INTRODUCTION

Gradient methods are common in the gravity and magnetics surveys. One big advantage of using field gradients is that temporal changes in the field are less of a problem than when using the fields themselves. A second big advantage of gradients is that the anomalies are "sharper" and more closely tied to the sources. The most significant disadvantage is that gradient anomalies are more sensitive to near-surface inhomogeneities than are field anomalies. A second disadvantage is that the gradient values may only be approximate if they are derived from field anomalies rather than measured directly.

Magnetic gradients are expressed simply as nanoTeslas per meter. For gravity gradients the traditional unit is the "Eotvos" which equals 10^-4 milliGals per meter.

GRADIENT ANOMALIES USING GRAVITY METER

Several authors have described projects using gradients approximated by finite differences) differential quotients).

In one example the goal was to map abandoned mines and tunnels under cities in Poland (Fajklewicz, 1976). With mines dating from the Middle Ages and with maps lost or destroyed in countless wars, these unknown cavities are severe hazards. The method has also been applied to map out karst cavities. In the experiment gravity readings were made at the base and the top of a ten meter high tripod. The difference between the lower and upper reading approximates the vertical derivative of gravity (strictly speaking the vertical derivative of the vertical component of gravity). Another tower type survey was tried in British Colombia with rather mixed results (Ager and Liard, 1982).

As we’ll soon see, the gradient anomaly is narrower than the field anomaly and thus gives a more precise indication of source location. One practical difficulty was shaking of the tower on windy days or at rush hour. As with seismic work in cities it’s a good idea to do the field work at night. Another problem is that the gradient anomalies are extremely sensitive to terrain variations and other irregularities near the stations.

Another project aimed to determine horizontal gradients of gravity in the mountainous Andean jungles of South America (Hammer and Anzoleaga, 1975). Because of the difficulties of road travel (no roads!) all the stations were reached by helicopter. At each site a triangle with sides of about one hundred meters was laid out and carefully leveled. The differences between readings at the three corners were solved to yield approximate values of the north and east components of the horizontal component of the gravity gradient (that is, dg/dx and dg/dy). Only exact relative elevations of the corners are needed; it is not necessary to level all along a profile.

GRADIENT INSTRUMENTS

For about half a century beginning around 1880 Eotvos torsion balances were used to measure various components of the gradient of the gravitational field (Nettleton, 1940). In most surveys these purely mechanical devices measured dg/dx and dg/dz. A typical measurement at one site took several hours. At each site very detailed measurements of local topography were made. Despite the slow and expensive field work demanded by these devices, they earned their keep on the Gulf Coast into the 1950’s.

Starting in the "Cold War" the military services became very interested in gravity as it influences the trajectory of incontinental ballistic missiles. Thus interest developed in rapidly surveying the gravity field from moving vehicles such as submarines, surface ships and even airplanes. Many factors combined to limit the accuracy of surveys to no better than plus or minus one milliGal, an accuracy unacceptable for all but the crudest purposes. Thus renewed interest in gradient instruments.

Skipping ahead to the twenty-first century we see that commercial and governmental organizations are flying detailed gravity gradient surveys (Vascp and Taylor, 1991; Jekeli, 1993; van Leeuwen, 2000). Typical specifications for a survey might be flight elevation of 120 m and grid spacing of 200m. Accuracy is advertised as plus or minus a few Eotvos units. Calibration includes mounting the whole two engine airplane on a big "shake table" and vibrating it for two weeks!

Ground, airborn and shipborn magnetic gradient measurements have been routinely made since the 1960’s or earlier. Instrumentation is much simpler than for gravity gradients and there is much less difficulty EARTH SCIENCES 734/834: CLASS 32, GRADIENT ANOMALIES

INTRODUCTION

Gradient methods are common in the gravity and magnetics surveys. One big advantage of using field gradients is that temporal changes in the field are less of a problem than when using the fields themselves. A second big advantage of gradients is that the anomalies are "sharper" and more closely tied to the sources. The most significant disadvantage is that gradient anomalies are more sensitive to near-surface inhomogeneities than are field anomalies. A second disadvantage is that the gradient values may only be approximate if they are derived from field anomalies rather than measured directly.

Magnetic gradients are expressed simply as nanoTeslas per meter. For gravity gradients the traditional unit is the "Eotvos" which equals 10^-4 milliGals per meter.

GRADIENT ANOMALIES USING GRAVITY METER

Several authors have described projects using gradients approximated by finite differences) differential quotients).

In one example the goal was to map abandoned mines and tunnels under cities in Poland (Fajklewicz, 1976). With mines dating from the Middle Ages and with maps lost or destroyed in countless wars, these unknown cavities are severe hazards. The method has also been applied to map out karst cavities. In the experiment gravity readings were made at the base and the top of a ten meter high tripod. The difference between the lower and upper reading approximates the vertical derivative of gravity (strictly speaking the vertical derivative of the vertical component of gravity). Another tower type survey was tried in British Colombia with rather mixed results (Ager and Liard, 1982).

As we'll soon see, the gradient anomaly is narrower than the field anomaly and thus gives a more precise indication of source location. One practical difficulty was shaking of the tower on windy days or at rush hour. As with seismic work in cities it's a good idea to do the field work at night. Another problem is that the gradient anomalies are extremely sensitive to terrain variations and other irregularities near the stations.

Another project aimed to determine horizontal gradients of gravity in the mountainous Andean jungles of South America (Hammer and Anzoleaga, 1975). Because of the difficulties of road travel (no roads!) all the stations were reached by helicopter. At each site a triangle with sides of about one hundred meters was laid out and carefully leveled. The differences between readings at the three corners were solved to yield approximate values of the north and east components of the horizontal component of the gravity gradient (that is, dg/dx and dg/dy). Only exact relative elevations of the corners are needed; it is not necessary to level all along a profile.

GRADIENT INSTRUMENTS

For about half a century beginning around 1880 Eotvos torsion balances were used to measure various components of the gradient of the gravitational field (Nettleton, 1940). In most surveys these purely mechanical devices measured dg/dx and dg/dz. A typical measurement at one site took several hours. At each site very detailed measurements of local topography were made. Despite the slow and expensive field work demanded by these devices, they earned their keep on the Gulf Coast into the 1950's.

Starting in the "Cold War" the military services became very interested in gravity as it influences the trajectory of incontinental ballistic missiles. Thus interest developed in rapidly surveying the gravity field from moving vehicles such as submarines, surface ships and even airplanes. Many factors combined to limit the accuracy of surveys to no better than plus or minus one milliGal, an accuracy unacceptable for all but the crudest purposes. Thus renewed interest in gradient instruments.

Skipping ahead to the twenty-first century we see that commercial and governmental organizations are flying detailed gravity gradient surveys (Vascp and Taylor, 1991; Jekeli, 1993; van Leeuwen, 2000). Typical specifications for a survey might be flight elevation of 120 m and grid spacing of 200m. Accuracy is advertised as plus or minus a few Eotvos units. Calibration includes mounting the whole two engine airplane on a big "shake table" and vibrating it for two weeks!

Ground, airborn and shipborn magnetic gradient measurements have been routinely made since the 1960's or earlier. Instrumentation is much simpler than for gravity gradients and there is much less difficulty correcting for motion of the platform.

GRADIENT ANOMALIES OF SIMPLE SHAPES

For gravity and magnetic gradient formulae we simply take the appropriate derivatives of the field anomalies. Compared to the field anomalies the gradient anomalies are "sharper" (i. e. narrower) and more precisely located over the source.

In most cases the vertical component of the gradient has the advantage of being symmetrical. Horizontal gradients, in contrast, tend to be "lop-sided" and harder to interpret.

HILBERT TRANSFORM

Sometimes we might have a vertical gradient profile but want a horizontal gradient profile or vice-versa. For a two-dimensional situation it turns out that we can readily calculate one profile (say dg/dz) given the other (dg/dx). It turns out that either component is the "Hilbert Transform" of the other.

REFERENCES AND READINGS

Ager and Liard, 1982, Geophysics, 47, 919-925.

Butler, 1984, Geophysics, 49, 1084-.

Fajklewicz, 1976, Geophysics, 41, 1016-1030.

Hammer and Anzoleaga, 1975, Geophysics, 40, 256-268.

Jekeli, 1993, Geophysics, 58, 508-514.

Nettleton, L., L., 1940, "Geophysical Prospecting for Oil", McGraw-Hill Company, 444 pp. (see chapter 5).

Vasco and Taylor, 1991, Geophysics, 56, 90-101.

van Leeuwen, 2000, The Leading Edge, 19, 1296-1297.