Derivatives products etc.

[Henry Lyatsky]

1.  The preceding discussions remind me why I like to work with gravity.  It's so much simpler, in so many ways.

2.  I always find the horizontal gradient (computed derivative) a lot more intuitive and easier for users to grasp.  It's also very revealing of many geological features vertical derivatives don't show.  I have also found it very fruitful to compute a 2nd-order horizontal derivative, which sharpens up many desirable features significantly (you treat your 1st-order grid as a base and run the horizontal-gradient procedure on it again).  Also, I have played with plotting horizontal-gradient maps as vectors - as attached.

3.  Let us please, please remember that our clients are GEOLOGISTS, for whom esoteric math is deeply off-putting because it smacks of the black box.  To be a successful geophysicist, one must first think like a geologist, and not like a mathematician or physicist.

Henry Lyatsky
* confidential

[Thorkild Maack Rasmussen]

Dear Henry,

I partly agree with your third statement that most of "our clients" are geologist and they do not care much about mathematics and physics, but that is not an excuse for not doing it right. 

I was recently at a conference and noticed that people who have been doing mineral prospectively mapping for many years did not understand the complexity of magnetic anomalies, and they tried to correlate and use magnetic maxima/minima with gravity maxima/minima plus geochemical anomalies. More specifically, correlate magnetic and gravity response peaks. It is a rather simple exercise to show that this is not an appropriate way to deal with magnetic data as input to correlation analyses with gravity data. Magnetic responses are always with both positive and negative peaks whereas this is not the case for gravity data. David Clark in his publication in Exploration Geophysics, 2014, 45, 271 -304 http://dx.doi.org/10.1071/EG14013 details how to proceed in transforming magnetic data to something that is more feasible, both for direct targeting, but also for input to mineral prospectively mapping. The figure below highlights the problem. Five spheres of which the one in the centre has remanent magnetisation orthogonal to the inducing field (here 90 degrees inclination of the inducing field).  Your suggestion in the mail is not going to provide something easy for interpretation.

 It is obviously not necessary that all geologist go through all the mathematics and check everything from first principle, but as a minimum they should be able to extract the information from the figures that explains the concepts. More suggestions for data processing can be found in https://onlinelibrary.wiley.com/doi/10.1111/1365-2478.13573  Enhancing magnetic source edges using the tilt angle of the analytic-signal amplitudes of the horizontal gradient by Pham et al. 2024.

Best regards

Thorkild

 

 

Thorkild M. Rasmussen

Professor in Exploration Geophysics

 

Luleå University of Technology
Department of Civil, Environmental and Natural Resources Engineering
Postal address: SE-97187 Luleå, Sweden
Visiting address: F-Building, University Campus, Porsön, Luleå
Phone: +46 (0)920 49 10 00 Fax: +46 (0)920 49 28 18

 

e-mail: Thorkild.Maa...@LTU.SE

Phone: +46 (0) 920 49 34 13 

 

To unsubscribe from this group and stop receiving emails from it, send an email to segmin+un...@aseg.org.au.

[Henry Lyatsky]

And that's precisely the reason I always try to explain geophysics basics to my clients from first principles, without math.

I was once told by a small oil company that mag is no good: they drilled every mag high on their land expecting to find basement highs, and found nothing.  I told them this fiasco could have been easily avoided by calling me ahead of their drilling program.

I am no fan of the analytic signal and tilt derivative because they combine the horizontal and vertical derivatives, which intuitively aren't apples and apples.  Math can bear out anything, but it's by its nature abstract, and we need our products to be geologically meaningful and intuitively relatable to the oroginal Bouguer and RMI anomalies.

A college prof told me long ago that his undergrad grav/mag course was all about the physics of potential fields.  I told him this material should take up no more than the first 3 lectures, and the rest should be spent on geologically minded survey design and data acquisition and processing with a view to geological interpretation.  He had no idea what I was saying, but I was very clear I'd never use grads trained that way.

Henry Lyatsky
* confidential

[S E Geoscience and Exploration]

Hi Henry, 


1. Yes. Gravity is a unipolar field (as is the electric field). It does not have issues with “gravity inclinations” or “gravity remanence”. 

The magnetic field is bipolar, but it can be simulated to a unipolar field with the RTP. Its intensity still decays at r^3 the distance to the source though.


2. You have shown the horizontal gradients in vectorial form before. That is a very nice visualisation. 

Now imagine the gradient vector field in 3D. Those vectors will always point to the source, no matter whether the field is unipolar or bipolar. 


3. When I joined mineral exploration 30 years ago, many geophysicists were talking about “how to bridge the gap to geologists”. 

Now, 30 years later, I am still hearing the same thing. 

If we do not change the way we communicate to geologists, in 30 years (I will be close to retirement), we will still be talking about “how to bridge the gap to geologists”. 


Back to the gradient vector field you showed, but in 3D, the vertical gradient (derivative) and the 

horizontal gradient (derivative) are components of the same single gradient vector in 3D. Stating that they are oranges and apples is not correct. They are branches of the same tree.

Are we aware that we use at least 4 completely different names to describe elements of the gradient vector field? 

- “Analytic Signal” to describe the magnitude 
of the gradient field. I heard even today something called “source strength - something similar to AS”. I still need to think about that.

- “First Vertical Derivative” to describe the vertical component of the gradient field.

- “Worms” to describe the horizontal component of the gradient field.

- “Tilt” to describe the ratio of vertical gradient over horizontal gradient. Only when plotting the gradient vector field in 3D, we will be able to enjoy the Tilt as an angle. Calling Tilt a derivative is definitely a misuse.

I think the Grav & Mag Committee must meet again to revise this terminology when talking to geologists. 

We can keep Hamilton, Lagrange, Laplace, Fourier, Hilbert and others for ourselves.

Regards

Sergio Espinosa, Ph.D., P.Geo

Director, Geophysics

S E Geoscience & Exploration

To unsubscribe from this group and stop receiving emails from it, send an email to segmin+un...@aseg.org.au.

[Henry Lyatsky]

1.  The magnetic field is dipolar.  Bipolarity is a human condition otherwise known as manic-depressiveness.

2.  RTP assumes all rock magnetization in the map area in induced and there's no remanence.  This may be the case on a local prospect scale but rocks first need to be tested extensively to see if that's actually the case.  On a regional scale, a default assumption that remanence is lacking across the map area is pretty sure to be ungeological.  I stay away from RTP, whose uncritical use is an example of poor knowledge of geology among geophysicists.

3.  In old European college programs in geophysics, the first 3 years were identical to the geology program and only in the last 2 years was geophysics covered.  Given the destruction of higher education since the 1960s and the inadequacy of a 4-year program, I took degrees in both geophysics and geology.  You can't be a successful geophysicist without being a geologist first.

Henry Lyatsky
* confidential

[Thorkild Maack Rasmussen]

Dear Henry and Sergio,

I noticed your reference to RTP and “source strength”.  I would like to put your attention to the paper by David Clark 2012: New methods for interpretation of magnetic vector and gradient tensor data I: Eigenvector analysis and the normalised source strength

 Exploration Geophysics, 2012, 43, 267–282http://dx.doi.org/10.1071/EG12020.

I assume this is what Sergio is referring to with respect to “source strength”. The method developed by David Clark does not make any assumptions on the amount and direction of remanent magnetisation and is therefore superior to the RTP method by (1) providing a transformed data set that peaks above the ore body irrespective of the amount and direction of remanent magnetisation  and (2) providing data for inverse modelling of the source. The transformation is therefore exactly doing what Henry is asking for; i.e providing an image that any geologist without interest in mathematics and physics can utilise for locating the ore body with respect to location horizontally. For depth estimation, you obviously need to do some modelling.

Best regards

Thorkild

 

Thorkild M. Rasmussen

Professor in Exploration Geophysics

 

Luleå University of Technology
Department of Civil, Environmental and Natural Resources Engineering
Postal address: SE-97187 Luleå, Sweden
Visiting address: F-Building, University Campus, Porsön, Luleå
Phone: +46 (0)920 49 10 00 Fax: +46 (0)920 49 28 18

 

e-mail: Thorkild.Maa...@LTU.SE

Phone: +46 (0) 920 49 34 13 

 

From: seg...@aseg.org.au <seg...@aseg.org.au> on behalf of Henry Lyatsky <lyat...@gmail.com>

[S E Geoscience and Exploration]

Thank you for explaining. I see I have lots to catch up. No time to get bored.

The Gradient (we call its magnitude “Analytic Signal”) also peaks on top of the ore body, no matter the magnetic inclination or the direction of any existing remanence. Does this normalised source strength do a better job?

Isn’t the measured TMI or AMF data enough to model the source?

The magnetic field intensity decays at the r^3 distance from the sensor to the source. Its gradient (in vectorial or tensorial form) 

decays at the r^4 distance. Does the normalised source strength also decay with the distance? If so, why is it called “source strength”. 

I am just comparing this with the concepts of magnitude and intensity in Seismology. While the magnitude describes the source (e.g. the magnetic susceptibility), the intensity is a value that decays with distance (e.g. the magnetic intensity or its gradient). If it is a value that decays with distance, calling it “source strength” is rather confusing, because it does not really describe the property of the source, but the value at a point X, Y, Z in the field.

I will have a read. Thank you for sharing.

Sergio Espinosa, Ph.D., P.Geo

Director, Geophysics

S E Geoscience & Exploration

[S E Geoscience and Exploration]

Question to Thorkild and to anyone else, 

I will start reading 

• New methods for interpretation of magnetic vector and gradient tensor data I: eigenvector analysis and the normalised source strength

But before I get into this, …

Question: does the Normalised Source Strength decay with the distance sensor-to-source in the same way that 

- the field intensity decays: r^2 for gravity and r^3 for magnetics, and 

- its gradient (the magnitude, called Analytic Signal) and corresponding components (Horizontal; and Vertical, called First Vertical Derivative) also decay: r^3 for gravity and r^4 for magnetics. 

The benefit of applying the gradient ratio (Tilt) is that the ratio diminishes the effect of this decay (amplitude equalising effect), and we can therefore see details and nuances in depth. The application of the arctan ratio to that ratio normalises the amplitudes from -1.56 to +1.56 radians.

Thank you

Sergio Espinosa, Ph.D., P.Geo

Director, Geophysics

S E Geoscience & Exploration

On Fri, 20 Feb 2026 at 21:36, Thorkild Maack Rasmussen <thorkild.maa...@ltu.se> wrote:

[Thorkild Maack Rasmussen]

Dear Sergio,

The answer to your question can be found in the paper by Clark (2012) on pager 273. An excerpt is shown below. The paper contains a lot of details so please take your time to read carefully. Various comments in the paper will guide you with respect to applications and use of the developments. 

In my view, it is important to emphasise that these developments are not a replacement for other old methods such as tilt-derivatives etc. for “qualitative" interpretations and standard inversion techniques with TMI data as input. It is complementary and may e.g. guide you with respect to choosing between vector inversion or scalar inversion, thereby avoiding unnecessary complications of e.g. selecting vector inversion if this is not required by the data.

Best regards

Thorkild

 

 

Thorkild M. Rasmussen

Professor in Exploration Geophysics

 

Luleå University of Technology
Department of Civil, Environmental and Natural Resources Engineering
Postal address: SE-97187 Luleå, Sweden
Visiting address: F-Building, University Campus, Porsön, Luleå
Phone: +46 (0)920 49 10 00 Fax: +46 (0)920 49 28 18

 

e-mail: Thorkild.Maa...@LTU.SE

Phone: +46 (0) 920 49 34 13 

 

[S E Geoscience and Exploration]

If the normalized moment "miu" refers to the dipole source, then it is not correct to compare it with the Gradient Magnitude (called in our daily language Analytic Signal), since the Gradient Magnitude does decay with the distance source-to-sensor, as do their components Vertical Gradient (called in our daily language First Vertical Derivative) and Horizontal Gradients. 

We normally plot the AS at the measurement surface, but the AS fills the whole 3D space. I did that exercise with Excel.

Thank you for sharing. 

--

[S E Geoscience and Exploration]

Furthermore, Tilt is not used for a “qualitative interpretation”. It quantifies the magnetic or gravity response with hard values.

Let’s imagine following scenarios:

For a shallow source: 

- vertical gradient: 100 nT/m

- magnitude of horizontal gradient: 100 nT/m

- Gradient Ratio: 1

For a deep source: 

- vertical gradient: 1 nT/m

- magnitude of horizontal gradient: 1 nT/m

- Gradient Ratio: 1

Therefore is Tilt an amplitude equalising value. 

Ah, sorry, someone wants to apply the arctan function to that value? 

There we go, it is 45 degrees or 0.79 radians.

Therefore, when applying the arctan function, Tilt is also an amplitude normalising value.

I will ask myself until my retirement (30 more years), and beyond, why people keep calling Tilt a derivative. 

Regards 

Sergio Espinosa, Ph.D., P.Geo

Director, Geophysics

S E Geoscience & Exploration

[Kim Frankcombe]

Sergio

At the risk of extending this unnecessarily...

r^3 is what we were all taught at uni and applies to much of what we are interested in as explorers but it is shape and to a lesser extent, inclination dependant - see Breiner p30. This is the structural index "n" of source parameter imaging.



When inverting mag tensor data we'd typically use r^4 but if we have r^1 or r^2 geology we get a better result using r^3 or r^(2.5-3.5) decay with distance. Horses for courses. For example basement will nearly always be r^0 to r and if that is your target you'd need to change the fall off rate to avoid underestimating its depth.

For clarity the Analytic signal does not have a maxima over the source for most inclinations - It is close but the biggest offset is at I=+/-45 deg and the least at 0 and +/-90. It just does a better job than most other commonly used tools and is invaluable at the equator.

Also in some rare perverse cases a local TMI high can become an AS low rather than high over the body. That is much more common when computing John Paine's ASVI and care needs to be taken to avoid making obviously magnetic bodies appear less magnetic. Deriving VRMI (Dransfield) or T Modulus (Gerovska) via phase rotation in the frequency domain also suffers from this problem

Gordon Cooper and Duncan Cowan have written some good stuff about this as have others. Some propose taking the AS of the RTP but to me that misses the distortion RTP will have over remanent bodies and at low latitudes. Taking the RTP of the AS however does restore the peak to closer to the vertical projection of the body at mid latitudes where the departure in AS is greatest. 

Gordon and Duncan suggested using the normalised standard deviation and that also works OK as a bump on the source method although like AS there is an offset at mid-latitudes. There is a good summary of some of the edge detection methods in GEOPHYSICS, VOL. 86, NO. 4 (JULY-AUGUST 2021); P. G23–G34, 10.1190/GEO2020-0165.1 - note the typo in their formula for the VDR in table 1.  Fedi and Abas extended the use of the THD of Tilt (=wave number) at non integer derivatives to derive a source parameter which recovers both n and depth and that works very well in many cases.

Cheers
Kim

--

Kim Frankcombe

Senior Consulting Geophysicist

ExploreGeo

PO Box 1191, Wangara, WA 6947 AUSTRALIA

Unit 6,10 O’Connor Way, Wangara, WA 6065, Australia

Phone +61 (0)8 62017719 - if your call goes to voice mail, leave a message. It converts to an email which I'll get where ever I am!

Email k...@exploregeo.com.au

[S E Geoscience and Exploration]

That is true. I never said anything different. 

But I was always referring to a dipole as a source. And the Earth is a dipole.

Since I am a bit lazy to write, I asked my Copilot to write it down for me. I just double-checked that it is correct.

(1) For an unipolar field, such as the Gravity or the Electric field, this is valid

(2) For a bipolar field, such as the Magnetic field, this is valid

[S E Geoscience and Exploration]

And you are also right when you say that the Analytic Signal (I call it "Magnitude of the Gradient Vector") does not necessarily pick over the source. My mistake. 

What I meant is that the gradient vector always POINTS to the source.

 

See, please, attached figure below (the values are not to scale. It is just to visualize).

* The red vectors represent the Magnetic Field Intensity Vector (with direction and magnitude)

* The blue vector represents the gradient of the magnitude of the Magnetic Field Intensity Vector.

You can quote as follows: 

Espinosa, S (2025), Training Course on Geophysics Applied in Mineral Exploration, 

GEOMIN, Society of Exploration Geophysicists, 

Jeddah, Kingdom of Saudi Arabia.  

 ()

[re...@exploregeo.com.au]

Sergio,

 

Your “Tilt Derivative” (and “Analytic Signal”, etc.) quest is our “Koala Bear”. A Koala is a marsupial, not a bear. Its technical name is “Phascolarctos cinereus” (yes, I had to Google that). Same goes for the “Tasmanian Tiger” (not a tiger), the “Bin Chicken” (not a chicken), the “Flying Fox” (not a fox), the “Tasmanian Devil” (not evil) or the “Sea Lion” (that’s right, not a lion…). Yet everyone knows what these animals are (at least we do here in Australia). I guess what I am saying is in science it’s acceptable to have a “purist” technical name specialists use amongst themselves to make them feel smarter, and a “common name”, more public-friendly that the other 99.9% of the planet use. I certainly would not go around saying I got urinated on the foot at the Canis lupus familiaris park…

 

So, by having a few different names for the same thing, including technically not-so-correct common names, I don’t think we are making things more confusing, I actually think we make them more accessible. I am certain everyone in this group understands what a tilt angle filter does and what it means, and virtually debating again and again and again (and again) how it should be called is a waste of resource and everyone’s time. If that is something that you feel passionate about (and something tells me that you are), then please start a sub-group with a few like-minded people, come up with a better nomenclature, distribute for review, and publish it. I would happily review and contribute to that paper. But sending countless almanacks criticising what other people wrote in past emails and reports is not doing any good to anyone.

 

I can see how a universal language in our field would be beneficial, but I must confess I believe we have much bigger fish to fry as a profession. I personally don’t really care if some data are called “First Vertical Derivative” or “Vertical Gradient” as long as there is appropriate metadata with the file. But for example, I am appalled by the fact that these “open file” datasets are only made available in a proprietary binary file format… I had my fair share of traumatic experiences trying to recover old corrupted Geosoft databases downloaded from Canadian geological surveys websites and that to me is a real problem hindering science and exploration, not whether I feel like naming my images “_dxy”, “_THG”, “_THD” or “_slope” on any given day…

 

Cheers,

 

Regis.

 

Regis Neroni, P.Geo. Consulting Geophysicist   T: +61 (0) 458 458 079 E: re...@newgengeo.com.au W: www.newgengeo.com.au

 

 

 

From: seg...@aseg.org.au <seg...@aseg.org.au> On Behalf Of S E Geoscience and Exploration
Sent: Monday, 23 February 2026 1:36 AM
To: Thorkild Maack Rasmussen <thorkild.maa...@ltu.se>
Cc: Henry Lyatsky <lyat...@gmail.com>; seg...@aseg.org.au
Subject: Re: [SEGMIN] Derivatives products etc.

 

Furthermore, Tilt is not used for a “qualitative interpretation”. It quantifies the magnetic or gravity response with hard values.

 

Let’s imagine following scenarios:

 

For a shallow source: 

 

- vertical gradient: 100 nT/m

- magnitude of horizontal gradient: 100 nT/m

- Gradient Ratio: 1

 

For a deep source: 

 

- vertical gradient: 1 nT/m

- magnitude of horizontal gradient: 1 nT/m

- Gradient Ratio: 1

 

Therefore is Tilt an amplitude equalising value. 

 

Ah, sorry, someone wants to apply the arctan function to that value? 

There we go, it is 45 degrees or 0.79 radians.

 

Therefore, when applying the arctan function, Tilt is also an amplitude normalising value.

 

I will ask myself until my retirement (30 more years), and beyond, why people keep calling Tilt a derivative. 

 

Regards 

Sergio Espinosa, Ph.D., P.Geo

Director, Geophysics

S E Geoscience & Exploration

 

On Sun, 22 Feb 2026 at 14:46, Thorkild Maack Rasmussen <thorkild.maa...@ltu.se> wrote:

 

Dear Sergio,

The answer to your question can be found in the paper by Clark (2012) on pager 273. An excerpt is shown below. The paper contains a lot of details so please take your time to read carefully. Various comments in the paper will guide you with respect to applications and use of the developments. 

 

In my view, it is important to emphasise that these developments are not a replacement for other old methods such as tilt-derivatives etc. for “qualitative" interpretations and standard inversion techniques with TMI data as input. It is complementary and may e.g. guide you with respect to choosing between vector inversion or scalar inversion, thereby avoiding unnecessary complications of e.g. selecting vector inversion if this is not required by the data.

Best regards

Thorkild

 

[S E Geoscience and Exploration]

The "Tilt Derivative" is not mine. Maybe yours? :)

If you are able to integrate the "Tilt Derivative" and find a root function and a constant, please call it a "derivative". 

For me it is primarily a ratio. And, as a ratio, it can see nuances in depth. Similar to the ratios applied in Radiometrics. 

I will repeat an example for two different cases:

(1) Shallow Source

- Vertical Gradient: 100 nT/m

- Magnitude of the Horizontal Gradient: 100 nT/m

- Ratio: 1

(2) Deep Source 

- Vertical Gradient: 1 nT/m

- Magnitude of the Horizontal Gradient: 1 nT/m

- Ratio: 1

Ratio is the same for both cases = 1. Therefore, the Tilt is amplitude-equalizing.

You prefer to have it in "angle format"? That is fine, go ahead and apply the arctan function.

Ratio is the same for both cases = 45 degrees. Therefore, the Tilt is also amplitude-normalizing (between -90 degrees and +90 degrees).

I will try to plot the gradient vector field in 3D, similar to what Henry did for the horizontal gradient vector field in 2D.

Regards

--

[Chris Wijns]

C0 - PUBLIC

Hear hear, end of discussion, thanks all.

Chris

---

From: regis via SEGMIN ASEG <seg...@aseg.org.au>
Sent: Monday, February 23, 2026 8:00:51 PM
To: 'S E Geoscience and Exploration' <se.geoscience....@gmail.com>; 'Thorkild Maack Rasmussen' <thorkild.maa...@ltu.se>
Cc: 'Henry Lyatsky' <lyat...@gmail.com>; seg...@aseg.org.au <seg...@aseg.org.au>
Subject: RE: [SEGMIN] Derivatives products etc.

 

[James Macnae]

There is an error in Breiner's diagrams and simple formulae that seems to be perpetuated:  when you integrate 1/rcubed you get 1/rsquared.  When you integrate 1/rsquared you det 1/r,  When you integrate 1/r you get natural log of r and NOT 1 (or not 1 over r to the power zero).  The decrease of index n by 1 with each integration used in Breiner's (and others) analysis is not valid for n=1.

Jim

James Macnae

j...@c3dmap.com

CD3D Pty Ltd.

4 Allenby Ave, Glen Iris, 3146

Victoria, Australia