water towa rds south a mpl ified by hurricanes (Chillit , 2018 ) and global warm ing (Feng et al. , 2013) , where it meets and mixes with the relatively cool and strong Western Australian current coming from the south. The m ixed current continues via Reunion to Africa under heating by tropical sunshine. Therefore , debri s coming from this area is settled most ly by tropical b arnacles and will experience such a temperature p rofile when s tarting in March . The temperature drop is intens ified and prolonged by the onset of winter . Therefore, the barnacle results add roughly another order of magnitude preference for the south- eastern solution. The only we ak point of this simp lified analysis is that the barnacles seem to be much too young (only a few m onths) judging from their size. However, one must remember that their grow th - rate depends on two parameters – temperatur e and available nutrients. During the initial pa rt of their l ife the tem perature was ideal for growth , bu t with only small a mounts of nutrients leading to a relatively slow growth. During the middle part , they practically went into hibernation due to tempe ratures appro a ching their growth limit. Finally , they drifted towards Reunion and grew increasingly rapidly. This explanation also clarifi es another issue , namely why so me of the barnacles grew above the water line. Most likely, there were tem porarily a much larger number of trop ical barnacles on the flaperon weighing it down and preventing settlement of other barnacles during the winter . However, during the cooler period most of the initial barnacles d ied and later fell off leaving a few alive above t he average water line. Most likely the same happened for other debris, which explains why it wa s much cleaner than usu al, leading som e investigators t o believe tha t the debris was planted. In case the cr ash had happened at the Inmarsat position , a variety of different bio fouling with origin in incr easingly war mer c lim ate -zones would grad ually have covered the flaperon, resulting in much high er biodiversity . However , the line of circumstantial evidence does not stop h ere where some readers m ay already be convinced. Depending on the finer details the official investigators estimated (Holland, 2017) that the crash starts between 00:19:29 and 00:19:37 where the plane is losing height somewhere between 15700 feet/min (fpm) and 25300 fpm indicating insignificant pilot control. The average downward speed will mo st likely be below the middle of this interval s ince a crash at 25000 fpm would probably give smaller pieces of deb ris than observed (Wise , 2018 ) . In any case an uncontrolled crash gives rise to a random horizontal w alk near the end - point of the lin e reaching the sea surface 1 - 2 minutes later (Steel, 2016) . This is in excel lent agreement wit h a sound feature record ed at the nuclea r arms listening de vice HA01 near Cape Leeuwin (34.892˚ South, 114.153˚ East) at 00:49:42. With a water - sound- speed of 1.484 km/s expected along this passage (Steel, 2016) the plane should crash 114 s after 00:19:37 at our south- easte rn solution to match the recording. This corresponds to a downwards speed of roughly 15000 fpm and a crash t ime of 1:54 min. New contrail data point to a cras h 20 -4 0 s later. A moderately rapid crash fi ts the average fragment siz e. In contrast, no sound features agree with the southern routes. Finally , the eye witness ( T ee , 2014) des cribes a large airplane diving to low height and flying slowly west of her boat located north of Bandar Aceh at (6.628˚ North, 94.438˚ E ast, plus 15 km east - northeast due to later time) on the night MH370 disappeared. It came from north, made a mo derat e turn nearby towards her and disappeared somewhere south without landing. The plane had a red halo around it and the normal lights and windows could only be seen in the cockpit while the rest looked strange. Consideri ng diffraction of red warning light an d small windows in the cabin versus white (green) lig ht in large cockpit windows puts the diagonal distance between her and the plane around 2.5 km . She estimates a 3- km horizontal d istance . These observations agree with our south- east ern solution , where t he plane makes a 6.3˚ turn 17 k m north of her pos ition while diving and causing delay, and passes 2 km west of her boat. Only the time does not fit. She puts the closest approach at 19:20 while our model says 20:59. This is also essential to get spatial agreement due to the movement of her boat perpendicular to the predicted flight route. H owever, she was particularly uncertain ab out the time, so it is n ot an unlikely error in an area where l ocal time -zones in India and Indonesia deviate two hours w ithin a few km of her position. 6. DISCUSSION. For the optimum Christm as Island solution, w e estimate that the model - placing- uncertainty is ±35 km along the sixth arc. The maxim um error will theref ore be roughly ±70 km (2 σ ), which we use as the (half) - length of the search zone. For the transverse extend we propose ±15 km, since the largest contribution to this uncertainty is second - order placing error (found experimentally to be 1/6 leading to ±6 km) followed by random walk uncertainty after 00:19:37 (±3 km) and fitt ing uncertainty (±3 km) giving a total of ±7.5 km transverse uncertainty, and again choosing the double for the extend of the search zone. It is worth noting that all data point to positions within the central 10 % of the search zone with the largest deviation coming from Kate’s observation leading to a point 4- 8 km south of the centre of the zone. As a funny coincid ence a 5 - km shift sou th will perfectly a lign the st raight flight w ith an end - point at Coral Bay Airport and roughly remove the 8 - km offset at MEKAR (Iannello, 2017) . We therefore guess there will be a relatively high chance of f inding the plane within 350 km 2 out of the 3500 km 2 . There i s also something speci al about the Christmas Island route going through th e intertropical convergence zone (Schneider et al . , 2014) where satellite detection and long - range radar are hampered by tropical thunderstorms, indicating intelligent planning. Most likely , the perpetrator (s) also knew about the ha nd shakes and deliberat ely directed and timed the f light to get close to the worst possible mathematical data -entanglem ent with satellite movemen t through spatia l correlation , making it almost i mp ossible to find the plane because this a llows for a multitude of solutions with si milar fitting quality . This w as achieved by flying a route resembling a magnification of the sate llite gro und projection curve with the SDU restarting as the plane left Malaysian radar covera ge and entered this route. T he U - turn was ca refully aligned to match the top of the satellit e projection curve , and the i mmediate contin uation was slightly curved , fo l lowed by a straight flight t o the end , pointing to perpetra tor(s) with k nowledge of entanglement (Wikipedia- 3) and (Kriste nsen et al. , 2012) . Interestingl y , f light simulation data found near the captain’s priva te computer (Steel, 2016), (Iannello, 2017) and (Wise, 2018) resembles a classical analogue of a quantum Singlet - entanglem ent (anti - correla tion) with the satelli te motion while the actual route matches a Triplet - entanglement (correlation) providing optimum hiding (Kristensen et al. , 2012) . However, c om bination of several scien tific metho ds with topology optimization (Plougmann and Kristensen , 2004) and (Bendsøe and Sigmund, 2003) allowed discovery of the Christm as Island route and its ident ification as the best solut ion . In case one would al so go for a repeated s earch near the end of the southern route, the placing- uncertainty is 2.5 times larger due to non - Euclidean effects. This leads to a search zone area of 20 000 km 2 which is close to the original official estimate of 25000 km 2 (ABC, 2017). Howe ver, most of it was already pa rt of previous search zone s so a much smalle r area will be sufficient . T o further strengthen evidence for our solution it is possible to do one of the following things: 1) Ask how the received signal strength can go up with increasing atmo spheric travel from 20:41 to 00:11 with reference to (Inma rsat, 2015 ) coverage and an antenna model? 2) Look for coincidence with sound recorded at Scott Reef or HA08 near D iego G arcia. Unfortunately, data for the relevant ti mes is not publica lly avail able (and Diego Garcia data compromised by local noise from a military exerc ise as shown after our publication). If we find one such coincidence, classical triangulation p inpoints th e exact crash - site to a few km. Ho w ever, i t is the local in - coupling in the sound - guiding layer of the ocean, which is most importan t – not the distance. Underwater mountains north of the expected crash - site may cause most coupling in southern direction and add some confusi ng echo. For a weak signal , this potentially prevents id entificatio n at Scott Reef . After effectively c ompl eti ng thi s manuscript , a method w as published for analysis of sound propagation in water to de termine impact - distances with only one detector (Kadri et al., 2017) in cluding appl ication for a p artial re - analysis of the data from HA01 on the night MH370 disappeared . There is a deviation of 2 minutes between their t ime axis and previously published da ta from HA01 (Steel, 2016) . If the new axis is c orrect, MH370 crashed exactly as the last handshake was in te rrupted ra ther than 114 s later, an d the distance to the satel lite projection would be roughly 11 km shorter due to smaller h , predominantly via equation (3) . However we suspect there is an error in (Kadri et al., 2017) since the distance calculate d for the stron ger neighbouring peak seems to be off by roughly 200 km as pointed out by some bloggers (Iannello, 2017) consistent w ith a 2- min ute error. Alternative ly it is possible tha t the two signals accidentally coincided, which may explain the distance- discrepancy by inte rference . 7. CONCLUS ION. In conclusion, we have used a novel combination of methods from science and engineering to disentangle and discuss all four solutions to the model of the disappearance of flight MH370. For the two stable solutions , we have delivered rigorous proof of their placing and validity by using a non - Euclidean version of Pythagoras. All other publically available data point to the Christmas Island solu tion, while we rule out the other three. The southern route is second best bu t still unlik ely , and the decision to stop searching was correct (ATSB , 2017) . Resumed searching at that location makes only little sense, since the probability of finding the airplane is below 1 % according to our estim ates . We propose instead a new, focused search zone of 3500 km 2 centred at (13.27 9˚ South, 106.964˚ East) with slightly elliptica l shape along the seventh arc and a total length of 140 km and width of 30 km. The probability of finding the plane there is above 90%. After com pleting the manuscript, we found additional evidence for the Christmas Islan d solution in 2019. In 2024 , we found black shadow contrails and a mushroom - cloud from the crash as described below. This sh ifts the crash position to ( 13.53° South, 107.11° East ), reduces the search area t o a 10-km radius, and increases the probability to 99 %. 8. NOTE ADDED IN PROOF IV . Based on discussions with several bloggers (Iannello, 2018- 2) and other independent investigators after the publ ication of th e first , second and third version s of our manuscript on ‘How to find MH370?’, and additional research into specific deta ils , we decided to rewrite the ‘Note added in proof’ paragraph to update the informatio n a third time in 2024, including an appendix on con trails . The rest of the original manuscript rem ains practically unchanged with a few minor updates and corrections . Initially we reanalyzed the connection between different routes and the obs ervations by Kate Tee using the speeds found by V ictor Iannello for different end - positions at the seventh arc (Iannello, 2017). Hereby we found that only routes ending between 26° S and 32° S could be consistent with her observations if they took place betw een the particular gybe s she pointed out. Only routes leading to latitud es between 11° S and 16° S would be cons istent , if her observations took place after th e last gybe. All other possibilities lead to inconsistency with her observations. As part of the same analysis, we looked at possible agreement with the s ound - features at Cape Leeuwin. The most likely feature is only consistent with end pos itions between 12° S and 15° S, while one of other two (much less likely) peaks could move the end -positio n down to 16.5° S (maximum). All other pr oposed end - positions are inco nsistent with the observed peaks. Trusting the input f rom Kate Tee (except the before/after gybe issue ) this makes our solution at 13. 5 3° S most likely, and practically rules out any solution between 20° S and 25° S. A stat istical analysis inc luding information from e.g. the unsuccessful seabed searches (detail s no t included here) leaves only 2 % pr obability of finding the wreckage between 20° S and 25° S. New contrail data from 2024 redu ce this to 0 %. Secondly, it became clear from discussions with bloggers (Iannello, 2018 - 2) that th e method we presented for calcu lating the BFO for diff erent routes than the Inmarsat s olution was inaccurate. It needed an additional correction for the l o cal speed of the airplane. Furthermore, it is unclear if the method we used for handling a pressure - induced shift (presented in the first version of this paragraph) i s correct. Concerning both these issues, it is extremely important to keep in mind that th e errors are rela tively small (in particular the pressure shift) , and that due to the sm all weight put on the BFO in our topology- optimization , these error s will only lead to sm all shifts of the final positio n. However, it is still im portant to ma ke the correc tions properly for two re asons. Firs t of all , there is a risk that the effects may exceed the perturbative regime, s o another one of the four possible end positions could overtake the role as the best fit. Secondly, even if the corrections only lead to shifts within the measurem ent uncertainty, they m ay still add significant ext ra search time and cost for a resum ed sear ch if handled incorrectly. The procedure for the speed correction is simple to first ord er. Because the speed on the route towards 13. 5 3° S is 3.1 % lower than for the Inmarsat solution, the net (inter nally calculated) Doppler shif t compensations reduce by this amount. The first - order correction s for this bring our BFO values very close to t hose calculated by Victor Iannello (Iannello, 2018- 2). There is still an insignificant deviation around 2 Hz due to smaller effects. In the follo wing, we will ignore this since it is far below the statistical unce rtainty. How ever, the 3.1 % correction is clearly significant, and in case one only looks at the BF O ( letting the weight- factor on BTO go to zero), one finds an optimum fit somewhere between Victor Iannello’s original 27° S solut ion from 2016 a nd the most recent independent group recommendation of a position near 34.4° S (Iannello, 2019). Since the BFO values are m uch less accurate than the BTO values, and their de tailed interpretation is somewhat uncertain, this is clearly not a good approach. By instead choosing to keep our original 1000 - fold lower weight on the BFO, one gets a much sm a ller shift (around 20 km south) near our original 13.3° S solution (fitting 13.53° S) . This seems like a complete ly insignificant shift, but it is unf ortunately still so large that it beco mes marginal to use a first - order perturbative approach, and the chi - squared values for the two best sol ut ions get closer to each other. It is unclear how to handle this situation without a complete re analysis fro m the bottom , but before doing this, it is ess ential to con sider the impact of other perturbations from changes in pressure and tem perature in the cabin. Looking at the 13. 5 3° S solution it is obvious that the entir e solution only makes sense if an attempt to bail out by parachute took place while flying slowly and at low height near Bandar Aceh. This would have left one of the doors behind the wings open, leading to significant d rops in pressure and temperature as the airplane return ed to 11 km flyin g height. It is unclear how to handle the pressure change, but sin ce it is most likely around 2 Hz (average value ), we choose to ignore it using the same arguments as for the 2 Hz deviation above. However, the temperature change is clearly significant. Bloggers (Iannello, 2019) found a shift for the temperature - stabiliz e d oscillator of 0.3 Hz/K. In addition, there is an expected shift in the power- out put for the SDU of -0.05 d B/K . One of the bloggers at (I annello, 2019) calcula ted a temperature drop of 60 K inside an open cabin . W e largely agree on this number provided the heat supply system s are ‘off’ . However, one also needs to know the detailed temperature curve. We found this by assuming a linear coo ling rate and a thermal time constant K -1 around one hour. The exact value of K was determined by solving the thermal differential equation   =  (      ) and fitting t o the result, w here T 0 = 25°C and T 11km = -45°C, giving K = 0.87 hour -1 from  =    70 °  1    (   )  If the cabin is cooling for 3 hours with a one -hour thermal time constant, this gives around 65°C temperature drop. Below we describe how to find the exact value of K . In order to perform the therm al fitting w e use d the additional (and practically overlooked) parameter measured by the Inm arsat system, namely the received power. In order to use this, it is necessary to d evelop a model for the power transmission to 3F1 . We d eveloped such a model based on a 3 - step procedure . W e estimate the diffractio n loss using the Fraunhofer approximation. We use angular momentum projection to estimate the circular polariz ation overlap , and we estimate the effective antenna area using geometr ic projection . In an earlie r first - order atte m pt, w e used an em ission pattern calculated by (Harvey, 1 963) under the assumption that the antenna was of the oldest ( static ) design type as described in the paper by (Fu, 2012) . After c ommunicati ng with bloggers at (Iannello, 2019) we were informed that the airplane used a mechanically ad justable phased array antenna for pointing the beam. We therefore developed a model for this antenna type using the method described above, while now ignoring pointing errors. This model fits muc h better to the data than the old static antenna model, confirming the validity of the information. By comparing to th e received power at the previous flight between Beijing and Kuala Lumpur , we observed an additional 1- dB deviation from front- back mirror symm etry (highest output in forward direction) , which seems reasonable given th e detailed geom etry visible in pictures of this antenna type . We fi tted the complete model to the last two handshakes before the airp lane disappeared , the two points at 19:41 and 20:41 and t he four middle points from the previous flight (Beijing to K uala Lumpur ), meaning 8 points in total, with the power - offse t as the only free parameter . For fitting of the previous flight , we used flight parameters fro m (Davey, 2016) and exclusively va lues measured in channel 4 . However, w e ignored points measured very close to tak e - off or landing (w here the exact climbing /decent and tilt angle s are important but unknown) , and the points directly af ter reboot (where the tem perature is unknown). All 8 points fit within 0.2 dB , which is nice considering that the expe ct ed uncertainty is around 0.3dB . It is important t o notice that the m odel corroborates that the airplane was approaching 3F1 at 19:41 , since this point is only 0.1 dB off fro m the mode l when the 1 - dB direction asymmetry is included (if not , it is 3 sigm a off , - equal to 0.9 dB ). We used the antenna model al one to determine the expected power for all the following points after 21:00 and found poor agreement . However, by including the thermal model with roughly one hour time constant the agreement dramatically improved. We completed the work by allowing the thermal constant to vary freely and f ound the best fit with K = 0.87 hour -1 . We have plotted the results in Figure 6. Figure 6. Illustration of the agreement between m easured power at 3F1 (large black circles) and predicted power for t h ree different cases. The (s mall ) blue circl es conne cted wit h blue l ine is our model i ncluding t he antenna t ransm ission an d cabi n cooling due to a n open d oor l eading to increa sed emi ssion powe r ( - 0.05 dB/ °) . The last point is corrected in accordance with figure 8. The grey curve is the expected power assuming a curved fl ight as in the first estimate by INMARSAT (hig hest proba bility version) . The red curve is the power expected along the final INMARSAT route , F (and /or the slightly sh ifted proposal from (Iannello, 201 9) ). Maximum d eviations are 2.5 dB (8 sigma) for the red curve and 1.5 dB (5 sigma) for the grey curve. In principle, the grey curve improves by shifting it furt her north, but this quickly pushes the flying speed below the stall limit and makes such a route impossible to follow. The red curve has un acceptably po or agreement with the measured values, while the agreement with the blu e curve is practically perf ect. Furthermore, some of the bloggers at (Iannello, 2019) criticized our mo del for having too large BFO deviations. While these deviations may be due to unusual tail - heavy statistical behavior of the SDU oscillator aft er exposure to low pressure and tem perature between 17:07 and 18:22, it may also be due to cooling after 21:00. In order to test this we calc ulated the expected osci llato r shifts ( 0.3 Hz/°) f or the SDU during the cooling found to match the received power levels in Figure 6. W e used the syst ematic (3.1%) BFO shifts fro m our model results and plotted these two data sets agai nst each other in Figure 7 . Figure 7. Illustration of the agreement between systematic (predicted) BFO errors and re sults deduc ed from our therm al model wit h an open ca bin doo r. The last point i s correcte d in accor dance wit h figure 8 (grey) . Notice the smo oth decay of the small e rror indicating that the r emaining deviation is potentially due to a memory effect fr om the previous tempe rature and pressu re cycl ing. It is also im portan t to notic e that t his figure ignores th e random deviati ons from table 3. T he lar gest one o f thes e @21:41 i s mos t likely due to passage through a tropic al thunderstorm in the inter tropical converge nce zone . The plot shows good agreement between our thermal model and the systematic BFO errors . This corrobora tes that t he deviations a re most lik ely due to a dramatic cooling of the cabin from an open door. The only larger systematic deviation (4 Hz ) occurs at 20:41. This may either be due to the tail -heavy drifting effect or indicate that the cabin door opened just before 20: 41 while the airplane was near 2 - km flying height ( leading to a sm all initial cooling). W e find the drifting explanation most likely, since the deviation slowly decays to zero o ver the following 3.5 hours, while the power output shows no s uch behavior (as expected , since th e power amplifier has no memory effect ). Finally, we u se the calcu lated tempera ture drop to estimate the average o scillator shift for all points af ter 21:00 to -15 .5 Hz. This means that we must subtract 15.5 Hz from the calculated (negative) D oppler shift to correct for the average effect. As part of the development of the original version of this paper, we derived a simple formula for the end - position shift due to a 1 Hz shift for all the last four points. This reads one - degree latitude (towards north) to com pensate each negative Hz added. The Inm arsat paper comes out with a 25% larger value, but that is for one point individually, and the continuity and straightness of the curve reduces the collective value by roughly 25%. This means that any solution calcula ted based o n BFO (alone) must be shifted 15.5 degrees (latitude) north to correct the effect of the temperature drop. Therefore, the optimum for a purely BFO based solution without temperatu re drop would be around 29° S, which is not far from proposals fro m the independent group (Iannello, 2019) . The remaining di fference could be due to drifting, random errors or low pressure. At 00:11 th e firs t engin e had flamed out, so the airplane automa tical ly turned attempting to compensate the (maximum skew) engine push. A simula tion at (Iannello, 2019) in d icates a flame ou t for the first engine around 00:08. We use d this to correct th e microwave power loss for the last point at 00:11 . New 2024- data gives o nly small additional correction s. Now the big question is if this is a wild speculation or a fact. First, we l ooked at the reboot at 18:25. A couple of experts (in particular the blogger DennisW at Iannello’s b log) had been complaining vigorously about the normal booting for the SDU. If the left power generator had been off to allow faster flight a t high altitude along the T hai- Malaysian boarder, the SDU oscillator would have been extremely cold (since its heater was also off) and at low pressure during the SDU booting, which would lea d to severe o scillator d rift for a prolonged period. Never the less the booting is normal as shown by Holland. A si mple explanation is if the SD U was re - conn ected at a later (conve nient) time when the cabin was back in normal condition b ut now manually to one of the power supplies . T his is consistent with Radar observations between Penang and MEKAR of an increasingly normal flight along waypoints after the fast, high, and slightly noisy (manual?) flight before Penang. While this is in g ood agreement with the SDU connected back to a generator, there a re still some re maining detai ls to settle . T he described succession of events leads to a clockwise crash spiral sim ilar to case 4 in the l ist of Boeing sim ulations if the SD U is connected to the le ft generator (in contrast to most of the Boeing simulations ending in counterclockwise spirals), and places the debris roughly 13 km further south and somewhat west. We initially guessed 10 km west, partly f rom arc curvature, partly due to height los s at 00:19. In combination with our previous es timate of a 5 -km ge neral position error , this gives an 18 - km south and roughly 10 - km west shift co mpared to our original position if we ignore the turn in direction before 00:11. Including this turn and a long er than expected straight section adds roughly 18 - km additional shift towards west gi ving a total of 28 km. However, there would be no normal handshake at 00:11 with the SDU connected to the right generator , and the straight flig ht segment af ter 00:0 8 would be too short to fit the second engine flameout with too little rem aining fuel . However, so me experts still feel t his entire explanation is somewhat speculat ive . In order to deliver a definitive proof , we therefore used our refined knowledge of the end- scenario to look for matching contrails fro m the airplane in satellite p ictures. New evidence from 2024 indicates that the cr ash fits bette r with the major ity of the Bo eing simulations. We found several aligned contrail segm ents matching our solution unt il briefly afte r 00:00 in two consecutive pictur es from the METEOSAT - 7 weather satellite (Weather Graph ics, 2014), followed by turn ing at 00:08, and ending in a spiral. Figure 8 illus t rates this con trail with the interesting added feature that it becomes abruptly thinner and more intens e at 00:08, exactly as the right engine flames out, th e power of the lef t engine increases to ma ximum and the t urn takes place. The satellit e pictures al so contain a c ontrail from another flight coming from southeast and flying d irectly over the airport a t Christmas Is land. Ba sed on the measured speed, timing and angle (extracted from contrails in consecutive pictures ) we have identified this contrail i s either coming from a flight leaving Melbourne at 5:00 am local time heading fo r Dubai (EK409) operated with an Airbus A380 explaining the more intense contrail than the one form MH370, or alternatively a cargo flight from New Zeeland . A cargo fli ght is most likely also using a l arge airplan e type with s imilar intens e contrail . Figure 8 . Illustratio n of two contrail s near C hristm as Island shortly be fore MH370 crashed in the region. The dashed white line may be flight EK 409 from Me lbourne to Dubai or a cargo flight from New Zeeland. The orange line r epresent s a fli ght comi ng from north -n orthw est, making a moderate turn at 00: 08, and ending i n a spira l south -s outheas t of Chri stmas Isl and. T he tim es and posi ti ons of this flight agree roughly with our model ( white stars 5-7 ) for MH370 . T he tur n time agrees with predicted flameout for the first engine wi thin 19 sec onds (Ianne llo, 201 9), and the simult aneous re duction i n thickne ss and increase in strength a gree with one of the Boeing simulations . In 2024, we found new e vidence usi ng shadow contrails as descri bed in the appe ndix ‘B lack Swan ’. The a bsence of a sha dow contrail for the last part of the r oute above (aft er 00:08) indicates thi s s egment was a false contrail, most likely indu ced by transformation of the satellite pictures to rectang ular coordinates. Instead , there is anoth er contrail - segment connected to the upper pa rt of the route near Chris tmas Is land (com plete ly unchange d) with corresp ondi ng shadow l eading to a position 16 km on the ot her side of the 7 th arc ( after tu rning left instead of right @ 00:0 8) . The new crash position is at (13.53° S, 107.1 1° E) 375 km from Christm as Island, where t he shadow contrail merges with its corresponding n orma l part , f ollowed by t he appea rance of a white mus hroom -c loud under the dash - arrow a few km south west , and r emaining visibl e for more tha n one hour. T he ‘Death s piral’ is qualitatively similar to the one abo ve , but tighter and shifte d 50 km east - sout heast aft er passing nicely throug h P 6 and P 7 . The end position in the middle of the spiral is marked with a yellow star. The scenario l eading to crash is si m ilar in the 2024 update (concerning spiral orientat io n), but the turn at 00:08 goes east (left) , the spiral is so mewhat tighter and it is moved 50 km east - southeast . Two seismic detectors (XMI, 2024) con firm both contrails passing Christmas Island in Figure 8. The airplane from the south passing directly ove r the CI airp ort gave large signals at the expected time in the seismic detector near the main building of the airport. We did no detailed analysis of this data, but concluded from the low frequency that it was a large, heavy airplane (bigger than MH370) flying somewhat faster than MH370. B oth seismic detectors recorded the passage of the airplane from no rth -northwest , providing a lot of detailed information. The center frequency (around 12 Hz) is typical for airplanes near the s ize and weight of MH 370. The exact signal arrival tim e from its shortes t approach to Christm as Island fits our m odel to better than 2 seconds precision (completely outstandi ng). The Doppler curve from the nearest detec tor at a low -noise, isolated loca tion in the southern part of the island fits with the speed to better than 4 %. The sign of the internal delay between the two detections prove that it was on the correct side of Christmas Island. The signal of the detector at t he airport ha d its first part cho pped of f by a hill nea r the airport, proving that the airplane came from th e north. Finally , co rrelation betw een the two Doppler curves and the observed contrail confirm the distance of the flight to better than 6 % (3 k m) and the angle to be tter than 10 ° agreement with our model. This entire collection of evidence from the seism ic signals prove without any signif icant doubt that the airplane must have been MH370 , since airplanes rarely pass o n that side of the island . In hindsight, this detailed solution also solves a number of smaller contradictions in the data. First, the downward acceleration found by the Holland paper around 00:19:37 seemed inappropriate compared to the Boeing simulations at that stage of a crash. K nowing from the power le vel at 00:11 a nd the contra ils that the first engine f lamed out a f ew minutes earlier than initially expected (around 00:08) brings this in better agreem ent with the Boeing simulations. Secondly, the straight f l ight from 00:08 to roughly 00:19 i s longer than the Boeing simulation. We think this is because of energy saving for the second e ngine since the pressurizat ion and heating syste m s were switched off shortly before 21:00. Thirdly, the termin ation in a tight sp iral dive after the contrail-end rules out an active p ilot ditching the airplane far from the seventh arc. Fourth, the 15 .5 H z temperature -induced (average) shi ft partly ex plains why the initial BFO plots from the Inmarsat paper deviated 10- 15 Hz from the measu re d values. Finally, the crash after a long flight to f uel exhaustion practically proves that i t was a deliberate act, and not a technical accident. As an added curiosity, the contrail from MH370 crosse s the northbound contrail at an angle of 13 degrees roughly 240 km northwest of Christmas Island, and the two airplanes might have be en close enough for visual contact in light from the rising sun. It would be highly interesting having a chat with the pilots of the other flight, s ince they might have seen MH370. Concerning agreemen t with the peak recorded at Cape Leeuwin , t he appearance of the mushroom-cloud indicates the crash happened 20 -4 0 seconds later than initially though. It also happened c loser to Cape Leeuwin moving things 20-25 seconds the opposite way. The remaining net shift towards later time (7.5±11 seconds) is unlikely to have significant impact, so the agreement rem ains good within the u ncertainty . Concerning the beaching pattern of debris , the westward shif t due to prevailing currents after the crash would bring the debris close to the maximum wind - field of hurricane G illian two weeks later . We are unabl e to calculate the detai ls precisely , b ut qualitativ ely it will improve agreement with observations due to some debris trapping in the hurricane , and thereby dragg ing several pieces to near 20° S where the hurricane decay s . A few open questions remain. Most im portantly, wh ether the perpe trator(s) kn ew leaving a door open shifted the BFO so it roughly matched the simulation in captain Shahs computer, and knew to eliminate suspicious oscillator drift and inconvenient handshake timing by turning the SDU power supply back on at an optimum ti me. If this were the case, we are dealing with a h ighly sophisticated, we ll prepared, and delibera te act of mass killing with scientific input to manipulate the data and m islead the investigators, not a simple attempt to h ide a suicide or an act of terrorism. The motive and fate of the perpetra tor(s) remain open questions. However, the nice agreement between our thermal modelling and the measured microwave pow er levels practically proves that he/ th ey attempted to bail out by parachute near Ban dar Aceh. The thermal modelling also explain s the unusual sta tistical distribution of BFO values recorded during the second attempted tele phone call at 23:14. The distribution looks like the upper half of a G aussian, indicating that low temperature and pressure far from normal operating conditions affected the SDU frequency reference. One final p iece of circum stantial evi dence also points toward a carefully planned act. The exact crash position is alm ost perfect to maximize the time for debris to reach f ligh t - or shipping-r outes. Looking at a n integrated plot of flights recorded by Flightradar24 across the entire w orld , we find that th e first debris wil l cross fligh t - or shipping -route s mo re than two months later , and most of it would beach in Soma lia (Iannello, 2019) in the absence of hurricanes, and never becom e available for the investigator s . The passage of hurricane Gillian furth er masked the crash po sition and helped misle a d the inves tigation , since it made the beaching pattern look as if the debris originally came from around 20° S . The Flightradar2 4 data also co nfirm few flights p ass on the re levant side of C hristmas Isla nd. 9. APPENDIX: SHADO W CONTRAILS – THE BLACK S WAN. During recent re - analysis of weather satellite data in 2024, we d iscovered several shadow contrails. The phe nomenon is not new, but because of its general importance for airplane detection, in particular for cases like MH370, and because of its optica l similarity with the shape of a black swan neck in satellite pi ctures , we dec ided to name the phenomenon the ‘ Black Swan’. In the following , we describe several new contrail observations related to MH370. We focus on details of shadow contrails, but also include other observations. In pri nciple, any airplane above the clouds during daytime hours make a shadow cont rail. However , shadow contrails can also appear during the night in infrar ed satellite pictures. There are two reasons for for mation of s hadow contrails. The first is simply the o ptical shadow they cast on clouds below. In m ost cases, this is extremely diff icult to see unless co nditions are good. Good conditions occur if the cl oud top s below are horizontal , well defined and with smooth surface. In addition, the sun must be at relatively low inclination. In addition, the shadow leads to local cooling of the top of the clouds. If an infrared camera is the observer, the contrast g reatly improves, since there is both dire ct infrared optical shadow and reduced thermal emission due to lower tem perature at the s ame time and pl ace . The thermal part of the effect also works at night, if the con trail is under thin clouds above a warm surfac e (typically h ot tropical sea ) casting an upward heat shadow . However , the thermal effect dem ands one more c o ndition. The wind speed must be low or moderate , not with too turbulent wind pattern , and predom inantly along the contrail to minimize therm al mixing . We noticed this was the case several places along our proposed route for MH370 because of its location r elative to the intertropical convergence zone and the related trade winds . Simultaneously, the clouds below (or in one cas e above) looked nice and smooth. Finally, we observed p erfect shadows f rom some stratospheric cloud s moving almo st perpendicular to th e normal clouds. As a first exercise, we calculated th e height of these clouds. They were at 19 km ab ove sea level in agreement with typical values. The result remained unchanged for different sat ellite pictures of the same cloud as long as we corrected fo r the changing solar incl ination angle and the satellite observation angle. This told us that Black Swa ns have two independent applications. They ma ke contrail hunting easier because of better contrast. Seco ndly, if the corresp o nding norm al contrail is visible, it is possible to de termine th e flying height above the clouds. This is valuable information, in particular for ascending or descending airplanes. H owever, it comes with a minor drawback. If the Black Swan is predominantly due to tem perature change, it takes roughly 15 minutes to develop fully. For ideal usage , the c on trail must remain stable for minimum 15 minutes, typi cally for 30 minutes , if stan dard wea ther satellites are u sed . We started sys tematic contra il hunt ing from the northern end of our proposed flig h t route. Initially we thought that iden tification of a U- turn (fina l major turn f or MH370 ? ) would be most important, but we discovered much more, including proof of K ate Tee’s observation . Figure 9 presents an overview of all relevant contrail featu res in the north ern region. Figure 9. Overview of c ontrail features related to MH370 in the n orthern sector. W e found evidence of a U - turn at the correct place and time. A low - flying infrared Chinese weather satellite observed a U - turn along the edge of a cloud far up in t he Andaman Sea. The position agrees within 5 km with our theoretical prediction of a U - turn for MH370 as shown in detail in Figure 10 . It is a n ice picture since the airplane is around 80 % through the U - turn , so the direction including some of the straigh t part lead ing up to the t urn is visible (indicated with arrow ) . This kind of navigation is extremely rare under normal flying conditions over open sea, - in part icular crossin g an airway w ith risk of a collis ion . Next , we found two sm all feature s in a p icture from a Japanese w eather satellite sta tioned far out over the Pacific O cean (M TS , 2024 ) . These confirmed the middle parts of the long trip up to and back from the U-turn. The important detail i s that the angles are correct, and that a third contrail se gment (between the two ‘legs’ ) cle arly identi fies as coming from another airplane by the sa me Chinese IR satellit e further south. Another satell ite picture IR@2 2:00 from the French weather satellite Meteosat 7 over the Indian Ocean delivered extremely important information near Bandar Aceh . The previous pictures (between 19:00 and 21:30) are unfortunately absent because the satellites so lar panels pass ed through the earth shad ow, so a small correctio n for an ho ur wind drif t is necessary. Slightly north of the position from where Kate Tee observed an airplane diving towards her there is a so - called fall - streak h ole in the thin, hazy cloud - cover, proving that an airplane did actually dive at t hat position . In addition, the contrail passing through the hole ‘inverts ’ in infrared , becoming a Black Swan . Before the hole, it is thin and white in IR (normal fo r a plane fl ying above the clouds at night) , while after the hole it became thick black (clearly visible) indicating that the airp lane was now below the clouds where the contrail absorbs and scatters heat from the ocean, cooling the cloud above and m aking the IR i mage turn black. When the contrail reached land in Aceh, there was no longer any cloud cover, and it turned white again. This is practically a fingerprint. The only problem is that Kate Tee got the time one gybe wrong as we already suggested in an earlier vers ion of our paper. If her gybe- number were correct, there would not have b een any trace of contr ail or drop- hole left at 22:00. Instead, there might have been something in the Jap anese sate llite picture at 20 :31 , but there is nothing at that position at that tim e. The southbound contrail in Himaw ari -7 stops shortly after passing the second Andaman Island as one would expect. Until recently there w as also a Japan ese satellite picture av ailable at 21 :00 (unfo rtunately deleted before we took a copy) showing a small er fal l - streak hole and white contrail until a few km into Bandar Aceh co nfirming the exact tim e the airplane passed. Th is means that it could only be MH370, thereby confirming a fingerprin t of the events. There is no longer any doubt Kate Tee saw MH 370 flying low and slow over Bandar Aceh , but it took place one gybe lat er than she rem embered . As a last detail, we w ould like to point out tha t with a norm al white con trail below a thin, hazy cloud at night , the visibl e optical contrast becomes very different, so K ate Tee thought it was smoke instead of just a no rmal contrail . Figure 10 . Chinese i nfrared satellite picture with the relevant pa rt of our proposed U - turn indicated with dashed blue line in agreement with the observed contrail from shortly be fore unt il 80 % into the t urn . Figure 11 displays recorded contrails and related phenom ena near Bandar Aceh observed by Meteosat 7 in infrared @ 22:00. T he sequence of events displayed in the figure , including the fe w- degree turn (6.3 degrees in our model), agree nicely with observations made by Kate Tee, but shifted the equivalent of one gybe forward in tim e. Figure 11. Whi te contrail, fal l - strea k hole, and bla ck shado w contrail observed ne ar Banda r Aceh. Additional inf ormation appears using an inte ractive display , since toggling betw een infrared and a near in frare d ( water vapour) pi ctur e show near infrared contrail s shortly outside Figure 11, both above the top and below the bottom of the figure . This conf irms that an airplane flew high ab ove the clouds in both places, and dived down in between . In addition, it allows the use r to dete rmine which of the lines above the Aceh province came from the airplane contrail, and which were due to effects from wind and tall mountains. The most obviously chosen (dark - grey) line agrees with th e near infrared contra il s as well as with our model prediction, and the IR contrails. This confirms the whole scenario. F inally, we are going to d iscuss contrail ev idence from the end of the flight near the seventh ar c. Several yea rs ago , we found contrai l segments near Christm as Island left by two d ifferent airplanes a s shown in F igure 8. Some of it appeared so ea rly after sun rise that shadow contrails are difficult to identify (insufficient solar h eating). Fortu nately, dat a from two seismic detectors at Christmas Island c onfirm the contrail s and secur e an ext remel y nice agreement with our model for MH370 for the contrail segment southwest of the i sland . Figure 8 describes a presumed continuation of the contrail recor ded at 00:30. N o other sources confirm this part , and we are now convinced it was a fake signal generated by a data- transform ation of the satel lite pi cture to rectangu lar coordinate s . The time and position for a heading change is correct (because the first engine ran o ut of fuel), but the ne w head ing was incorrect. In 2024, we decided looking for shadow co ntrails to find the correct end - scenario. Figure 12 shows a clear Bl ack Swan po inting to the opposite side with comparable angle in an untransformed IR picture from METEOSAT -7 . This means the heading change did appear at the expected time and position, but the airplane turned east instead of west in agreement w ith the most likely e lectrical config uration of the airplane according to Bo eing simulations . Next to the Black Swan , we also found the prim ary white contrail (fain t) . Figure 12. Shadow co ntrails r ecorde d in IR at 00:30 by Meteosa t - 7 after Christmas Island are marked with dashed bl ue (not ice 90° t urn). A faint white contrail observed in the same picture is marked with blue line. The normal white contrail also appears in the correspondin g visible sa tellite pictu re with better contra st as s hown in Figure 13. The most important detail is that the two contrails merge in one point, and no contrail is leading away from this point. Figure 13. Whit e contrai l recorde d in vi sible light at the same time and place marked with blu e line. Notice again the 90 ° picture - turn and agreement with the faint white con trail in IR above , and a white feature just after the contrail. It is also v isible as a bl ack spot in IR. That is the top of a mushroom - cloud from the cra sh. For shar p eyes, t here are spokes going fr om the whi te top t o an o uter rin g and clo uds belo w appear blurred. Pr obably the blur red ring is the s hock wave from the thermoba ric expl osion of MH370 hitti ng the ocea n. Instead, we found evidence of an explosion a few km southwest of the contrail end ing (in both Figures 12 and 13). The position is at ( 13.53° S, 107.11° E ). In a picture from 01:00 , the explosion - feature d ev eloped into a complete mushroom - cloud of the so -c alled Wilson type (Mushroom, 2024) with a ‘crown’ around its top as shown in Figur e 14 . This type of mushroom-cloud typic ally develops after underwater nu clear test explosio ns. We made an estimate of the total amount of energy transferred to the atmosphere during crash of an airplane like MH370. There are 3 different contributions: Kinetic energy of the airplane, energy from exp losion of the lithium b atteries it carried (no fu el is le ft, so its contribution is zero), and finally energy transferred from the hot water splashed hundreds of meters up in the couple of degrees cooler morning air. We found a total amount of energy equal to roughly 3 % of the Hiro shima b omb, explaining that a m ushroom - cloud could form. The fact that it only just made it above the clouds points to a cloud top around 3 km. As fur ther evidence , the left over of the mushroom - cloud is still cle arly visible in b oth infrared and visible satellite pictu res recorded at 01:3 0, but shif ted slightly north by wind drift. Figure 14. Visible picture at 01:00 at the same place (with a sm all wind corre ction) marked with a dashed blue ring, a gain 90° turned, and show ing devel opment of a Wils on cloud from t he origina l mushro om. With this information in mind and by including a small segment of Black Swan near Christmas Island, we can now measure the flight height during the e ntire end - scenario. Figure 15 displays the result s togethe r with estimates of the horizon tal speed of th e airplane with 10 - 15 % relative uncertainty. W hen passing closest to Christmas I sland, the flying height was 10.9±1.0 km in nice agreement with our model. When the contrails merged at the end, the height had dropped to 3.0±0.5 km. This height - loss appea red within roughly 11 minutes, and simultaneously the forward velocity dropped alm ost a factor 4. The Holland paper tells us that soon after the end of the plot , MH370 accelerated downward by 0.68 G and developed a l arge vertical speed. All t his evidence tell us t hat the airplane could only have been MH370, since no conscious pilot would have allowed such crazy navigation more than 1000 km from the nearest airport in mainland Australia. Any normal pilot would immediately after the first en gine - failure have tur ned around and m ade an emergency landing in the nearest a irport at Ch ristmas Is land to avoid a catastrophe. Since no ot her airplanes dis appeared tha t night, it must have been MH370 loosing height and crashin g. Figure 15 . Measur ed height and estim ated for ward spee d duri ng the las t 24 mi nutes of fli ght abo ve the cl ouds. Development of the first part of the mushroom - cloud takes only approximately 1.5 minutes , so we can practically ignore wind drift. Therefore , determination of the centre of the mushroom-cloud is so p recise th at the debr is wa s within a 5 km radius of this position. 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