The more I learn about geodesy, the less I understand.?ÿ Somebody help me with this one, please.
I thought GRAV-D was flying instruments at perhaps 20,000 ft and deriving from that data a measure of the gravity on the ground.?ÿ
But the earth has varying density as you move around.?ÿ If there is a lump of different density, relatively close to the surface, then 1/R^2 for that lump is much more sensitive to height above the surface than is 1/R^2 for the majority of the earth.?ÿ So when you calculate the contribution to gravity from that irregularity as measured on the earth's surface, it is much different than the contribution you get as you fly over at a few miles height.
I did a very simple spreadsheet with some made up numbers, and estimated that a change by 1e-12 of the earth's mass (a lump that might be on the order of a a fraction of a mile thick by a few square miles), centered a mile and a half below the surface, could affect the measurement by 6 mgals more at the surface than the effect when flying at 4 miles up.?ÿ If you assume a constant mgal/meter correction for the plane's altitude, that corresponds to moving the geoid by meters from where the measurement would have put it if the earth were symmetrical.
When you fly over a mountain or the Kennecott copper mine, this should affect the results more.
Am I totally lost, or how does GRAV-D deal with this effect?
To visualize this draw a circle representing the earth, then draw a larger concentric circle representing the flight path. Then draw two point sources of gravity at different depths inside the earth. Draw gravity force vectors from each point source.
You will see that as the plane moves on the outer circle the gravity is acting straight down through an equivalent place on the surface of the earth. The plane feels exactly the same force as the surface, just reduced by the extra distance.
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This stuff is way over my head, but I think the answer is that they don't use a constant correction for altitude. A description of the various adjustments and corrections made to the raw data and some insights to the raw data collection itself can be found here (look particularly at the material beginning on page 16):
?ÿ?ÿ https://www.ngs.noaa.gov/GRAV-D/data/NGS_GRAV-D_General_Airborne_Gravity_Data_User_Manual_v2.1.pdf
The total number of corrections and adjustments made in post-processing boggles my mind, but it is fun to contemplate.
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Each object generates a gravitational field. From tiny objects to galaxies all generate that field. So you are correct the mountains and more dense objects exert more force on a satellite in orbit as it passes over. But remember, that satellite isn't just orbiting the earth. It's also in orbit around the sun, and the earth is a binary system and therefore it orbits the moon as the moon orbits the earth. The center of mass of the earth-moon orbit is under the earth's surface so the earth "wobbles" as the moon circles. The moon and sun exert more distortion to the satellite's orbit (I think) than the lumpiness found in the earth's mass. Imagine the earth, moon and sun all lined up as the satellite passes across the dark face of the earth, the moon on the light side. Add to that other extra-terrestrial bodies, Jupiter, the center of the galaxy, you get into some very hairy equations.?ÿ
It's best to track the orbits and update the satellite's positions, I doubt they will ever get a perfect handle on where any satellite will be at any time in the future, but they are sure getting close. It's amazing that they get it as right as they do considering what they are up against.?ÿ
To visualize this draw a circle representing the earth, then draw a larger concentric circle representing the flight path. Then draw two point sources of gravity at different depths inside the earth. Draw gravity force vectors from each point source.
You will see that as the plane moves on the outer circle the gravity is acting straight down through an equivalent place on the surface of the earth. The plane feels exactly the same force as the surface, just reduced by the extra distance.
The problem with that is the distance between the "lump" and ground is vastly different than the distance between the "lump" and airplane.?ÿ Since it depends on 1/distance^2 and you don't know its actual distance below the surface, the reconciliation is a whole lot more complicated.?ÿ You can't solve it with just the airborne gravity value and the altitude of the aircraft.
Here is a web link to an NGS discussion on physical geodesy that may answer some basic questions for you Bill.
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If you want to wade into the mathematical underpinnings of the topic, you may wish to download the textbook, "Physical Geodesy" by Heiskanen and Moritz, 1967.?ÿ Warning:?ÿ The first chapter reminds me of a brutal grad course I took over 30 years ago in mathematical physics.?ÿ I'm not sure why the book is available for free download on Microsoft's Internet Archive.?ÿ I searched on "Heiskanen and Moritz" and was able to download a PDF.
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You may also wish to contact Dr. Dan Roman at the NGS.?ÿ He used to be the lead on developing the geoid models and is listed as a contact for the geoid team.?ÿ I see that he is now the Chief Geodesist.?ÿ There is also this link you may find of use.
If I understand correctly, you are concerned with how the impacts of masses are accounted when determining the ellipsoid-geoid separation. This is the downward/upward continuation problem. Pages 18+ of this reference discuss it:?ÿ https://pdfs.semanticscholar.org/8e2c/03953ddbc2503efa748cd36ebc39d37dbfd5.pdf
Some suggested authors beyond the classics (Heiskanen, Moritz, et al) are Will Featherstone, Lars Sjoberg who write clearly and have supervised a number of good papers.
One addition comment, when pursuing the topic remember that geophysics and geodesists frequently use different terminology. Prof Featherstone (Curtis University) has a nice paper on the topic. His on-line publications are found here:?ÿ https://staffportal.curtin.edu.au/staff/profile/view/W.Featherstone
Enjoy,
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DMM
More specifically on airborne gravity, here is a good paper:?ÿ https://ir.nctu.edu.tw/bitstream/11536/10891/1/000245953500001.pdf
To visualize this draw a circle representing the earth, then draw a larger concentric circle representing the flight path. Then draw two point sources of gravity at different depths inside the earth. Draw gravity force vectors from each point source.
You will see that as the plane moves on the outer circle the gravity is acting straight down through an equivalent place on the surface of the earth. The plane feels exactly the same force as the surface, just reduced by the extra distance.
The problem with that is the distance between the "lump" and ground is vastly different than the distance between the "lump" and airplane.?ÿ Since it depends on 1/distance^2 and you don't know its actual distance below the surface, the reconciliation is a whole lot more complicated.?ÿ You can't solve it with just the airborne gravity value and the altitude of the aircraft.
But it doesnt matter how deep any particular lump is, it only matters what the net force is at any point on the earth's surface is. Since there is no significant source of gravity between the plane and the surface, the plane reads what the surface does, just reduced by the square of the distance. No matter how deep the gravity source is the difference between what the plane and the surface of the earth feel is the same.
Of course there is a lot of math that goes into adjusting for things like irregularities in the shape of the surface, the acceleration of the plane ect..but the math gives a very elegant solution for the basic theory.
Maybe I dont understand what you are asking though....
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"Reduced by the square of the distance" The reduction for plane vs ground for the lump is very different from the reduction for the bulk of the earth.?ÿ It depends strongly on the depth of the lump, which you don't know.
After some more reading, I think the single-measurement problem i posed cannot be solved.?ÿ
When they have essentially complete airborne coverage of an area, they can compute a gravitational potential and then refer it to the ground and geoid.?ÿ Extensive measurements can work for an area, but measurements at one point can't resolve it for that point.
I hope someone well qualified will comment on that assertion.
I too hope that someone ??well qualified? will respond to the tread. Unfortunately, the subject is not likely to be addressed in short, easy answers.
I still think Hwang??s paper has many answers (see page 5 of 14, section 5).
On the geophysical side see:?ÿ https://rallen.berkeley.edu/teaching/F04_GEO594_IntroAppGeophys/Lectures/L03_GravCorrAnalysis.pdf
Slide 6 of 11 in the above might be of some interest.
Have you reviewed Prof Jekeli??s presentation at the 2016 Summer School on Airborne Gravimetry? Unfortunately, I have only reviewed his slides and not watched the videos yet. It??s only been available for two years 😉
Lots to learn, too little time, too many distractions.
Posted by ??not an expert? nor ??well-qualified? DMM
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Slide 6 is definitely relevant.?ÿ It demonstrates that the gravity variation over a range of points is what allows one to learn something about what is below.
Airborne Gravitometry was a 5-day course - no wonder you haven't watched it all.?ÿ I hope to work on some of it in pieces.?ÿ How much I watch depends on the extent I have to remember Stokes Theorem, Poisson's Integral, and such math I haven't used in the 46 years since I saw it in college.?ÿ I still do ok on algebra, trig, and simple calculus, but have forgotten most of what follows those courses.?ÿ I find the Physical Geodesy textbook beyond me, except that I can pick up a few points on those occasions when they pause the barrage of equations to explain what one of them means in practice.
http://fgg-web.fgg.uni-lj.si/~/mkuhar/Zalozba/Geodesy_Gravity_Wahr.pdf see Chapter 6.
Prof John Wahr??s Class Notes linked above have a higher text to equation ratio than the Physical Geodesy text. I find his writing to be clear and he uses plenty of examples. Quite a remarkable scholar. Unfortunately, he died in 2015. The class notes above were for a three-course sequence.
BTW, are you reading the first or second edition of PhysicalGeodesy?
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Wahr appears to be the first priority for study.?ÿ He gives lots of intuitive descriptions along with his math.
My copy of Physical Geodesy is from 2005, described as a major revision from the original.?ÿ There also appear to be slightly newer printings with the same cover.
Dear Bill93,
I'm the former Chief Geodesist, before Dr. Dru Smith & Dr. Dan Roman,
and I think I can help answer your questions.
Yes, indeed, you are looking at "gravity gradients", whose units
are mgal/m.?ÿ For a ellipsoidal normal gravity model, the gradient
term depends on latitude. This can be found in (2-216) of pg.82
of Physical Geodesy (2005) Hofmann-Wellenhof & Moritz.
Note that this equation not only has a height term, h, but also
ends with an h-squared term. This term includes the variation
of gradient with height. For best accuracy, that end-term is
also included in classical gravity anomaly computations. That said,
the approximate value of the Earth's mean gradient is about
0.3 mgal/m (see top pg.84). And, yes, you can get significant
differences due to surface (or near surface) density contrasts.
The process of "moving" surface gravity to gravity at altitude
is "upward continuation. And, converting airborne gravity to
surface gravity is done by "downward continuation", or DC.
Bill93, your intuition that the single-measurement problem cannot
be solved is correct -- if you mean solved exactly. But, if
we allow more elaborate Earth/gravity models, then we can do
better analytic DC. But, for the case of unknown density contrast
of an unknown extent, at an unknown depth, a single gravity
measurement can't resolve those 3 unknowns.
This suggests two different approaches to better and better
DC. Take more gravity measurements; and use better Earth
models, which can include spherical harmonic models (like GRACE
and EGM2008) and digital terrain models (which, ideally,
could include surficial density estimates based on geology).
The utilization of more gravity measurements does *not* mean
that one must do geophysical inversion to try to estimate
detailed models for density contrast and source form, depth,
size and orientation. One can "short circuit" the process
by invoking Laplace's Equation (1-20) pg.7. First derivatives
of potential give gravity, and the second derivatives give
gradients. So, conceptually, a detailed horizontal field of
gravity provides a detailed field of horizontal gradients --
which is two thirds of (1-20). And since Laplace imposes
the condition that the terms sum to zero, we may recover
the remainder -- a field of vertical gradients. And, again,
this is done without geophysical modeling/inversion.
In practice this is done with Poisson's Integral formula,
(2-274) pg.99, which holds for any harmonic function
(harmonic functions, by definition, satisfy Laplace's Eq.).
Gravity anomalies are not harmonic, but can be continued
by means of (2-282) pg.100. And, if desired, gradients can
be continued by (2-394) pg.120.
The fly in the ointment of this is that Poisson's Integral
is written for upward continuation, not DC. However, one
may take a local area, discretize onto a grid, build
a linear system relating the gravity at surface to the
gravity at altitude, and invert the system. If one does this
then it is seen that upward continuation is quite stable.
On the other hand, DC is stable for a certain distance
of down, then it slides into numerical instability. Tiny
changes at altitude, lead to massive changes at the surface.
This instability is reflecting the fact that DC is
a mathematically "ill posed problem" (as defined by Hadamard).
In essence, as one tries to DC further and further, there
are more and more answers that can be reported as the solution.
To get around this fundamental problem of ill-posedness, additional
information is added to the problem. This is called "regularization".
One common type of regularization is that the solution should have
a condition of smoothness. The imposition of of regularization
condition does bias the solution. But it is biased in a reasonable
way and avoids the chaotic results of unregularized DC.
The end result of this is that even if you had a perfect
airborne gravimeter, you can only get a certain level of
resolution. And, that resolution is limited by how high you fly
your gravimeter. The higher you fly, the less resolution of
the DC-ed surface gravity. Therefore, the line spacing of
GRAV-D is coupled to the altitudes at which they fly. This
is one way that GRAV-D deals with ill-posedness.
The airborne gravity of GRAV-D is designed to provide a middle
tier or length scale of gravity information. The new satellite
fields provide the foundation, and digital terrain corrections
give us the shortest wavelengths. This general approach is
broadly described in the 2010 FIG abstract:
"The Application of High-Degree Gravitational Models to Processing
Airborne Gravity Collected under the NGS GRAV-D Project" by
Holmes and Roman (current Chief Geodesist).
https://www.fig.net/resources/proceedings/fig_proceedings/fig2010/papers/fs01c/fs01c_holmes_roman_abs_4057.pdf
And, all this gives a quick background to the abstract of a paper
presented at the 2018 EGU by NGS and NRCanada:
"A Performance Analysis of Several Downward Continuation Methods
for Airborne Gravimetry" by Li et. al.:
https://meetingorganizer.copernicus.org/EGU2018/EGU2018-10223.pdf
The bottom line is that gravity at altitude is smoother than
gravity at the surface. And, tiny surface (or subsurface) variations
will get "washed out" in the noise. When one downward continues
gravity, one magnifies the measurement. And a DC process will amplify
the smaller gravity signals more than the broader signals --
leading to instability. Many approaches can be taken to regularize
the DC process. NGS has favored a Spherical Harmonic Analysis
approach, as described in a 2013 Journal of Geodesy paper on the
Geoid Slope Validation Survey of 2011. However, NGS does seem to
be actively researching alternative DC methods (per the 2018 abstract).
Perhaps a different method may be favored (?) -- stay tuned to the
2019 Geospatial Summit, May 6-7, 2019.
