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\line{\hfill \bigbf LAB 2 \hfill}
\vskip 1 em
\line{\hfill\bigbf LATITUDE, FREE AIR \hfill}
\line{\hfill\bigbf BOUGUER CORRECTIONS\hfill}

\vskip 2 cm


\vskip 1 em\item{} Gravity at the surface of the Earth varies by about 5
gal due to the centrifugal force of the earth's rotation and the
attraction of the resulting ellipsoidal shape of the earth. This is a
relatively large variation. At mid-latitudes it is about a mgal increase in gravity for every km
moved in the direction North. Fortunately this effect is well known and
easy to correct for. The current standard is defined by the world geodetic
system (1984)


$$
\gamma(\theta)=978.03267714{ 1+0.00193185138639cos^2\theta\over
\sqrt{1-.00669437999013cos^2\theta}} 
$$


\item{} where $\theta$ is the co-latitude ($90^o$ minus latitude). An older standard is the 1967
International Gravity Formula

$$
 \gamma= 9.78031846(1+0.0053024 cos^2\theta -.0000058 sin^2 2\theta) m/s^2
$$




\item{} One hundredth of a mgal (the reading level of the most common
exploration gravity meters) is the seventh decimal place
9.780318\underbar{4} 6, so this formula is accurate enough for most
exploration purposes, but note that there is almost a mgal difference
between the two standards.

\vskip 1 em \item{1)} Correct your data for the latitude effect by
subtracting the international gravity formula from your (tied to absolute)
values. 

\item{} There are two ways to do this.  If you have the latitude of each
station, you can simply use these in the IGF and subtract the result from each of your tied-to-absolute gravity values. Alternatively, if you have
your station positions in \it m \rm relative to the origin of your coordinate system,
you can proceed as follows. Calculate the
IGF at the latitude of the origin of your coordinate system, and differentiate the formula at this
latitude as well,  so you have the IGF at
one point and its local rate of change. The latitude effect is then 
$$
 978.03267714{ 1+0.00193185138639cos^2\theta\over
\sqrt{1-.00669437999013cos^2\theta}} +{d\gamma\over dx} X
$$
where $\theta_o$ is the co-latitude of the zero of your coordinate system and X is the distance a station is north of the zero. Subtract t5his from your tied-to-absolute values.
The latter method is good to $\pm 0.01 mgal$ over $\pm 25 km.$  You need Northings to $\pm 12 m$, so you need
to find the latitude of your origin to $\pm 0^o.0001$. The GPS monument is the origin of our local coordinate system. The latitude $52^o.1962$, or co-latitude $37^o.8038$, and longitude -$106^o.3983$. 

 \item{} The rate of change of the IGF67 (close enough to WGS84) with co-latitude is
$$
{\partial\gamma\over \partial \theta}
=-978.04900\times(0.0052884sin2\theta +0.0000118sin4\theta) gal/rad
$$
At colatitude $45^o$ this corresponds to 
$$
\Delta \gamma =-5.1722 gal/rad \Delta\theta
$$
or
$$
\Delta \gamma =-.8118 mgal/km \times S
$$
where S is the distance south of the base station in km. At S'toon this is 
-.7805 mgal/km.

The derivative for WGS1984 (in gal/rad) is
$$\align
{\partial \gamma\over \partial\theta}= &-978.03267714 sin2\theta\bigl[ {0.00193185138639\over \sqrt{1-0.00669437999013cos^2\theta}}\\
&+0.00669437999013{( 1+0.00193185138639cos^2\theta)cos\theta sin\theta\over {(1-0.00669437999013cos^2\theta})^{3/2}}\bigr ]
\endalign
$$
\item{} or
$$
\Delta \gamma =-978032.67714[{0.00527904138145\over 6378.137}sin(2\theta) 
        +{0.00001293254726\over 6378.137}sin(2\theta)cos(\theta)^2]  \quad mgal/km
$$

$$\Delta\gamma=-0.81306 mgal /km \quad \text{at} \quad 45^o \quad\text{and}\quad -0.78668 mgal/km \quad \text{at Saskatoon.} 
$$
\item{}WGS 1984 includes the mass of the atmosphere (the earlier IGF definitions do not). That is, to construct WGS 1984, the atmosphere was condensed on the surface of the ellipsoid, so that WGS 1984 is actually gravity at the surface of the ellipsoid, but outside the atmosphere. The reason for this is that WGS 1984 is used in satellite gravity and satellites are outside the atmosphere.  The convention is to correct WGS 1984 (GRS1980) by subtracting the correction below, or equivalently correct observations by adding it. For a station at sea level we would need to add the attraction of the full atmosphere (at a point just outside) and for a station outside the atmosphere we would need to add zero. For all other elevations the correction to be added to observed gravity (in mgal) is:

$$\align
\delta g_A=0.874e^{-0.000118h^{1.047}} mgal \quad & \text{for } h \ge 0 \\
\delta g_A =0.874 mgal \quad\quad\quad\quad\quad\quad &\text{for } h < 0\\
\&\text{or}
\delta g_A=0.874 -9.9\times 10^{-5}h +3.56 \times 10^{-9} h^2\\
\endalign
$$
\item{} where $h$ is in m.
 
\item{} So the maximum effect is just under a mgal. 
 
\item{} The Free Air Effect is the gravity variation that results from the
vertical motion of the gravity meter in the nominal gravity field of the
earth. The IGF and the world geodetic system also define a vertical
gradient. Including corrections for the ellipsoidal shape of the earth this
is IGF(1930) :

$$\align
{\partial\gamma \over \partial h}=& -.308766[
1-.0014665cos^2\theta]mgal/m\\
&+ 1.442\times10^{-7} mgal/m^2 \times h\\
\endalign
$$

For comparison IAG 1975 is:
$$\align
{\partial\gamma \over \partial h}=& -.308769[
1-.0013797cos^2\theta]mgal/m\\
&+ 1.442\times 10^{-7} mgal/m^2 \times h\\
\endalign
$$

and WGS80 is:
$$\align
{\partial\gamma \over \partial h}=& -.3087691[
1-.001424cos^2\theta]mgal/m\\
&+ 1.442\times 10^{-7} mgal/m^2 \times h\\
\endalign
$$

\item{}In IGF(1930), the vertical gradient has a constant part
(-.308766 mgal/m), and adjustments for the variation with latitude, and
height. In exploration surveys, usually only the lead term is used. In the
Saskatoon area (colatitude 38$^o$) WGS80 is 


$$\align =& -.308422 +1.442\times10^{-7}\times (580 m) \\ 
=& -.308464 mgal/m \\
\endalign $$

\item{} which differs by only 0.03 percent from the standard value -0.3086 mgal/m.

\item{} This (-.308464 mgal/m) is the free air \it effect. \rm It causes
observed gravity to decrease by .3 mgal per m of elevation above the
ellipsoid.  To correct for this we subtract the free air effect from raw
gravity measurements, that is, add .3084 mgal for every m of elevation
above the geoid. Note that the gradient is calculated at some height above
the ellipsoid,
but the correction is for height above the geoid. This convention arose
because heights above the geoid are more readily available than heights
above the ellipsoid (at least before GPS). It has only become possible since the 1970's to
estimate the difference in height between the geoid and the ellipsoid to
better than a few meters, but geophysicists still needed to do some sort
of free air correction, so the use of height above the geoid in the free air
correction became standard.



\vskip 1 em\item{2)} If we measured a vertical gradient during field school, compare
this value with the above, and comment on any difference. Despite the fact
that the measured  gradient may differ from the standard value, we must
still use the standard value in the free air correction. 

\vskip 1 em \item{3)} Correct your gravity data for height above the local
geoid.

\vskip 1 em\item{} The Bouguer correction is a correction for the
attraction of the mass between the gravity station, and the local geoid.
A  simple  Bouguer correction approximates the attraction of that
mass with the attraction of a slab of the same mean density. The
attraction of a slab of thickness $h$ and density $\rho$ is  $
g_b=2\pi G\rho h. $ If the density is $2.67 gm/cm^3$ then this is
$g_b=0.1119 mgal/m \times h. $ Since Saskatoon is about 580 m above sea level, the
Bouguer slab is entirely in the sedimentary column, so the mean density of
material between the physical surface and the geoid is slightly less than
the standard 2.67 gm/cc that would be appropriate in shield areas.
However, it is common practice to use a standard Bouguer density of 2.67
gm/cc, and treat any variation from this in the modeling stage. 

\vskip 1 em \item{} A Bouguer anomaly calculated from a simple slab model is
called a \bf simple Bouguer anomaly. \rm A \bf complete Bouguer anomaly
\rm also includes corrections
for the curvature of the earth, and  corrections for the attraction of the
local topography (the terrain correction). The curvature correction is
also called the Bullard B correction, and it is achieved by replacing the
flat Bouguer slab with a curved spherical cap of the same thickness, and
radius 166.738 km. The correction is normally done with the aid of
tables, but a polynomial in height is adequate for most situations

$$ \align
BB\approx & Ah+Bh^2 +C h^3\\
&A= 1.464\times10^{-3}\\
&B=-3.533\times 10^{-7}\\
&C=4.5\times 10^{-14}\\
\endalign
$$

\item{} At elevations below 4 km the spherical cap has more downward
attraction than the slab, so the simple Bouguer correction is too small.
At greater elevations the attraction of the spherical cap is less than the
Bouguer slab, so the Bullard B is negative. BB as calculated above would
be added to the attraction of the slab to get the attraction of the cap,
and subtracted from simple
Bouguer anomalies. 

\vskip 1 em
\item{4)} Do the Bullard B correction. 

\vskip 1 em \item{}  The terrain correction can be done in a variety of
different ways, but the easiest and most reliable, is to include the
terrain as part of the modeling. We will return to this in a later lab.  
The traditional way to do terrain corrections is with a Hammer chart, 
although this is so tedious, I won't make you do it.




\vskip 1 em\item{} The terrain effect
of short horizontal scale topography within 2 m of a station, is
usually handled by an inners correction done in the field. In this context
'inner' means inside the smallest Hammer zone. If the elevation changes by
$h$ (always positive) over the inner Hammer zone, then the inners correction is:

$$
\rho Gh
$$

\item{} which should be added to the Bouguer anomaly. Note that this is a
factor $2\pi$ smaller than the Bouguer correction for a slab of the same
thickness. Thus, inner topography that varies by a m over the inner hammer
zone would have an inners correction of about 0.01 mgal. In the context of
the field school inners corrections may be required at stations very near
the road embankment. Whether these need to be done or not depends on their
magnitude, and the repeatability  of the survey.

\vskip 1 em \item{} Make a table of Free Air gravity and Bouguer gravity at each of your stations. 

\bye


\vskip 1 em \item{5)} Using a 1/50000 scale map and a Hammer template
and tables, estimate the terrain corrections for the outer zones A to H.
