CALCULATING BOUGUER ANOMALIES FROM OBSERVED ABSOLUTE GRAVITY VALUES

Last week we worked on converting raw gravity data, as read directly from a gravity meter, to absolute observed gravity values. The process involved calibrating the data, applying tidal corrections, averaging repeated values and their times of occupation, dedrifting the values, obtaining gravity differences between the base and the stations referred to it, and finally converting the relative values to absolute values using a base at which absolute gravity is known. These absolute observed vakues, which are the actual values of gravity at particular sites, constitute the fundamental value that is to be subtracted from a modified theoretical value to produce the so-called Bouguer anomaly. The Bouguer anomaly is named after a French scientist (Pierre Bouguer), who derived the formula for an infinitely extending slab, the basis for the so-called mass correction (discussed below). In simple terms, the Bouguer anomaly is the difference between what gravity is at a particular location, and what it ought to be at that same location; the "ought to be" value is based on a value of gravity calculated from a theoretical model of the earth (reference ellipsoid) that predicts gravity values on the reference ellipsoid (near sea level) based on latitudinal positions.

The absolute observed value or actual gravity value is affected by several different factors: elevation of the location above some arbitrary reference point (oftentimes sea level), mass between the location and the arbitrary reference point, terrain around the location, latitudinal position on the Earth's surface, and even the curvature of the Earth if a survey has been conducted over a large area. This gravity value is also affected by any lateral deviation from a simple concentrically-layered density configuration of the Earth. The lateral deviations in density are the target of a gravity survey, but their effects are obscured by the previously-cited factors of mass, elevation, terrain, latitude, and curvature. Therefore, the density-change portion of the gravity value must be isolated from the rest of the gravity value by mathematical procedures; this process results in the Bouguer anomaly.

Two types of Bouguer anomalies are recognized: (1) simple Bouguer anomalies, in which the various factors have been corrected, with the exception of terrain effects, and (2) complete Bouguer anomalies, for which terrain effects have also been removed. In areas of flat-lying or gentle topography, the terrain effects are small, and may be safely ignored, if such omission does not impair the required precision of measurement.

Three steps are followed in calculating the Bouguer anomaly (presuming the observed absolute value is already available):
  1. Calculate the theoretical value for a station, based on its latitudinal position (formula derived from the mathematical expression for the reference ellipsoid).
  2. Modify the theoretical value, as if it had been obtained with the same elevation, mass distribution, and terrain conditions as are present at the measured station.
  3. Subtract the modified theoretical value from the observed absolute value.

The second step above differs slightly from the procedure normally cited in textbooks of correcting the observed absolute value, but it results in the same answers, and is somewhat easier to understand.

CALCULATING THEORETICAL VALUES

The theoretical value of gravity at any position is predicted by the International Gravity Formula, based on a so-called reference ellipsoid. This figure of the earth is one that tries to account for the changes in gravity on Earth, caused both by its departure from sphericity, resulting in an elliptical shape, and its rotation (causing centrifugal forces that affect the gravity field). The formula currently being used was developed in 1967, and is based on both theoretical and empirical considerations. This formula replaced a 1930 version, used to produce a great many Bouguer anomaly maps. Although use of the newer formula versus the older one does produce a discrepancy of several milligals, this discrepancy is systematic for localized surveys, and the changes in Bouguer anomalies (reflecting lateral density variations) are consistent whether one uses the old formula or the new one. If you are ever in the position of integrating sets of Bouguer anomaly values, it is critical to make sure that all values were derived using the same formula. The formula for theoretical gravity (1967) on the reference ellipsoid is:

g = 978.031846(1 + 0.005278895(sin2phi) + 0.000023462(sin4phi),

in which phi is the latitude angle at the station. This produces a gravity value in Gals; you will need to convert to milligals when modifying this theoretical value (next step). The theoretical value needs to be calculated for the base and for each station in the survey (the base serves not only as a useful reference point, but is a bona fide station as well.

MODIFYING THEORETICAL VALUES

Four steps are needed to modify the theoretical value, so that it can be compared with the observed absolute value:
  1. Apply an elevation modification (so-called free air correction)
  2. Apply a mass modification (so-called Bouguer correction)
  3. Apply a terrain correction
  4. Apply a curvature correction

The first two modifications will be discussed below. The terrain modification is more complicated, and either requires a manual exercise for each station using a template and topographic map, or requires a sophisticated computer program using a digital elevatio model of the local topography. Terrain corrections will be the subject of next Monday's lab exercise. Curvayure corrections, while very real, can be ignored for localized surveys, since the differential effect between the two most extremely-positioned stations is negligible. The Pine Mountain Club survey falls into this category, and curvature corrections will not be included in your data reduction procedures.

Elevation Modification. Gravity changes with vertical position, decreasing with increased elevation above Earth's surface. The rate of change, or so-called free-air correction, is 0.09406 milligals per foot of elevation. Sea level is the reference for elevations on topographic maps, which are still mostly expressed in English system units (feet rather than meters). Consequently, this formula is most useful to us in the form given. Sea level is the most common reference used for making free-air corrections, although in some local surveys, it may be advantageous to use another reference elevation. The value of the modification is found by multplying the rate of change times the number of feet above (or below) the reference value. Since gravity decreases woth elevation, a theoretical value obtained at a hypothetical station above sea level would also have a lower value. To modify the theoretical value, the calculated free-air correction is subtracted from the theoretical value if the station lies above the reference plane.

Mass Modification. If a gravity station lies above a reference plane, it not only is affected by the elevation, but by the mass that lies between it and the reference plane. The effect of that mass is to increase the gravity value. The amount of increase is proportional not only to the distance (elevation), but also to the density of the intervening rock. The mass factor, in milligals per foot of elevation, is 0.01277x(rho), where pho is the density expressed in grams/cm3. This expression is multiplied times the elevation above or below the reference plane to obtain the modification factor. The choice of density is of course very important, and its value is known as the reduction density fore the survey. For regional surveys, a value of 2.67 gm/cm3 is usually used, since it is an average value for crustal rocks. For local surveys, the choice of density is usually dictated by the local geology, and the intent of the survey. For example, if one wishes to do a gravity survey of serpentinites in the San Luis Obispo area, one would find out that these bodies are encased in graywackes that have a density of 2.60 gm/cm3. The anomaly caused by a serpentinite is one due to the lateral change in density from graywacke to serpentinite, and it is thus most appropriate to use the 2.60 value as the reduction density.

Above the reference plane, the effect of the mass is to increase gravity, so the mass factor (also called the Bouguer correction) is added to the theoretical value to modify it. The mass correction and the free-air correction discussed above can be combined into a single formula for modifying the theoretical value, to cut down on the calculation work.

If the modified theoretical gravity value is now subtracted from the observed absolute gravity value, a simple Bouguer anomaly value is produced, provided that curvature can be ignored. The more usual practice is to also to apply a terrain modification to the theoretical value; this is the topic for Monday's lab exercise.

LAB ASSIGNMENT

Using the values obtained from the laboratory on calculating absolute observed gravity values, obtain the simple Bouguer anomalies for both the base and station.