Physics & Math applied to TPL data can save millions in FM370 search

Discussion in 'Physics & Math' started by Billy T, Apr 19, 2014.

  1. Billy T Use Sugar Cane Alcohol car Fuel Valued Senior Member

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    Physics & Math applied to the Towed Pinger Locator data record can save millions of dollars in FM370 search effort. I know how, but need help to do the math, which can be split into two separate problems. This post suggests in considerable detail, with real values, how to do the first half of the problem. I.e. how to build a model of the "Sound Field" (defined later) the black box pinger emitted as a math description is the starting point for the second half of the problem.

    Consider a Omnidirectional 37Khz point source on the floor of a flat ocean and let the straight up vertical be the “z-axis” of a Cartesian coordinate system. Let the “y-axis” be parallel to the assumed straight line path of the Towed Pinger Locator, TPL with trajectory ( X, y(t), H). I.e. H is the constant height of the TPL's path above the ocean floor (tacitly assumed to be flat). And X is the constant distance of that path from the y, z plain.

    When the TPL is at (X, 0, H) the Doppler shift is zero. I.e. then the TPL is receiving 37Khz ultra sound waves and to be definite, a lower frequency as y(t) becomes positive. There exists at least two records of exactly how the frequency changed as the TPL moved and was receiving sonic signals. This data with math and physic I now describe can located the sound source in the Cartesian frame much more accurately than the “triangulation” currently be used, greatly reducing the duration of the searching with the “BlueFin” detector which has little if any directional sensitivity (only range).

    If the speed of sound were constant then the surfaces of peak acoustical pressure are a set of n concentric hemispheres centered on the sound source. I'll call that set of pressure peaks, outwardly moving away from the sound sources, the “Sonic Field” regardless of it shape.

    The speed of sound in water is much higher than in air and in general increases with both temperature and density. The TIP was surely well below the sonic channel layer whales use to communicate even when separated by dozens of kilometers. Probably the water temperature, T, in centigrade over the entire path the sound traveled from source to TPL was constrained by 5 < T < 10, so compared to T in Kelvin, to first order, the temperature variation in the speed of sound can be neglected.

    Here is good approximation for the speed of sound in sea water:
    c(T, S, z) = a1 + a2T + a3T2 + a4T3 + a5(S - 35) + a6z + a7z2 + a8T(S - 35) + a9Tz3
    where T, S, and z are temperature in degrees Celsius, salinity in parts per thousand and depth in meters, respectively. The constants a1, a2, ..., a9 are:
    a1*=*1448.96, a2*=*4.591, a3*=*-5.304×10-2, a4*=*2.374×10-4, a5*=*1.340,
    a6*=*1.630×10-2, a7*=*1.675×10-7, a8*=*-1.025×10-2, a9*=*-7.139×10-13
    From: http://en.wikipedia.org/wiki/Speed_of_sound, but note the a6 term is the main pressure correction and equal to 16.3 (m/s) per Km of depth increase. Also at same link is this graph for ocean north of Hawaii:

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    The nearly linear deep section has speed of sound increasing by ~ 20(m/s)/km but is real data with all effects included; however I will assume the speed of sound in the deep Indian ocean off west Australia is given by: S(d) = 1550 +17(d – 5.5) where d is the depth in Km and S in in m/s. For example, S(6.5) = 1567 m/s. This because the salinity there is surely less than 35 and thus that factor alone is reducing S as fresh water from melting Antarctic ice is flowing there. I.e. I only let the graph's deep water slope only slight increase the mathematical model's 16.3 pressure correction term.

    I suggest that the simple spherical surfaces of the Sonic Field description equations have the z-axis replaced by a reduced z' to reflect the fact that the wavelength – separation between two constant sound pressure surfaces in the Sonic Field is decreasing along the z-axis as the speed of sound decreases. I e. The constant peak pressure surfaces are given by (x^2 + y^2 + z' ^2)^(1/2) = any of a set of integer constants n, n+1, n+2 ... times the wave length, L, near the source at depth d = D, a constant.
    Now L = S(D) / 37,000 as speeds were in m/s so frequency must be in Hertz to give wavelength in meters. Or, L = {1550 +17(D – 5.5)} / 37,000.

    The parameters (unknown variables we hope to find when second half of the problem is solve – see discussion in next post) in this description of the Sonic Field are D, X, and H.

    I have only suggested an approach to creating a math description of the Sonic field, but leave the actual doing of that to those with greater math skill. What is desired is a math description of the Sonic Field in terms of: (x, y, z' and n) or better as function of (x, y, z and n). I.e. n is an integer designating which the "flatten spherical" sound wave peak pressure shells we are on. They are always one wavelength separated, but the wavelength is not a constant as we have reasonable correctly stated what the speed of sound is and how the speed of sound is changing with depth.
     
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  3. Billy T Use Sugar Cane Alcohol car Fuel Valued Senior Member

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    I will soon post what the second half of the problem is, but first request that someone who can* copy the OP and re-post it at Physic.org or PhysisForums, etc. where there may be others, both interested and skilled in math, and of course giving a link back here for them to read my ideas / statement of the second half of the problem.

    * I am without posting privileges at other Physic and Math forums with experts who may be able to solve the problems. After I have made that promised "Second half of problem" post, I will PM a few of the very good in math posters active here to direct their attention to this new thread.
     
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  5. Billy T Use Sugar Cane Alcohol car Fuel Valued Senior Member

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    This post concerns the “second half of problem about physics & math applied to TPL data can save millions in HM370 search

    Once one has a Sonic Field mathematically described (first half of the problem discussed in the OP), one needs to compute how the Doppler shift of signal received by the TPL is changing as it moves on its straight line (assumed) at constant (assumed) speed, K, (for Knots, but m/s is to be used). That speed is known, but not by me, but one knot = 0.514 meters per second , so I will suggest use the speed of the TPL as 2 m/s until the actually value is told us.

    Recall from the OP, that we modeled the trajectory of the TPL as purely a straight line parallel to the y-axis of a Cartesian coordinate system and traveling in the direction to increase the y coordinate with time, with the pinger (ultra-sound source 37Khz, assumed) at the origin and the z-axis was vertically up. This trajectory was H meters above the assumed flat ocean floor and X away from the y, z plain.

    Also the OP noted: “ When the TPL is at (X, 0, H) the Doppler shift is zero. I.e. then the TPL is receiving 37Khz ultra sound waves and a lower frequency as y(t) becomes positive. There exists at least two records of exactly how the frequency changed as the TPL moved and was receiving sonic signals.”

    The Doppler shift is due entirely to the speed of the TIP and we can greatly simplify the problem by falsely assuming the Sonic Field (defined in the OP and hopefully known when first half of the problem is solved) is static. I.e. ignore the fact the sound pressure peaks do move away from the source. I. e. Every 1 / 37,000 seconds, the pressure peak surface that was with designation “n' moves to be come identical with the current n+1 pressure peak surface.

    All the illustrations I could find of the Doppler effect have either the source or receiver moving so I am forced to mis-use this one:

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    The compressed or shorter wave length on the right side of drawing are due to their source assumed to be moving in that direction, but our source is stationary; However, we can consider that right side to be the vertical direction up from a stationary source in deep water because as the waves travel upward, the speed of sound is decreasing - the frequency does not change but the wave length is becoming shorter. The distance between the n and n+1 pressure peaks is becoming shorter. I.e. I can use this drawing for further discussion.

    Now imagine the TPL going across this diagram from bottom to top and cutting thru the circles. For example almost tangent (slight miss) to the right side of the smallest circle which I will call “n =1” (and the largest circle has n = 6). Note that a receiver moving on this trajectory will cross each circle twice, but the time between circle crossings P(n, m) is not constant. For example P (6, 5) is much shorter than P(2,2) the two crossings of the next to smallest circle. Note also that ALL these Ps are inversely proportional to the speed along the trajectory. I.e. the FREQUENCY received is inversely proportion to the period between the receiver encountering wave fronts or pressure peaks.

    The first of the two crossing of each circle when the distance to the sound source is decreasing is a positive Doppler shift and the second is a negative Doppler shift. I.e. while the distance is decreasing, {y(t) < 0}, the received frequency is slightly higher than 37Khz and conversely when y(t) > 0 the received signal is slightly less than 37hz. The larger X is, the smaller will be the maximum Doppler shift observed. I.e. a "lucky pass" right over the sound source will have the maximum Doppler effect, but that maximum will decrease for larger values of H.

    In the real case these wave fronts are moving very fast compared to the speed of the TPL so even thought the spacing between the n and n+1 is not constant the frequency will be very near the 37Khz radiated but we are interested in the small frequency difference received as the TPL moves thru the Sonic Field – I.e. the changing Doppler shifts. These Doppler shifts are directly proportional to the inverse of the spacing between the circles, but of course the real problem is a 3D one not the 2D one illustrated and discussed here. I.e. by falsely treating the pressure peak circles, the "Sonic Field", as stationary, we have drop out the 37Khz and are left with only the Doppler shifts we desire to know as function of time (or the y location of the TPL) the speed of the TPL and its trajectory wrt the black box sound source.

    Mathematically, I suspect it is much easier to compute the temporally changing Doppler shift function given the speed of the TPL, its height above the ocean floor, called H in the OP, and the separation of the trajectory line from the y, z plain called X in the OP than to do the inverse. I.e. from the measured Doppler shifts, and known speed, of the TPL learn the values of H & X, but if this inversion equation is too difficult to obtain, one can compute the Doppler shifts expected for a set of many H & X and see which one of the set replicate the measured Doppler shift as a function of time. I.e. the entire record is fit, by trial and error if need be.

    Note if X' is a the best fit then -X' is too. There is a "right left" ambiguity which can be removed by processing the second longest measured record of Doppler shifts. I did not really invent this methodology. I worked at APL/JHU when Sputnik was launched. The Doppler shift observed at APL was processed and gave the most accurate orbital parameters. Much better than others got from triangulation of its position at two or more points in it pass "over head."
    But for more than two decades, the method I am suggesting for locating the Black Boxes told the location of US Navy ships at sea to a circle of probable error of only a couple of dozen meters with data from a single long pass the transit satellite from horizon to horizon. I won't be more specific as the accuracy possible from a single pass may still be secrete.

    Using this method is a very good first cut at locating the Black boxes but with measurements of the currents and temperatures in the ocean below the trajectory of the TPL in the area this method suggests, and then making corrections for the refraction of "sound rays" can perhaps define their location so precisely that the Bluefin would need only a day to find them and other debris from flight HM 370. Note the vertical ray (normal to the sound wave front) is not bent (refracted) but one leaving the source at say 45 degrees from the vertical will bend some towards the vertical, but I doubt making this "2nd order" refinement is worth the effort.
     
    Last edited by a moderator: Apr 19, 2014
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  7. Aqueous Id flat Earth skeptic Valued Senior Member

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    BillyT,

    Can you give us a statement of the problem? As far as I can tell you are thinking of a way to improve the location technology by relying on Doppler instead of whatever method they presently use to triangulate. I don't understand the system in use and therefore I can't fully appreciate how this should be done. Here are some questions that came to mind as I read your posts:

    1. Can you describe the pinger signal? I believe it's 180 dB (not sure what that's referenced to), 1 Hz pulse (presumable square wave).

    2. Can you describe how the transponder works? As I understand, the ship sends a request for data to the submersible, and it replies with bearing information via low bit rate modem (probably amplitude shift keying). This would have to be done on another frequency. I wonder what the message information contains. Does the sensor on the submersible rotate, and simply reply with its best estimate of the bearing relative to the submersible, or is it fixed, and it just replies with amplitude (received power) information?

    3. Do we need to make assumptions about the depth of the pinger in order to calibrate for the difference in depth of the submersible? As I understand it, that's your intent for discussing the different wave velocities as a function of temperature.

    4. Does the transponder tell the ship its depth, position and attitude?

    5. If you came up with a better technology, would it be possible to use it in the present search, or does it require modifying the pinger? Does it require modifying the transponder? Or is this just an algorithm to be used on the ship?

    6. As you know the military is very knowledgeable about direction finding and geolocation methods. AWACS comes to mind, as well as similar systems, which detect air-to-air and ground-to-air radars. Analogous systems are used to geolocate and/or fix the bearing on signals of all communication bands in the electromagnetic spectrum. On top of this marine acoustics is an area of military science that's very well developed. Why can't a high-tech military sub locate a pinger with whatever equipment it has on board? They should already be able to read the amplitude, carrier Doppler and pulse phase shift with good sensitivity and accuracy. They should be able to do this close enough to the sea floor to overcome thermoclines and/or to compensate for velocity change at depth--although it's not clear to me why this matters, since for direct bearing information all you need to do is pick the angle of the loudest ping. I suppose it's cumbersome maneuvering a sub to match heading to bearing, but that's an unknown to me. I don't understand the problem.

    7. As for the math, my suggestion is to go with polar coordinates and rely on sea floor maps to look up the depth, or else forget that and go with spherical coordinates. So rather than the rectangular coordinates you suggest, it's more directly useful to give the three measurements as Az-El-Depth. (Azimuth and elevation). Azimuth would be 0° at magnetic North, although this requires the transponder to employ a magnetic compass. That gets back to the question of how it knows its position, attitude and velocity. (Attitude would be the heading in both azimuth and elevation). Elevation would be negative below the ship reference level (probably the water line) but I don't know their convention. What I'm suggesting is an Earth-centered reference frame which is cumbersome since it's hard to work with latitude-longitude at this scale. (You have to know the datum of the Earth geoid). I think they are probably using a transponder-centered reference frame. In that case 0° bearing is ahead of the submersible, and 90° would be to its right side. Do you have any idea how this is currently being done?

    8. I think once we have a reference frame in place, and using just one submersible, I think you'll discover that there are an infinity of possible locations that would produce a given Doppler. In TDOA (time difference of arrival) three submersibles would be needed. It gives the highest precision theoretically available. The idea is to time tag each received pulse and send the results back to the ship. There, the time differences (A-B) and (A-C) are calculated. Each of these two differences describes a hyperbola of rotation. By calculating the intersection of the two, a curve is deduced, and by estimating the difference between submersible's depth and depth of the pinger the curve can be narrowed down to an approximate location. For that though, we need to be working in Earth-centered coordinates. Which system is best for locating the pinger? I imagine the military ships prefer azimuth and range to steer to the location. But that requires a coordinate system which is ship-centered.

    9. One last thought. If the intent is to improve this for the future, I would think the smartest thing to do would be to have all aircraft constantly sending their flight data recorder information out to a global network -- encrypted for security -- in real time. That would eliminate the need to search for it at all, and would get the rescue teams as close as possible to the place any survivors might be found. In any case a pinger should send the last known GPS coordinates, using the ping as a baseband modulated modem type signal.

    10. If the transponder were connected directly to the submersible steering system, it could just keep steering for the direction in which the Doppler shift is greatest. Conversely, on it passes the pinger, the Doppler would cross zero and reverse, so the the steering could be made to stop the point the front of the submersible directly down to the sea floor and that's where the pinger should be found. (With a few maneuvers to correct for slip during the turn.)
     
  8. Billy T Use Sugar Cane Alcohol car Fuel Valued Senior Member

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    Thank for numbering your questions - I don't repeat them to keep this post shorter. Yes your introduction is what I am trying to do; tell a much better way to use the data ALREADY collected by the TPL.

    (1) I think you have pulse nearly correct, but I have no direct knowledge of pluses that were radiated. The dB or strength of the received signal varies. I believe at least one of the recorded events lasted more than an hour. My suggested method is described for continuous radiation but pulses could be considered just periodic samples of that. Yes the pulses is no doubt grossly speaking a "square wave" ever second I think, but the "on time" is probably on the order of 0.1 second duration, which I will assume for replies. I.e. the energy required is only ~10% of what a continuous ultra-sound would require. (30 days instead of 3 days before the battery goes dead) Thus, at one second intervals you get 0.1 second long train of ~37Khz signal or about 3,700 cycles and that is enough to tell that the distinguish the received (Doppler shifted) signal 37, 003 Hz from the un-shifted frequency (37, 000); however if the pulse duration is less or erratic in duration simple counting of the number of cycle received will fail. What is needed is a Fourier analysis of the recorded pulses which will have strongest component near, but not exactly at, 37,000 Hz. This will allow for large variation in the pulse duration.

    (2) No, I can't but assume the TPL sensor is not directionally sensitive. I do however assume it can be set to hold a desired depth. I.e. has a pressure sensor and some control surfaces that make rise or fall to keep that depth, somewhat independent of the ship's speed.

    (3) Yes, sort of as the analysis does have variable H, the distance of the TPL's trajectory above the ocean floor. What the "first order" analysis I suggest actually tells is the direction that the sound ray has as it arrives at the TPL (which is of course a continuously changing as the TPL moves). I. e. to first order we ignore the refraction by the water. By extending these arrival angles, for every pulse with measurable Doppler shift, back wards to the ocean floor, we get the first order "circle of confusion" - sort of like looking thru a imperfect lens. Then the temperature as function of depth must be measured and the effect of the real refraction taken into consideration to reduce the size of the "circle of confusion."

    As I suggested in post 3, it may be too difficult to calculate X and D directly form the available record of Doppler shifts. It is much easier, but still too tough for me, to assume H and X an then calculate the expected Doppler shift as function of time (or the y coordinate of the TPL). Once this calculation is in computer code, 100,000 pairs X and D can be quickly done and tested against the measured and recorded Doppler shift record with some "best match" criteria. to find which D & H are best.

    (4) I don't know, but getting those would be an easy and cheap facility to have so I bet it does. What is also should have, but probably does not, is a crude active pulse sonar to tell every few seconds how far it is above the ocean floor - nothing as complex as the Bluefin has that actually profiles the floor.

    (5) No mods needed but some parameters could be set differently to make this approach more applicable. The main thing I would do differently is make the Back boxes into triggered transponders. I. e. don't have then pinning away ever second when no one is listening. If they only pinged when acoustically prodded, they could still be functioning after five years with no more battery volume required. That would require they have microphone, but they can be made very rugged. Hell during WWII APL made a vacuum tube (no transistors then) the reliably survived being shot out of Navy's guns at Kama Kazi attacking planes an basically saved the Pacific fleet.

    (6) No comments.

    (7) I used Cartesian coordinate as that is easiest for most to understand and does make the mathematical description of the TPL location a function of time in 3D space very simple. I.e. it is at (X, st, H) where "s" is its speed and t is time with t = 0 being when it crosses thru the x, z plain. Use of cylindrical (not polar) coordinates would make that motion description more complex, but probably make the first half of the problem (specifying what I called the "Sonic Field") more simple. The transformation of z into smaller z' would still be just a useful. I judged /guessed the first half of the problem is easier to solve than the second so wanted the simple (X, st, H) form for the TPL motion there. However, any coordinate system can be used by who is going to try to solve the two part of the problem. I think I made a great advance by realizing the Sonic Field could be taken as static and then the "circle crossing periods" give (the inverse) of the Doppler shifts directly. I.e. the 37Khz "drops out" of the problem.

    (8) No, you are trying to use a GPS approach with three time delays compared and need precise clocks that are all synchronized. The beauty of the Doppler or "Transit" system is there is only need of ONE frequency stable oscillator and relative motion between it and the detector.

    (9) Yes that seems to be the idea that is gaining strength, but will be very expensive. I for one would rather see the minor modification to the black boxes made at very low cost, that make them into transponders that only send out signals when there is some one listening (who has prodded them to do so) and then have battery lives measured in years, no days. I and wife fly from Sao Paulo round trip to NC beach (Outer banks rented house) annually at cost of nearly $4000, for air fare in economy class - I don't want to see that go well above $4000 especially as planes being lost are such rare events and normally without any crash survivors. I do want the black boxes to be recovered in all cases to learn what went wrong for suggestion of how to make them even safer.

    (10) The search system requirements are very different from those that can do recovery work. I don't think it a could idea to try to combine them into one unit. I worked nearly two decades in APL's space department (but mainly on energy systems and implantable biomedical devices, not space craft) I was, however, not infrequently present in discussions about how to make the instruments for a new space craft, as I am a well educated physicist and easily "think outside the box." At almost every one of these planning meetings some one would suggest that the weight could be reduced if we combined instrument A with instrument B. The head of the department was usually opposed - liked to re-use the already tried and space proven instruments again. He had a favorite, but some what vulgar way, to express his disapproval:

    "There is only one instrument that does two very different jobs perfectly - and God made it."
     
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  9. Billy T Use Sugar Cane Alcohol car Fuel Valued Senior Member

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    I now realize that the solution to problem of accurately locating Flight HM370 by Doppler processing already exists - Probably both at APL and the US Navy's Transit program files. This is because the Doppler frequency shift does not depend upon which is moving. (The constant frequency radiator or the receiver of the Doppler shifted signal). Thus the computer programs, developed at APL, which very accurately located a submarine by processing the Doppler shifted signal from one Transit satellite's constant frequency radiation can be directly applied to accurately find HM370's black box location. ("Very accurately" means a circle of probable error of less than 100 meter radius.)

    I. e. If the Towed Pinger Locator, TPL, were radiating a constant frequency and the Black Box were receiving a Doppler shifted signal, as submarines received the Transit satellite's radiation, it would be exactly the same record of changing Doppler shift that ships searching for HM370 recorded. The location of the Black Box with this existing data and computer program yields will not be exactly correct due to water refraction, but that is an easy correction to make as surely the TPL was below the SOFAR channel.*

    Below the SOFAR channel sound rays (Normals to the sound wave fronts) bend to be more nearly vertical than they were when emitted by a Black Box on the bottom, except for the vertical ray which remains vertical. Thus if the existing data and computer program predicts that the Black Box is laterally located by the same horizontal distance as the TPL is above the ocean floor (I.e. X = H in prior notation, then the TPL is receiving a ray at 45 degrees from the vertical); but in fact that ray left the Black Box with a slightly greater angle between it and the vertical, say 50 degrees just to be clear. Thus, the actually lateral distance along the ocean floor, X, from point directly below the TPL is slightly more than Height, H, of the TPL above the ocean floor for this case of a 45 degree ray received by the TPL, when it is closest to the Black Box - with zero Doppler shift at that one point).

    * The SOFAR channel is at depth where the speed of sound is least (typically about 1 Km deep). Whales use it to communicate long distances. Sound traveling horizontally within the SOFAR will remain there - If some rays try to escape, they are refracted back into the SOFAR channel. Thus they weaken mainly by absorption and linearly with distance (not inverse square).
     
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