On 18 October, the geophysicist Hosein Moghaddasi posted a comment on LinkedIn about the use of magnetics in mineral exploration in which he used the term ‘Tilt derivative’. The vast majority of geophysicists working in that area would have passed that by without comment, but aroused the ire of at least one, who wrote
Why do you call Tilt a derivative?
The derivative of the function
y = f(x) is y’ = f’(x) = dy/dx.
Does Tilt represent such a derivative? I do not think so.
Unwisely, I entered the conversation with a comment of my own, I said
I think you are being unduly picky here.
Yes, derivative has a specific meaning in calculus.But in plain English a derivative is something derived from something else.
And with that definition, tilt is a derivative
NB the finance people talk about derivatives when dealing with things like stock options. They have a value derived from something else.
Not, certainly, the most incisive of comments, and it unleashed a response in which it was suggested that I was the sort of person who would confuse dogs with cats or pyrite with economically attractive minerals. A twelve-part course in very basic calculus was also provided.
All of which set me thinking a bit more about terminology in exploration geophysics, where there is no final arbiter of correct usage, and in particular about ‘tilt’ in magnetics, because ‘tilt derivative’ is not the only term circulating, and confusing the issue, on this subject. There is also ‘tilt angle’ and ‘tilt depth’, neither of which is quite what it seems. Arguably, it is not ‘tilt derivative’ but ‘tilt angle’ that is the most misleading of these terms, since it would seem reasonable, in the context of magnetic fields, to assume that ‘tilt angle’ is synonymous with ‘magnetic dip’.
Amongst geographers it is, but that is not how it is used by potential-field geophysicists. For them it is the tilt of the gradient of the magnetic field, and is obtained by determining the magnitudes of the gradients in the vertical direction and in the horizontal plane, and dividing the former by the latter and taking the arc tangent. Mathematically, these gradients are the vertical and horizontal partial derivatives of the field, and the objection was therefore formally correct; the ratio of these two derivatives is not itself a derivative, if that word is being used in the way it is defined in calculus. If, however, objections are to be raised, ‘tilt angle’ seems to me to be more objectionable than ‘tilt derivative’. The latter at least suggests that calculus derivatives are involved, a fact that ‘tilt angle’ conceals very efficiently.
The situation became still more complicated in 2008, when Ahmed Salem and his colleagues presented a paper with the title “Interpretation of magnetic data using tilt-angle derivatives”, in which the two terms were combined and in which the reader’s eyes were assaulted almost immediately with a page and a half of equations involving partial differentials. First there were three equations for the vertical and two horizontal gradients involved in computing the tilt angle, and then the discussion shifted to the Euler equation, for which an equation was given in terms of differentials. There followed three equations obtained by differentiating this equation with respect to the three coordinate directions, and these were then multiplied by expressions involving another five partial differentials. One of the ‘horizontal’ equations obtained in this way was then subtracted from the ‘vertical’ one, and the process was then repeated for a second pair, and the two resulting equations were then added to produce an equation involving no fewer than thirty first- and second-order partial differentials. These were then divided by yet another partial differential, the total gradient of the magnetic field.
The second page of Salem et al. (2008) “Interpretation of magnetic data using tilt-angle derivatives”
One of the ‘horizontal’ equations obtained in this way was then subtracted from the ‘vertical’ one, the process was repeated for a second pair, and the two resulting equations were then added to produce an equation involving no fewer than thirty first- and second-order partial differentials. These were then divided by yet another partial differential, the gradient of the magnetic field.
By this time the readers of the paper may well have lost the will to live, or at least the will to carry on reading, but help was at hand. Salem and his colleagues had waited until this point to ‘suggest a simple strategy to automate the method and reject poorer solutions‘ via ‘the total horizontal derivative kh of the tilt angle’, claiming this to be ‘sharper‘ than the total derivatives of the field and therefore capable of generating maxima that were better defined and centred over the 2D body edges’.
So, were all those other partial differentials invoked in vain? Reading on, it seems that they were, and even the use of ‘kh’ was questionable, because very soon a statement appeared that the method ‘uses the strategy … of applying a moving data window to the magnetic anomaly’ and, a little further on, that ‘the choice of data window size is a function of the data quality and the degree of interference of anomalies from nearby sources’, and it was also made clear that the method was to be applied to gridded data. The use of gridded data means that calculus has been abandoned (as it always eventually is in geophysics) for the mathematics of finite differences and for digital processing. With such data, the size of the grid cells determines the shortest wavelengths that are preservedl, and the size of the window determines the characteristics of what is, in effect, a low-pass filter. These are parameters that must ultimately be chosen by human beings, and Salem et al. actually stipulated that before a solution could be considered acceptable, it had to be examined in terms of the following criteria
‘1) An accepted range based on the structural index _e.g., 0 to 2 for 2D sources_
2) An accepted range for the depth _e.g., source depths that the interpreter may accept as a possible solutions.
3) Acceptable estimates of horizontal source coordinates that should be close to the location of the detected vertical derivative peak
4) An acceptable threshold of the estimates of the standard deviation from the least-squares solutions depending on the quality of the data.’
This is, in a way, comforting. To examine acceptability in these terms, someone will have to look at the original data, before the algorithms get hold of them (and, hopefully, before the reduction-to-pole algorithm that seems to be an essential part of the process is applied). And not merely look, but think.
And so, to return to what started me down this road. Was the complaint about the use of the term ‘tilt derivative’ justified?
Certainly, yes, but on more grounds more fundamental than those suggested. Not on whether a map based on ratios of derivatives can be properly described as a derivative map, but on the fact that we should be honest and admit that we don’t ‘do’ derivatives at all in exploration geophysics. We are wedded to digital processing and we should avoid blinding clients, and students, with the symbology of calculus, and talk about gradients instead. Almost everyone understands what that word means.