Gradients or derivatives of magnetic and gravityfields are more sensitive than the measured TMI and normal gravity to changes in the physical properties of the subsurface, so they are a detail-enhancementfilter. Derivatives emphasise shallow bodies in preference to the deeper-seated broader features, which produce small changes (gradients) in the fields. Magnetic and gravity derivatives are calculated from the TMI and gravity data (see Gradients and curvature in
Section 2.7.4.4), if they were not directly measured with a
gradiometer as described in Section 2.2.3. The use of the vertical gradient is especially common. It can be visualised as the difference between the upward- and downward- continued (see Section 3.7.3.2) responses at equivalent locations, and normalised (divided) by the difference in height between the two continuation responses.
Vertical and horizontal gradients are very sensitive to the edges of bodies and are‘edge detectors’. Note how the relevant responses inFigs. 3.25and3.26are concentrated at the edges of the prism. For vertical dipping contacts, and with vertical magnetisation in the case of magnetic sources, the derivative responses coincide with the contact. They are displaced slightly from the body when the contact is dipping or the source is narrow, although this displace- ment is only normally significant for very detailed studies (Grauch and Cordell,1987).
Figures 3.25d and3.27d show the first vertical deriva-
tives of the gravity model and the gravity data, respectively, with the equivalent responses for the magnetic data shown
in Figs. 3.26b ande, and Figs. 3.28c andd. As shown in
Figs. 3.25and3.26, the relationship between the derivatives and their respective sources is quite simple for gravity, but can be more complex for magnetics. The form of the
magnetic derivatives depends on the direction of the source magnetism and is usually multi-peaked and dipolar unless the magnetism is vertical. The asymmetric responses due to non-vertical magnetisation can be over- come by applying the derivatives to pole-reduced data (see
Section 3.7.2.1).
The first vertical derivatives of the Las Cruces and Broken Hill data are dominated by much shorter- wavelength anomalies than the Bouguer anomaly and TMI data. Features in areas which are smooth in the TMI data now contain recognisable anomalies (e.g. 1) and details of the structure of an intrusive (2) and folded stratigraphy (3) are easier to resolve. The first vertical derivative of the Las Cruces gravity data shows a slightly spotty appearance; this is noise caused by the poor defin- ition of the short-wavelength component (shallow sources) of the gravity signal due to the comparatively large distance between stations. This is nearly always present in ground gravity data. Note the northeast–southwest linear feature traversing the southeast part of the image (1), which is much less obvious in the original data.
A useful enhancement for gravity data (and also pseu- dogravity data; seeSection 3.7.2.2), is the total horizontal gradient which peaks over vertical contacts (Fig. 3.25e), or forms a ridge if the source is narrow. This enhancement creates a circular ridge at the Las Cruces anomaly
(Fig. 3.27e), but since the edges of the source are not
vertical these do not correspond with the edges of the deposit. The magnetic data from Broken Hill clearly emphasise source edges (Fig. 3.28h).
3.7.4.1 Second-order derivatives
Second-order derivatives (see Gradients and curvature in
Section 2.7.4.4) can be an effective form of enhancement.
The zero values of the second vertical derivative coincide with the edges of sources if their edges are vertical, but more importantly, the response is localised to source edges increasing their resolution. Figures 3.25h and3.27hshow the second vertical derivatives of the gravity model and the Las Cruces gravity data respectively, with the equivalent responses for the Broken Hill magnetic data shown in
Fig. 3.28e. The improved resolution is clearly seen in the
Broken Hill magnetic data, e.g. (4). However, second and higher order derivatives are very susceptible to noise and so are only useful on high quality datasets. As shown by the ‘spotty’ appearance of the Las Cruces image, they are rarely useful for ground gravity data unless the station spacing is very small.
3.7.4.2 Analytic signal
Combining the three directional gradients of the gravity or magnetic field to obtain the total gradient, cf. Gradients and curvature inSection 2.7.4.4, andEq. (2.5), removes the complexities of derivative responses. When applied to potential field data, the total gradient at a location (x, y) is known as the analytic signal (AS) and given by:
ASðx, yÞ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ∂f ∂x 2 + ∂f ∂y 2 + ∂f ∂z 2 s ð3:23Þ where f is either the gravity or the magneticfield. Where the survey line spacing is significantly larger than the station spacing, the across-line Y-derivative is not accurately defined. It is preferable then to assume that the geology is two-dimensional and set the Y-derivative to zero.
The analytic signal has the form of a ridge located above the vertical contact (Figs. 3.25fand3.26c), and is slightly displaced laterally when the contact is dipping. The form of the source prism is clearly visible in the transformed data- sets; the crest of the ridge delineates the edge of the top surface. The magnetic data from Broken Hill show distinct peaks above the relatively narrow sources which define the northeast–southwest trending fold (Fig. 3.28f). Being based on derivatives, the gravity data from Las Cruces are quite noisy (Fig. 3.27f). Another example of the analytic signal of TMI data is shown inFig. 4.25b, in this case the response is controlled by magnetism-destructive alteration which greatly reduces the short-wavelength component of the signal to which the analytic signal responds.
The analytic signal is effective for delineating geological boundaries and resolving close-spaced bodies. Since the magnetic analytic signal depends upon the strength and not the direction of a body’s magnetism, it is particularly useful for analysing data from equatorial regions, where the TMI response provides limited spatial resolution, and when the source carries strong remanent magnetisation (MacLeod et al.,1993).
The gradient measurements are susceptible to noise which can severely contaminate the computed analytic signal. This is especially a problem with comparatively sparsely sampled datasets. Higher-order total gradient sig- nals calculated from higher-order gradients, such as the second derivative, have also been proposed, e.g. Hsu et al. (1996). Like the derivatives, the response becomes narrower as the order increases, offering the advantage of higher spatial resolution. However, higher-order gradients amplify noise which leads to a noisier analytic signal.
3.7.4.3 Tilt derivatives
Shallow sources produce large amplitudes in the vertical and horizontal gradients. The large amplitude range pre- sents a problem for display. Ratios of the derivatives of each class of source have similar amplitudes, so the reso- lution of both classes can be balanced by dividing the vertical derivative by the amplitude of the total horizontal derivative. Furthermore, the ratio can be treated as an angle and the inverse tangent function applied to attenuate high amplitudes (see Amplitude scaling inSection 2.7.4.4). This is known as the tilt derivative (TDR) (see Miller and Singh,1994), and at a location (x, y) it is given by:
TDRðx, yÞ ¼ tan1 ∂f dz , ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ∂f ∂x 2 + ∂f ∂y 2 s 2 4 3 5 ð3:24Þ where f is either the gravity or the magneticfield.
For gravity anomalies and vertical magnetised bodies the tilt derivative is positive over the source and negative outside of it. Its form mimics the 2D shape of the anomaly source with the zero-value contour line delineating the upper boundaries of the source (Figs. 3.25g and 3.26h). Its shape is more complex for non-polar magnetisation. The amplitude scaling properties of the enhancement are well demonstrated in the Broken Hill magnetic data in areas of weak TMI, e.g. (5) and (6) inFig. 3.28g; com- pare these with the responses of the other derivative enhancements.