Low Cost, Time-of-Flight Gravimeter Arrays

Gravimeter array imaging requires building low cost, high sensitivity, time-of-flight (aka high sampling rate) sensors.

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For this project I wanted to design a low cost instrument that can be built by someone from high school onward, that is sensitive enough to track the sun and moon second by second using their vector tidal acceleration signal. The high sampling rate allows low resolution methods to use statistical averaging to improve resolution. There are MEMS gravimeters, atom gravimeter chips, "molecular electronic transducers" and a wide range of analog methods that can be upgraded with better position sensors, ADCs and small smart arrays.

The network of superconducting gravimeters has been collecting data for decades.  About 95 percent of the signal is the sun-moon signal.  I have an example of one month of data in an attached image.  You can just see a few spots of blue where the actual signal peeps out from underneath the calculated signal.

The calculation is quite simple, if you have a ready source for precise positions of the sun, moon and earth in station centered coordinates.  Luckily NASA's online Horizon system provided by the Jet Propulsion Laboratory (JPL)  Solar System Dynamics group has made it easy.  I will post how to generate and download the necessary data to calculate the signal to be expected, and post some programs on GitHub to take the positions, calculate the tidal signal, and help you do the linear regression needed for each axis of the instrument.

This is a very forgiving method for getting started.  If you have gaps in the data, or periodic noise, earthquakes or other random interruptions, each measurement of the three axes in time can be individually compared to what is expected.  You will need to try to get your instrument aligned carefully to North, East and Vertical unit vectors.  You will need the exact GPS location of the instrument.  If you are off a bit, you will see that in the regression calculations, and can use the regressions to solve for the position and orientation. That is a bit advanced, but I will try to simplify it and post it on GitHub and here.

I have a few ideas myself.  I would like to try some of the MEMS, atom chip and other gravimeters that are getting sensitive enough to be called "gravimeter" rather than accelerometer.  In fact, "seeing" the sun moon signal is a good indication that any new technology has reach a certain level in capability.  Large networks can potentially provide feedback to the JPL ephemeris process to help refine the values for GMsun and GMmoon.  Longer term for these "second" instruments they can probably try to track Jupiter and Venus.

My more immediate goal is to find the instruments that are sensitive enough and can be logged at high sampling rates, in order to be able to apply "time of flight" and correlation methods to arrays of gravimeters for things like 3D imaging of earthquakes, the earth's interior, ocean currents, and atmospheric currents.  It will probably take one or two generations of people learning about and using gravitational fields on a daily basis for these things to be possible.   I thought I would try to share what I have learned by calibrating the SG and seismometer networks this way, and what I am learning from the many new device manufacturers.

The vector tidal acceleration at an instrument location on the earth is a simple Newtonian gravity calculation.  For now, it only uses the sun and moon.  The sun's gravitational potential at the instrument has a gradient that is the acceleration we measure.  But because the sun also accelerates the earth too, you take the sun's acceleration at the instrument location and subtract the sun's acceleration at the center of the earth.  Then add to that the moon's acceleration at the instrument, minus the moon at the center of the earth.  There are xyz values for each of these.  So you are taking the x value of the suns acceleration at the instrument and subtracting the sun's x value at the center of the earth.  You also have to calculate the vector centrifugal acceleration at the instrument.  It sounds complicated, but it is mostly software and keeping data organized.

Once an instrument is calibrated by comparing it to what is expected, then it can begin reporting on what it measures.  The measurements can be solved for the position of the sun, assuming standard values for the moon, and solved for the position of the moon, assuming standard values for the sun.  These...

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First Interference Ring Setup.jpg

This is a picture of the setup used for the "first interference ring" setup. The batteries and laser diode at the bottom, the two mirrors, the screen, camera and postit note to block the direct beam. The laser is pointing to the right edge of the farther mirror, the beam bounced back toward the bottom, hits the right edge of the mirror closest to the bottom, then boes to the right edge of the yellow postits. The scattered bean goes across the full space between the yellow postits and the camera. It is probably the front surface reflection, but I don't know for sure yet.

JPEG Image - 2.70 MB - 05/01/2019 at 13:42


Gravimeter Images.png

These are some of the images you might encounter when you are looking for analog sensors that will give you gravitimeter data streams. Laser and magnetic levitation, pyrolytic levitation, the signals and someones measurement of them. High eigenmodes of non contact atomic force cantilevers. Ocean waves and natural sources of gravitational change.

Portable Network Graphics (PNG) - 740.68 kB - 04/11/2019 at 16:39


Magnetometer and Gravimeter Networks Image 2.png

These is the second diagram from the video "Spatial Resolution of Magnetometer and Gravimeter Imaging Networks"

Portable Network Graphics (PNG) - 61.42 kB - 04/07/2019 at 23:00


Magnetometer and Gravimeter Networks Image 1.png

These is the first diagram from the video "Spatial Resolution of Magnetometer and Gravimeter Imaging Networks"

Portable Network Graphics (PNG) - 78.39 kB - 04/07/2019 at 22:59


Spatial Resolution of a Network of Sensors using Signals at the Speed of Light and Gravity.xlsx

This is the spreadsheet used in the Video "Spatial Resolution of Magnetometer and Gravimeter Imaging Networks" You can open it in Excel or google Sheets. If you change the sampling rate, the top uses a standard WGS 84 ellipsoid to estimate the global surface coverage at that sampling rate. If you look at the bottom portion, it calculates the number of 3d "spots" there will be in the region between the earth's surface and 1000 km out. If you want a different search area, change the height. Don't forget the height is in meters!!! :)

sheet - 12.87 kB - 04/07/2019 at 22:47


  • Low Cost Gravimeter Video 1 - My first interference picture and thoughts​

    RichardCollins05/01/2019 at 11:31 0 comments

    Here is a video of my first interference ring.  It is a 10 for $4.99 mini laser diode at 650 nm, two AA batteries taped together with a brass nail taped on one end and the laser diode wires to the +/- on the other end. The diode is laying on a book.  I have two 2" round mirrors (50 for $10.99).  The laser hits the edge of the first one,  bounced back and hits the one closer to the laser, then goes off and hits a piece of white paper about a foot away,  The two mirrors are about 5 cm apart. I have a $39.99 1600x1200x30 fps Crosstour sports cam that can be plugged in as a webcam, sitting a few inches from the screen.  The first dark ring is about 1 centimeter (cm) across.  I am using APowerSoft Screen Pro to record the video.  [ For the next video, I have Javascript/html program that I wrote to read the camera frames and collect statistics, display various pictures of the camera stream, and try to determine what all this wonderful data can be used for.  I have to try to use if for monitoring the pendulum, but I just love the statistics for their own right.]
      Oh, I have four 1 inch round magnets for a dollar, and a pack of 12 for a dollar half inch paper binding clips.  I clip the clip on the mirror, squeeze the wires and remove them, stick the mirror and clip on the magnet.  My table happens to be a wrought iron outdoor table so they stick, but they are heavy enough to sit on a table.  I had to learn to touch the mirror and only move the magnets carefully.  I will measure the angles and restrictions for getting this image because it took me a couple of hours to wrap my head around what it was doing.  Finally, I have a stack of sticky notes blocking the laser beams from hitting the screen, so the camera is not washed out.   The lights are out, except for the computer screens.  Looking forward to seeing what I can do with the data stream.  Will put a picture and diagram. Will record some data and play a bit.  I have other things I am doing, but will try to do more later today or tonight.

  • More Measurement and positioning techniques at sub-nanometer resolution

    RichardCollins04/21/2019 at 22:27 0 comments

    I am reviewing laser measurement techniques at nanometer resolution.  I need to go beyond that, but wanted to be sure I have a good foundation.  This article is helpful:

    "A review of nanometer resolution position sensors: Operation and performance"
    Andrew J. Fleming at

    It includes many low cost methods, and encourages the use of statistical measures that cannot be gamed or spun for marketing purposes.  My interest is on the "interferometer" and "encoder" categories, since the sensors need to be insensitive to magnet fields. When he wrote this in 2013, the sampling rate was in the kilosamples per second (ksps) range.  Now the same methods can use Msps and Gsps low cost solutions.

    [ "capacitive positions sensors" "nanometer" ] and [ non contact "atomic force" "nanometer" ] yield many useful efforts, but much reported is heavy on potential markets and dreams, then practical low cost commodity sensors you and I can use.

    I definitely need to look more at piezo devices and linear motors.  If my pendulum starts to swing I need to get out of the way, then track closer and closer as it settles down.

    A LOT of these things are the end result of people pushing older amplifier and ADC technologies, and they are about a thousand times more expensive than the new ones. But I cannot ignore anything.  I wish there were a way to help all the older instrument makers to upgrade.  There are still a lot of older instruments that do not use embedded processors, data sharing, modeling or statistics at all.

    This one at seems well reasoned and helpful, but one glance at the photos, and I know it is well beyond my meager budget.  They do mention "Simple capacitive sensors, such as those used in inexpensive proximity switches or elevator touch switches, are simple devices and in their most basic form could be designed in a high school electronics class. ", so I am going to be reading "Capacitive Sensor Operation and Optimization (How Capacitive Sensors Work and How to Use Them Effectively)" at to learn from a master craftsman.

    Go to run.  I am reading so much my sight keeps going out. Wish I had some help.

  • Monitoring still smaller masses for a gravimeter - atoms, electrons, photons

    RichardCollins04/21/2019 at 20:20 0 comments

    The mass of the object being monitored keeps shrinking.  The old seismometers had big masses on a spring, or supported by a wire and able to swing.  But as electronic techniques make the measurement of position faster, smaller and more precise, the big masses are stil useful.

    That is why I am trying to instrument a simple pendulum with precise sensors.

    But follow the thought all the way down.

    The "atom interferometers" make use of the fact that the wavelength for a particle is h/mv, where h is Planck's constant , m is the mass of the particle, and v is the velocity of the particle. The larger the mass of the particle being used, the smaller the wavelength.  

    h = 6.626070040E-34 Joules/Hertz

    But other than some experiments with atom interferometer chips, the cost and complexity of the "atom" methods seems difficult to implement.  I will keep looking, but will it is hard to find something I can adapt during this short period of this contest.  Oh, most of the "quantum" experiments with Bose Einstein condensates, including some superconducting configurations can be treated as simple atom methods.

    My note here goes to still smaller particles, the electrons.  These are used in large quantities in our current (pun intended) devices.  We store them in our capacitors, move them from place to place, modulate them, and find them generally fairly useful

    But we have not (so far as I have found yet) used the fact they have mass that is sensitive to gravitational potential (time dilation effects), and gravitational potential gradients (acceleration and velocity effects).  The atom interferometers make use of well studied internal states of atoms and molecules to manipulate and monitor them for use in sensing.  But there are very very specific interactions of electrons that can as well.

    I do not like the term "spin spin", because it doesn't tell me what is happening or what I can do with it.  I like the term "permanent magnetic dipole interaction".  I will put up with "hyperfine interaction" when it is applied to interactions of magnetic dipoles, magnetic quadupoles or any combination of Schrodinger states of atoms, molecules or particles.  If it can be represented as a "particle" whose field has multipoles, and these interact, I would say "multipole interactions".

    So I am looking at all the electron magnetic dipole interactions to see which phenomena might be "hacked" to make a low cost, small, precise gravimeter.

    One magnetic dipole interaction I have used for a long time is electron-electron magnetic dipole binding, where two electrons bind magnetically to form a stable pair.  I think that is the basis of supeconductivity generally, but I am trying to stay on track to solve this gravimeter problem in time. I am pretty sure the same thing is going on with proton-proton magnetic dipole binding in neutron stars and everyday nuclei here on earth.  I use magnetic dipole binding to estimate nuclear reactions where particles with permanent magnetic dipoles bind with nuclear energies. Sorry, just reminding myself of all the pathways I have investigated over the years.  I want one to help me here.

    So in a radio receiver, the fluctuations in the voltage of the electrons in a capacitor are related to "kT".  But part of that signal is gravity, part is the earth's magnetic field, part seismic, part human noise, and part kinetic fluctuations and phonons in the parts.  We distinguish "kT" in resistors, but it is just the same mix of signals coming through the potentials affecting particular devices in our circuits called resistors.  I tried to use a "kT" sensor (Johnson noise) sensor recently, but so far inconclusive, because you have to measure "everything" to sort out the source of the noise. 

    So, in parallel, I am gathering data from magnetometer arrays, seismometer arrays, gravimeter arrays, VLF ELF ULF and all frequencies of electromagnetic sources, power...

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  • Trying Optical Feedback Interferometer/Interference Sensors

    RichardCollins04/21/2019 at 19:33 0 comments

    Ben Krasnow has uncovered a simple method for laser measurement.  It will take some effort to convert it to a convenient tool, but he gave enough instructions to do that.  Here is his video: - "Laser diode self-mixing: Range-finding and sub-micron vibration measurement."

    He found some laser diodes that have an integrated "internal monitor photodiode" with feedback.  The feedback signal is what he is tracking.  You get "interference" because the reflected wave timing matches the timing of the outgoing wave.  So you should be able to get the same effect with an on purpose emitter and receiver pair. 

    I checked just now, and I think this "Self-mixing laser diode vibrometer December 2002" at is the same approach.  

    Looking more deeply, this paper 2004 "Self-Mixing Laser Diode Velocimetry: Application to Vibration and Velocity Measurement" by Lorenzo Scalise, Yanguang Yu, Guido Giuliani at explains "

    "Laser Doppler velocimetry  and laser Doppler vibrometry  are well-known measurement techniques widely used for the precise remote measurement of the velocity of fluids, and for accurate measurement of the displacement, velocity and acceleration of solid objects. With these types of instruments, it is possible to measure the velocity and displacement of the target surface, simply by using a light beam."

    So there is a rich source of useful techniques - once you know to google "velocimetry" "laser" and "internal monitor photodiode".

    Still further checking find "Microcantilever Displacement Measurement Using a Mechanically Modulated Optical Feedback Interferometer"  at  By 2016 they have broadened the concept from piezo to cantilever, and cleared up that it is an optical feedback loop. 

    Finally, "Optical feedback interferometry" today has over 10,000 results for the exact phrase."Optical+feedback+interferometry"

    Now to find one to adapt to my problem and move on.  I just need a data stream.

  • Gravitational Potential Type Detectors

    RichardCollins04/11/2019 at 17:05 0 comments

    This project's immediate goal is low cost acceleration field measurement and imaging techniques. 

    But the gravitational potential changes the rate of clocks, nuclear and chemical processes, at the surface of the earth - particularly, because of the changing distances and orientations of the sun, moon, earth and many things on earth.  These "direct potential" instruments derive from resonance measurement on the electronic and magnetic states of atoms - cesium and rubidum as a start - for use as precise atomic clocks.  As such they found the that clocks do change their rate at rest because of the absolute value of the gravitational potential (the acceleration is just the gradient of this potential), and can be "inverted" to report on the potential itself.

    Mossbauer effect (very precise accounting for recoil energies during state changes in atoms and molecules with narrow linewidths) can be used to measure the gravitational redshift, which depends on an integral of the potential between two points.  His innovation was to find materials where the lattice surrounding the emitting (and absorbing) atoms (molecules) could absorb the recoil energy on the timescale in which the state change occurs.  That is difficult because finding and characterizing materials to have precise atomic and nuclear properties is hard for bulk materials.

    With laser tools and fast ADCs and computing, it should be possible to use a wide variety of paired emissions and absorptions where the recoil can be tracked, accounted for, and compensated for.  There should be cyclotron versions and microwave plasma versions as well.  I am reviewing all those methods and possibilities, and will report later, or as I have updates.

    The importance of the gravitational potential detectors, is there are broad classes of instruments and experiments being proposed, just started, or going on, sensitive enough that they need to account for (1) earth tides, (2) small variation in station location, (3) changes in the rates of atomic and molecular rates, frequencies, and energies due to the changing potential due to the sun and moon relative to the station. The earth itself is changing shape, and its potential changes are tracked by the International Center for Global Gravity Field Models (ICGEM),, along with its temporal variations.

    This last affects many Bose Einstein condensate, quantum, superfluid, plasma and nuclear experiments.  I am trying to lay out the general rules, but my advice is that if you are saying "nano" and starting to whisper "pico" and "femto" and "atto", you need to check your local gravitational potential and acceleration field "weather report".  :)

    Now it is a general rule of thumb, that any system that has to account for these types of changes, can invert their models for correction to become reporting nodes in a global network of sensors.  Sensor monitoring the positions of the sun and moon, and the shape and gravitational events on the surface of the earth and in the solar system.  So if someone's G experiment is giving them fairly wide variation from place to place, and time to time, it might well be they have not accounted for the sun, moon and earth portions of the potential and its gradient.

    I do not know exactly how the signal is initated and travels from where the change is made, to "direct potential" or gradient sensors.  For a pendulum swinging nearby, used as a reference source, or just keeping video tracking and identification senors on poles of the nearby highway to identify them and correlate with gravity signals. Or using 3D video and imaging to get the shape of the ocean to calculate its gravity signal at your site.  All of these are quite complex.  Easy to do, but with noise and careful work required. The signals travel at the speed of (light and gravity) so you need to sample at rates appropriate to your needs.  If you want to use a gravity...

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  • Glasgow MEMS Gravimeter making progress

    RichardCollins04/11/2019 at 01:29 0 comments

    This MEMS gravimeter at the University of Glasgow uses an interesting "optical shadow" detector.  And they have added tilt and temperature monitoring and compensation. They still have not calibrated all three axis, nor are they doing routine sun moon calibration, but seem to be making progress.

    A High Stability Optical Shadow Sensor with Applications for Precision Accelerometers

    Field tests of a portable MEMS gravimeter

  • MagQuest $1.2 Million Dollar Contest to Improve Global Magnetic Measurements

    RichardCollins04/07/2019 at 23:19 0 comments

    The National Geospatial-Intelligence Agency is looking for people to find novel approaches to gathering and utilizing magnetic data in conjunction with their development and support of the World Magnetic Model (WMM).  They rely heavily on ths European Space Agency "Swarm" of satellites to measure and calibrate the World Magnetic Model.  But they would like to use sensor data fusion to improve and verify the space-based sensor network.  So anything is fair game.

    Personally I think they should at least upgrade the existing magnetic networks with decent ADCs, continous high sensitivity monitoring arrays, and get their Internet data sharing down to current practices at least.

    Here is what they say about the WMM:  "The WMM is embedded in thousands of systems. More than a billion smartphone users depend on the WMM to point them in the right direction when they use mobile navigation apps. Drivers rely on the WMM to power the compasses in their cars.  The WMM is also critical for military and commercial uses around the world. Among other applications, it supports navigation and attitude determination for submarines, satellites, and aircraft, while also informing operational logistics like the numbering of runways."

    So they have an existing user base, but are not collecting ground truth from them apparently.  If you could hack the cell phone to provide ground truth, or plug millions of low cost sensors into Internet so they can be used to constrain the solutions when they try to build their model, it will help them with their current plans.  And, presumeably pay for you to continue to help them.

    But it might be that there are other ways to provide better solutions to the fundamental problems and needs of their clients.  Perhaps there are better ways to solve for orientation and location using existing GPS, or by upgrading selected cell phones.  Or by letting people put in local nodes to provide very precise updates on magnetic field variations.  I can think of about 30 things to try. But, my main concern is they also upgrade sampling rates so whatever sensors are created or upgraded, can be used for magnetic time of flight imaging arrays.

  • Video: "Spatial Resolution of Magnetometer and Gravimeter Imaging Arrays

    RichardCollins04/07/2019 at 22:44 0 comments

    I made a video that you might find interesting, titled "Spatial Resolution of Magnetometer and Gravimeter Imaging Arrays".  I talk about ways to hack existing gravimeters, seismometers, and magnetometers by adding sensors and ADCs so you can chop the incoming signal (light and gravity and magnetic fields are all coming in at the speed of light and gravity) in time, and use that to narrow down where the source event is in space.  Using networks and correlating data from many sensors, you can narrow down which specific volume element is involved.  

    I did some simple calculations to clarify my ideas, using sound cards and inexpensive ADCs as examples.  I talk about localizing earthquake seismic waves, but the same considerations should apply to the magnetometer networke -- IF they can sample at high enough rates to get the spatial resolution where you can begin to image and localize magnetic events.

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