GSC data available as "1st Vertical Derivative" and as "Vertical Gradient"

[S E Geoscience and Exploration]

Dear all, 

I am going through some public data provided by the Geological Survey of Canada.

Can someone explain why the same data is available as "1st Vertical Derivative" and as "Vertical Gradient". It is exactly the same. This only confuses the end user.

Regards

--

Sergio Espinosa, Ph.D., P.Geo

Director, Geophysics

S E Geoscience & Exploration

[S E Geoscience and Exploration]

Furthermore. 

Sometime ago, someone was saying that a derivative is a calculation, and a gradient is a measurement. 😳

More recently, someone else was saying that a derivative is one thing, and a gradient is another thing. 😟

And now, I see on the data portal of NRCan two products as if they were different: “1st vertical derivative” and “vertical gradient”. 😱

I ask myself why this lack of rigor.

I do not wonder why so many geologists get frustrated with Geophysics. We like to make it complicated.

This is a nice video explaining:
https://youtu.be/tIpKfDc295M

Regards

Sergio Espinosa, Ph.D., P.Geo

Director, Geophysics

S E Geoscience & Exploration

[Joseph Reilly]

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

[S E Geoscience and Exploration]

Sorry, but that “core difference” is not supported either by Physics (Potential Field Theory) or by Mathematics (Vector Algebra). 

The definition of a “gradient” is a “derivative in space”. In the 2D and 3D cases, it would be a partial derivative. 

Perhaps I am wrong?

Sergio Espinosa, Ph.D., P.Geo

Director, Geophysics

S E Geoscience & Exploration

[Richard Smith]

I think the GSC know some people call it a vertical derivative and others a vertical gradient, so they have it with two names so the search engines will find it.

Regards

Richard Smith

 

[S E Geoscience and Exploration]

I am pretty much convinced that the so-called “1st vertical derivative” should be renamed to what it is: 

a “calculated vertical gradient”. 

A value plotted on the Im-axis of the gradient complex plane. 

And the measured vertical gradient should be called what it is: 

a “measured vertical gradient”. 

Has anyone had the idea to multiply the measured vertical gradient with -i ? Hopefully not.

Both values have exactly the same physical meaning: they represent the vertical component of the gradient vector field.

Perhaps the Grav & Mag Committee of the SEG is willing to review the vocabulary applied in Exploration Geophysics? It is about speaking a simple language that will certainly help the communication with geologists.

This is a print shot of the explanation by the Khan Academy about gradients (or spatial derivatives):

Sergio Espinosa, Ph.D., P.Geo

Director, Geophysics

S E Geoscience & Exploration

[Gmail]

What we often miss is a statement of the uncertainty in the geophysical products provided, possibly one source of disconnect with related discipline specialists (geologists & especially engineers) - in this example the errors or uncertainty in the "measured" vs "calculated" components will be substantially different, not just in scalar amplitude but as a function of frequency, and as mentioned previously, due to common mode rejection, geospatial locations & baselines between measurements.

 

Perhaps what we require is better, ancillary definitions of such errors, so that products are not used in an under-informed fashion.

 

Richard makes a good point about SEO, these should be able to index reports & metadata in suitably accessible formats & locations too.

Regards

Reece van Buren

On 16 Feb 2026 at 18:53 +0200, S E Geoscience and Exploration <se.geoscience....@gmail.com>, wrote:
 

I am pretty much convinced that the so-called “1st vertical derivative” should be renamed to what it is: 

a “calculated vertical gradient”. 

A value plotted on the Im-axis of the gradient complex plane. 

 

And the measured vertical gradient should be called what it is: 

a “measured vertical gradient”. 

Has anyone had the idea to multiply the measured vertical gradient with -i ? Hopefully not.

 

Both values have exactly the same physical meaning: they represent the vertical component of the gradient vector field.

 

Perhaps the Grav & Mag Committee of the SEG is willing to review the vocabulary applied in Exploration Geophysics? It is about speaking a simple language that will certainly help the communication with geologists.

 

This is a print shot of the explanation by the Khan Academy about gradients (or spatial derivatives):

 

<image.png>
 

 

 

 

Sergio Espinosa, Ph.D., P.Geo

Director, Geophysics

S E Geoscience & Exploration

 

On Mon, 16 Feb 2026 at 01:59, Richard Smith <geofi...@aol.com> wrote:
 

I think the GSC know some people call it a vertical derivative and others a vertical gradient, so they have it with two names so the search engines will find it.

 

Regards

Richard Smith

 

 

 

On Sunday, February 15, 2026 at 02:42:52 p.m. EST, Joseph Reilly <jmre...@sbcglobal.net> wrote:

 

 

 

<1771184559586blob.jpg>

 

 

On Sunday, February 15, 2026 at 12:05:05 PM CST, S E Geoscience and Exploration <se.geoscience....@gmail.com> wrote:

 

 

Furthermore. 

 

Sometime ago, someone was saying that a derivative is a calculation, and a gradient is a measurement. 😳

More recently, someone else was saying that a derivative is one thing, and a gradient is another thing. 😟

And now, I see on the data portal of NRCan two products as if they were different: “1st vertical derivative” and “vertical gradient”. 😱

I ask myself why this lack of rigor.

I do not wonder why so many geologists get frustrated with Geophysics. We like to make it complicated.

This is a nice video explaining:
https://youtu.be/tIpKfDc295M

 

Regards

 

Sergio Espinosa, Ph.D., P.Geo

Director, Geophysics

S E Geoscience & Exploration

 

On Sun, 15 Feb 2026 at 12:50, S E Geoscience and Exploration <se.geoscience....@gmail.com> wrote:
 

Dear all, 

 

I am going through some public data provided by the Geological Survey of Canada.

 

Can someone explain why the same data is available as "1st Vertical Derivative" and as "Vertical Gradient". It is exactly the same. This only confuses the end user.

 

Regards

 

 

<image.png>
 

 

--

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.
 

 

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

[Kim Frankcombe]

Sergio

The vocabulary you propose already exists - at least in draft form - attached. I'm not sure who's edits those are but not mine. I'm also not sure as to where it got to but the ASEG tech standards group reviewed this for adoption in 2023. It needs work before adoption but at least makes a start on the process.

You'll note a section defining gradient vs derivative which they then go and ignore and mix up to make confusing - no point arguing with me about it - its not my document.

However I would take exception to their description of TMI which perpetuates the myth that TMI=Magnitude of the field.

See Briener p 3 & 4 for detailed explanation but here is the last para.

also in Reeves 2005


That is Reeve's  bolding not mine.

So for the vast majority of the data we look at the anomalous TMI /= Magnitude as computed by the vector sum of the three components of the anomalous field and all valid modelling programs project the computed 3 components into the Earth's field direction to generate the anomalous TMI. BTW - This is not an invitation for another 6 months of pointless emails about TMI vs Magnitude.

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

[Stephen Reford]

The nomenclature was covered to some extent by Bates, Biegert and Reid in the April 2024 issue of The Leading Edge – “Magnetic data — What am I looking at?”. (Are we allowed to attach SEG articles on SEGMIN?).

 

Other references:

Gravity & Magnetics Exploration Lexicon by Goussev & Peirce (2000) – can be found online

Gravity and Magnetic Encyclopedic Dictionary by Goussev (2022), published by SEG.

 

Personally, I have always preferred gradient for a measured quantity and derivative as a calculated quantity.

 

The GSC used to measure the vertical gradient with its own aircraft (history lesson attached) so perhaps the nomenclature that Sergio discovered is actually differentiating between measured and calculated.

 

Cheers! Stephen

[S E Geoscience and Exploration]

Hello Kim, 

I know the vocabulary does exist. I am only proposing a revision to make it simpler, so that we can improve communication with the geologists and with other end users of our data. 

This is an example showing that we like to make things complicated. 

Let’s imagine we have a distribution in 3D of a scalar value. This scalar can be for example 

- the temperature, e.g. between 5 and 25 degrees C, 

- the magnitude of the gravity intensity field, e.g. between 9.8100000 m/s^2 and  9.8100500 m/s^2, or 

- the magnitude of the magnetic induction intensity field, e.g. between 0.0000450 T and 0.0000480 T.

Let’s now apply the Nabla operator to the field:  

(it will not take time until Ken Witherly changes my nickname from "Mr Anisotropy" to "Mr Nabla").

The result of applying the Nabla operator to the scalar field in 3D is a gradient vector field in 3D. 

(1) We call the magnitude of this gradient vector field 

“Analytic Signal”

(2) We call the vertical component of this gradient vector field 

“1st Vertical Derivative”

(3) We call the magnitude of the horizontal component of this gradient vector field 

“Gradient Magnitude” 

(4) We call the arctan function of the ratio (2) over (3)

“Tilt”, “Tilt Angle”, “Tilt Derivative” 

Four completely different names of four values that actually relate with each other and to the same gradient vector field.

(1) should be called “Magnitude of the Gradient”

(2) should be called “Vertical Component of the Gradient”

(3) should be called “Magnitude of the Horizontal Component of the Gradient”

(4) should be called “Gradient Ratio”

Also, proper naming leads to right understanding. 

Not long ago, I heard that the Analytic Signal is a "new magnetic intensity" or a “filter that fixes asymmetries”. Sorry, but the AS is not a magnetic intensity, it is not a filter, and it does not fix asymmetries. 

One property of the gradient vector is that it always points to the source of the potential field. In the case of the magnetic field of the Earth, it does not matter at what latitude we are ( magnetic inclination) and if the geological source contains remanence or not. The gradient will always point to the source.

  

I like many pleasures: reading Lev Landau is a pleasure, listening to Abdullah Ibrahim is a pleasure. But when a geologist I am working with tells me "now I understand, thank you for explaining" goes beyond any pleasure, it is a fulfillment. 

Stephen, 

I am not discovering anything. Did you watch the video of the Khan Academy? The definition of a gradient is simply a spatial derivative. It is written so in any Physics or Mathematics book. It has absolutely nothing to do whether it is calculated or it is measured.

About a year before Bates, Reid, and Biegert published the paper on the nomenclatures, I published the same topic on my Blogspot and made it public to anyone. I tried to publish the content in The Leading Edge, but it was rejected. Luckily the same topic was published a few months later. 

Cheers 

(sorry, here in the Kingdom I should say "Regards" instead)

Sergio Mr. Nabla

[James Macnae]

To the average geologist, my numerically expert accountant and also my next door neighbour (a musician),  gradient determines where they can drive.  They know that the average gradient along the road up a mountain is less than the direct gradient directly to the top, but the the elevation gained through the 'line integral along the road' is identical to the impossible straight way up.  They have no clue or interest that numerical or analytical derivatives d(height)/dz = 1 and d(height)/dx = 0.   They have no clue that their perceived gradient is actually d(height)/d(hypotenuse) rather than d(height)/dx since they believe in odometers and not linear distances. They also have no clue that at a pothole anomaly on a flat road that there are two equal and opposite delta functions in derivative d(height)/dx but do know there is no detectable change in average 'gradient' measured on either side of the pothole.. 

Temperature is a bit better.... they know dT/dNorth generally increases (I live in Southern Australia) and vice versa if you go to Tasmania or Antarctica, and that dT/dz drops or is negative  (they ski in the Australian Alps even though the ski fields are north of Melbourne) and see the screens on aircraft saying the outside temperature is -55C.

So, I suggest we just feel superior whichever terminology we use.  The average punter understands road gradients in percent (same distance units) or degrees per 100 (or 1000) km (of latitude) or degrees per 100 m (of temperature vs height).  Is 1 nT/m an anomaly? Depends on distance to source and direction and complexity.  Ultimately, magnetic derivatives are tensors, not vectors, and one gradient or derivative in one direction with undefined demominator limits is close to meaningless.

James Macnae

j...@c3dmap.com

CD3D Pty Ltd.

4 Allenby Ave, Glen Iris, 3146

Victoria, Australia

[Jayson Meyers]

How many glasses of wine did it take to form this email vector?

I estimate 4 large red wine glasses sipped within 20 minutes, forming a volume reduction gradient of 0.003 glasses per second.

Can a geologist or your artsie-accounting neighbour maintain that tensor while driving up a hill?

I reckon you got them beat!

And maybe the Canadian database is referring to measured mag gradients between two aircraft sensors?

Sent from my iPhone

On 17 Feb 2026, at 5:57 pm, James Macnae <j...@c3dmap.com> wrote:



<image001.png>

also in Reeves 2005

<image002.png>

[S E Geoscience and Exploration]

James, 

a gradient can be represented as a vector or as a tensor. It depends on the context. 

* The gradient of a scalar field is always a vector field, e.g. the gradient of the temperature or the gradient of the magnitude of the gravity intensity (a magnitude is always a scalar).

* The gradient of a vector field is a tensor field, e.g. the gradient of the vectorial gravity intensity.

If you review all chapters of Classical Physics, the Nabla operator always plays a major role, e.g. in the Maxwell equations. I can't say the same about Quantum Physics. I am still trying to understand that spooky science. 

Gradients are also a motor in social sciences, for example to describe migration. 

Have a look at this video by the Khan Academy. It is a nice refresher. It takes only five minutes. 

(2989) Gradient - YouTube

I am still scratching my head: how can people say derivative is a calculation, and gradient is a measurement?

Regards

[S E Geoscience and Exploration]

I can tell you how many glasses of wine it took me to form this email vector in Saudi Arabia. 

[S E Geoscience and Exploration]

Some will like this.

The first figure is the Calculated Vertical Gradient (also known as 1st Vertical Derivative). 

The second figure is the Measured Vertical Gradient. 

Measuring the Vertical Gradient is always better than calculating it. 

Isn't this beautiful ?

[Kim Frankcombe]

...and some wont.

It looks like you are comparing a grid based VD on an old poorly levelled dataset against a grid of line based vertical gradient. Not really apples and apples Sergio. Try computing a line based VD and then gridding that for a better comparison and please, level that TMI before applying fft filters to it😭. When computing the VD add a HF roll off (lancoz) near the nyquist to suppress noise.

As the papers from the '70/80s showed - in well sampled data with decent software it is hard to tell the difference between computed and measured vertical derivative/gradient - which is why we so rarely acquire it now days.

Cheers
Kim

[S E Geoscience and Exploration]

That is data downloaded from the GSC dataportal. It refers to the same survey. Check the first email of this thread. 

Back to the original topic: one is called "1st Vertical Derivative" and the other is called "Vertical Gradient". Both figures mean exactly the same 

My gut feeling tells me that measuring the Vertical Gradient is always better than calculating it. And I consume lots of probiotics.

[thorkild rasmussen]

Dear all,

The answer to the question "I ask myself why this lack of rigor" by Sergio is to be found as just another example among many other cases of inconsistent and/or imprecise use of various terms. In most cases, people would get the correct understanding without need to enter a detailed description of the terminology. I got somewhat surprised how the question by Sergio triggered responses from several people. I will add my comments below and maybe trigger some new comments of which some are related to gradients and some related to other inconsistent and/or imprecise use of different terms in geophysics.

 

Gradients and derivatives:

The gradient is by definition a vector with partial derivatives as elements in its components, and from a mathematical view point a derivative is the difference between two values in the limit where the distance between the observations goes to zero. An approximation to the derivative is often calculated as the difference between two measured values where sensors are placed “sufficiency close” to each other, and then simply dividing by the distance. This measuring approach has the advantage of eliminating diurnal magnetic disturbances. As mentioned by Kim Frankcombe, measuring is always better that estimating from total field data since this measured “gradient” or more correctly difference provides independent information. You can obvious also estimate the vertical derivative from both top and bottom sensors and then e.g take the mean, provided you have data sampled properly without aliasing in a grid and compare with the approximation based on simple normalised differences. 

Aliasing can be a severe problem in airborne magnetic and applying a proper interpolation algorithm prior to applying a FFT techniques for derivative calculations is essential. The data below illustrates this very well. The left panel shows TMI interpolated and gridded data provided by the geophysical contractor and the right panel the interpolation done by me (so far unpublished algorithm but I hope to get time to published when I get retired from my current position!).

 

 

 

Another comment in relation to gradient/derivatives. A standard QC procedure for airborne magnetics is fourth order (horizontal) difference calculations followed by envelope calculations. Thus, instead of talking about vertical gradients based on measured data, it is more precisely a first order difference calculation in vertical direction. It is however probably not a good idea to introduce another term to add more to the confusion.

Wave numbers, frequencies, change and spatial variation

As mentioned above, you will find many examples of inconsistent and/or imprecise use of various terms. One of my “favourite” examples is with respect to describing spatial variations of static magnetic fields. I am not a native English-speaking person so I might be wrong here, but here my view on this. 

It is very common (also in geophysical textbooks and publications) that short and long wavelength static data features are referred as high and low frequencies respectively. In my view, frequency is about how frequent, i.e. how often something changes with respect to time. Thus, I always refer to wavelength content when describing a static phenomenon. I furthermore also only use the term “change” when referring to time variations. The remote sensing community got it right when talking about “change detection” meaning variations from one instantaneous image compared to one obtained during the succeeding orbit of the satellite. The spatial variations within each instantaneous image should be referred as (spatial) variations and not changes. The distinction between change (variation with respect to time) and spatial variation is obviously of importance in the study of the global magnetic fields and its variation with respect to time and space. Similarly, when talking about changes within a static geophysical model derived from a specific dataset obtained at one instant of time does not make sense to me. Some of our colleagues doing geodynamic modelling need to make the distinction clear between what is variation with respect to time (change) and variation in space. I remember that I asked a native English-speaking editor if my interpretation was correct and he could not answer. We therefore asked a professor in English language, and he concluded that the most likely reason for the potential misuse of change with respect to spatial variation is related to an implicit time variation, i.e. somebody drive from southern England to Scotland and note a variation in temperature since it takes time to travel. Similarly, you change the focus when viewing a geophysical model and that involves implicitly a time variation. 

Resistivity method terminology

Another example (in my view) of inconsistent and/or imprecise use of difference terms is the use of the term “resistivity method” for the DC-geoelectric method. You can get resistivities based on EM methods also. I even once noted that somebody referred EM methods as “conductivity” methods. Why not just refer to DC-geoelectric method?

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 

[Rolf N Pedersen]

Allow me a thought:

Cover your derriere with E&O Insurance before saying/writing something.  No shortage of lawyers to help you.

Sergio, if in KSA, test the local rules and regulations, and drink some siddiqui.

Rolf

[S E Geoscience and Exploration]

Dear Thorkild, 

Thank you for your response. I guess many prefer not to deal with this topic and prefer to delete the emails.

Just to be brief: 

* A displacement is a vector. The first derivative (after time) of displacement is velocity, also a vector. The first derivative (after time) of velocity is acceleration, also a vector.

* The first derivative (after space) of a scalar is called gradient, also a vector.

     

     The gradient can be 1D, 2D, 3D. 

     In the 1D case, it can be plotted on the Real axis, and partial derivatives are not required.

     In the 2D and 3D case, partial derivatives are required.

     Many values in Exploration Geophysics are scalar. 

     The temperature is scalar. 

     The magnitude of the gravity field (in mGal) is a scalar. 

     The magnitude of the magnitude field (in nT or Gauss) is also a scalar.

     

I have never understood why we use so many VERY DIFFERENT names to describe products that are closely related to each other and are based on exactly the same gradient vector field built from the gravity or magnetic intensity magnitude. 

- first vertical derivative to describe the vertical gradient

- gradient magnitude to describe the horizontal gradient

- worms also to describe the horizontal gradient

- analytic signal to describe the real gradient magnitude 

- tilt to describe a gradient ratio (the arctan function normalizes its values)

I will continue working with geologists using simple words and simple explanations, so that they understand the physical meaning and how it maps geology in depth. 

Many prefer to "explain" by referring to chapter 123 and verse 456 of a published text. I prefer to explain, as it is explained to a child. This reminds me of a quote of Richard Feynman.

Regards

 

[Rolf N Pedersen]

Brief?

Looks like it was influenced by Mark Twain:

 

“I didn't have time to write a short letter, so I wrote a long one instead.”

Rolf