Can you please review these slides and comment?
Dear all,
as many of you perhaps know, I train geologists with a strong and genuine interest in Geophysics.
What can I still add to this set of slides? Check please attachment.
The geological meaning of this topic is another set of slides which I will be sharing later.
Regards
Sergio.-
Comments
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@Sergio reviewing your slides needs more context.
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Dear Elisabeth,
Here is for example more context.
A vector V in the 3D space can be decomposed into the three orthogonal components Vx, Vy, and Vz.
The magnitude (called mathematically modulus) of that vector is calculated as follow:
SQRT of ( (Vx)^2 + (Vy)^2 + (Vy)^2 )
Now, the spatial derivative of a scalar is called gradient, and gradient is a vector.
This means the gradient of the scalar TMI can be described as d(TMI) / dX, d(TMI) / dY, d(TMI) / dZ.
Last is normally known as First Vertical Derivative.
The magnitude (called mathematically modulus) of that Gradient vector is calculated as follow:
SQRT of ( (dTMI/dx)^2 + ( dTMI/dy)^2 + ( dTMI/dz )^2 )
I have asked myself all my life since I started using Oasis montaj, why Geosoft calls that "Analytic Signal". That should be called Gradient Magnitude.
Also, the ratio (Vertical Gradient) / (Magnitude of Horizontal Gradient) is a dimensionless amplitude-equalizing value. This means, it does not matter how deep or how shallow a source is. All sources will appear as coming from the same depth.
When we apply the trigonometric function arctan to that ratio, the ratio becomes an angle. This is an amplitude-normalizing value. It does not matter whether the project is related to archaeology or to exploration in the Gulf of Mexico, the values will be normalized between -1.56 and +1.56 radians.
I have asked myself all my life since I started using Oasis montaj, why Geosoft calls that "Tilt Derivative (TDR)". That should be called just Tilt.
And Tilt is an angle only when it is drawn on paper or only when we imagine the Gradient vector field in 3D space. But that angle has nothing to do with geological angles such as strikes, dips, and plunges.
Naming all these parameters wrongly only confuses the non-geophysicists (and also some geophysicists). I read recently in a text that the physical unit of "nT/m" was given to a parameter called Tilt Angle Derivative. As said, Tilt is neither a derivative nor a (geological) angle.
Would this be more context?
Thanks
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The way I see it is that your formula:
SQRT of ( (dTMI/dx)^2 + ( dTMI/dy)^2 + ( dTMI/dz )^2 )
is both the magnitude of the gradient vector and the amplitude of the analytic signal.
The terminology follow scientific publications, e.g.:https://www.eoas.ubc.ca/academics/courses/eosc450/ewExternalFiles/Blakely_PotentialFieldsText.pdf
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Hi Victor,
What I wrote in my original message is not "my" formula.
That is exactly the same as formula 12.60 (in vectorial form). It is the calculation of the magnitude, amplitude, norm, or modulus (four words to describe the same) of that vector A.
deltaT is the magnetic anomaly, a scalar in nT
d (deltaT) / dx is the gradient in x direction, vectorized by the unit vector i
d (deltaT) / dy is the gradient in y direction, vectorized by the unit vector j
d (deltaT) / dz is the gradient in z direction, vectorized by the unit vector k
A is the gradient (a vector in the 3D space) of the magnetic anomaly deltaT
Could you please describe with your own words why the magnitude, amplitude, norm, or modulus (four words to describe the same) of the gradient (a vector in nT/m pointing to the source) of the magnetic intensity (a scalar in nT) is called Analytic Signal?
Regards
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It's not exactly like you state.
The norm of the gradient so happens to have the same formula as the amplitude of the analytical signal (note the i in the 12.60).
Formula 12.60 so happens to have no negative frequency components (definition of analytical signal), it is constructed to have that property. That property has proved useful for interpretation in magnetic signal and in other domains as well.
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Again, what people call Analytic Signal is nothing more that the gradient (a vector oriented in the 3D space) of the field intensity (a scalar distributed in the 3D space).
I have the impression that many, not all of course, have forgotten that the field intensity (nT for the magnetic field, mGal for the gravity field) decays with distance to the source no matter the direction (r^3 for the magnetic field, r^2 for the gravity field).
The distance is given of course by deltaX, deltaY, and deltaZ in the Eucledian space. As it is in the definition of the Analytic Signal.
The unit vector in those three orthogonal directions of the gradient vector are i, j, k. As it is in the definition of the Analytic Signal.
The values we plot on Analytic Signal maps is nothing more than the magnitude or norm, or module, or amplitude, or length, or size (six words to describe the same) of the gradient vector. It does not just "happen to have" the same formula.
Another story is that the vertical component (known as First Vertical Derivative) and the horizontal component of the gradient vector relate with each other on the complex plane. Those values on that complex plane (Re & Im) are a function of the distance to the source. That is why the ratio (called Tilt) is useful to map deep sources.
I am not sure why you talk about frequency. We are not dealing with time series or functions of time A=f(t), what is called monitoring. We are dealing with functions of space A=f(r), what is called mapping. By the way, "A" stands for "Anomaly", not for the magnetic vector potential.
The gradient d(A)/d(r) is of course very useful, since it always points to the source, no matter if the source is bipolar (such as in magnetic fields) or unipolar (such as in gravity and in electric fields).
Also, the gradient is calculated with the Nabla operator. The Nabla operator applied to scalar fields and to vector fields, as divergences and rotations, is one of the most used (if not the most used) operator in Physics, e.g. in the wave Equation, Maxwell equations, Navier-Stokes, etc.
Please, correct me if I am wrong.
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Hi Victor,
I would suggest to do this mental exercise.
Imagine you have a fix-wing with four magnetometers: left wing, right wing, tail, and towed below the aircraft.
- left wing and right wing would be measuring horizontal gradient across flight line
- tail would be measuring horizontal gradient along flight line (two consecutive measurements)
- horizontal gradient across and along flight lines would form the (total) horizontal gradient. I do not think there is a need to call it "total". Since horizontal gradient is only only.
- tail and towed would be measuring the vertical gradient, what is also known as First Vertical Derivative. This gradient is now measured, not calculated using Fourier analysis.
- Horizontal gradient and vertical gradient would form the (total) gradient. I do not think there is a need to call it "total". Since gradient is only only. It is a vector with a direction (pointing always to the source) and a magnitude, what is also known as Analytic Signal.
Question: do you see any need to describe the vertical gradient as the Im part of a complex plane?
I am convinced that calling the magnitude (norm or module) of the gradient vector "Analytic Signal" is completely unnecessary and confuses only the non geophysicists, especially the geologists who normally pay our salaries. It also does not fit the mathematical definition of Analytic Signal.
https://en.wikipedia.org/wiki/Analytic_signal
I am an advocate of presenting Geophysics simple and easy to understand.
Thank you
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The analytical signal, which is just any formula which has no negative frequency components, is deemed useful because, among other things, one can take an envelope. This property is deemed useful before even talking magnetics
So this is reason why it is useful is formed independently from the reasons you are stating. It does fit the mathematical description, I think you are confused because frequency normally relates to time, but here we talk about spatial frequency (unit m-1 instead of s-1).
But nevertheless, I agree that formulating the 'useful formula' by taking the norm of the gradient is easier to understand and easier to see why it's useful.
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Am I confused? What you call here frequency is actually called wavenumber in Physics.
I think I understand very well what a difference, a differential, a derivative, a spatial derivative, a gradient, a gradient vector, and how the magnitude of a vector is calculated.
A final question: if the magnitude of the vertical component of the gradient vector (called in daily jargon First Vertical Derivative) is plotted on the Im-axis of the complex plane in the Analytic Signal description, why isn't the vertical component also described on the Im-axis when talking about Tilt?
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The reason why there is an imaginary component in the analytical signal is because then horizontal and vertical derivative forma form a Hilbert pair (with dZ calculated using the formula for upward continuation
)
The Hilbert pair is not sought after when talking about tilt.
.
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Amen
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