eps – 435 applied geophysics potential field methods - gravity · 2016-03-03 · eps – 435...

EPS – 435 Applied Geophysics

Potential field methods - Gravity

EPS435 – Fall 2008 Dr. Michael Riedel [email protected] EPS435-Potential-02-01

Overview of Methods and Technology

There are two principle types of gravimeters in use since the replacement of the pendulum-based instruments:

(a) Stable gravimeters (less sensitive)

(b) Unstable gravimeters (mostly in use nowadays)

Typical examples of stable gravimeters: Askania, Boliden, Gulf

Typical examples of unstable gravimeters: Thyssen, LaCost-Romberg, Worden

mailto:[email protected]




Principle operation of a stable gravimeter

Principle operation of an unstable gravimeter





Measurements in the field are influenced by many unwanted factors, disturbing the view on the real target:

We distinguish between:

(a) Instrumental drift, (b) tide-, (c) latitude-, (d) free air-, (e) terrain-, (f) Bouguer-, (g) isostatic-, and (h) elevation-corrections.

For gravimeters mounted on ships or planes/helicopters we have to apply an additional correction, called Eötvös-corrections, to compensate for the effect of the Coriolis force.





The most common problem is instrument drift, that is a slow change in the gravitational acceleration due to instrument artifacts mainly induced b temperature changes inside the gravimeter. This is normally dealt with by repeated measurements at a base station (where gravity should not change).

Mathematically you would either define a simple interpolation (linear or spline) or fit another “intelligent” function through the observed data points to correct your data from the measured profile.





Tidal deformation of Earth due to Sun and MoonThe shape of the Earth changes frequently due to tidal deformation caused bythe gravitational attraction of the Moon and Sun. The sea-tides are a well-known phenomenon; there are two high and two low tides every day.

Local variations in seafloor topography and shape in coast-line result in largevariations of the magnitude of the sea tides across the planet. The largest seatides are observed in the Bay of Fundy (Minas Basin) with a tidal height of over10 meter. The gravitational effect of Sun and Moon is not always aligned, but if they act in the same plane, the tides are much larger. We call those spring-tides. If Sun and Moon are at a right angle to each other and their effects partially cancel each other out, we get neap tides.





Consider the following system of Earth and Moon (similarly you can imagine the Sun-Earth system). The force FA is larger than FC, as the point A is closer to the Moon than point C.





Relative to the force exerted by the Moon on the center of the Earth, there is a differential force ΔF acting on the tow outer edges:

At point A, the differential force is towards the Moon, whereas at point C it is awayfrom the Moon. Relative to the Earth centre, point A will be displaced towards theMoon and point C consequently away from the Moon.





Now that we know how tides are generated, how would a tidal effect influenceour gravity measurements at a base-station?

At the base you move(depending on latitude) up to 30 cm up and down twice a day, just very slowly. Thus, the resulting gravitywould look like a sinusoid.On top of that we have to deal with drift. The combined effect of drift and tide is shown as the dotted line.





A latitude-correction might become important if your survey covers largeareas and crosses significant distances in a N-S orientation.Recall the international standard of gravitational acceleration as function oflatitude φ [NOTE: φ is in rad, not degree, multiply with π/180]:

g0(φ) = 9.78031846 [1+0.005278895 sin2φ – 0.0000023462 sin4(φ)] m/s2

(EQ 1.4)

Normally a local base-station is selected for which the horizontal gravitygradient (δgL) can be determined at a given latitude (φ, in rad). This correctionis always negative with distance increase northward in the northernhemisphere or with distance southward on the southern hemisphere tocompensate for the increase in gravity field towards the poles.

δgL = - 8.108 sin 2φ [g.u. per km N] (EQ 1.10)





Free-air correction

The basis of this correction is the reduction in magnitude of gravity with height,irrespective of the kind of rocks underneath. The free-air correction is thedifference between gravity measured at sea-level and at an elevation h.Commonly a value of 3.086 g.u. per meter [=0.3086 mGal] is acceptable formost practical applications.The free-air correction term varies with latitude from 3.083 g.u. per meter atthe equator to 3.088 g.u. per meter at the poles. With a normal measuringprecision of 0.1 mGal for most modern gravimeters, the station elevationneeds to be known within 3 – 5 cm.





Taken the Earth as a sphere (mass M, radiusR), the value of gravity at sea level is:g0 = GM/R2. (EQ 1.11)The value of gravity at a station with elevation his then:gh = GM/(R+h)2. (EQ 1.12)This can be expanded in a Taylor-series:

(EQ 1.13)

The difference in gravity at sea level (g0) and ath meters (gh) is the free-air correction (δgF):

(EQ 1.14)

Schematic illustration ofthe free-air correction.





Terrain correctionThe effect of topography (valleys, hills)can be quite substantial. The hill withexcess mass has its centre of massabove the level of the valley, where thegravimeter station is located. Theresulting attractive force by the hillresults in a slight reduction of themeasured gravity at the station.Similarly the effect of a valley on gravitymeasurements with a station on top of ahill can be visualized by defining themissing mass of the valley relative to thehill as –M. Thus, the measurement ofgravity on top of the hill is againunderestimated.





Terrain correction is in reality a very complicated task. It requires knowledge ofdetailed topography as well as a good understanding of the density of thesurrounding rocks (which may also be the target of the survey).Historically a Hammer-Terrain correction has been used, named afterSigmund Hammer (1939). This Hammer-correction consist of a series ofsegmented concentric rings, superimposed over a topographic map. Theaverage elevation of each segment is estimated and each segment is given aconstant background density, identical to the density used in the Bouguercorrection (see later).Nowadays, GIS (Geo-Information-system) can help digitize topography andcalculate average masses of the topography surrounding the stations. Moderngravity-interpretation packages usually come with a tool for terrain correction,that can be linked to standard GIS data formats.





Bouguer correction

When applying the free-air correction, we assumed that the materialbetween the station (at level h) and sea level consists of “air”, i.e. we totallyneglected the fact that it is actually made out of rock, with a density muchhigher than air.By applying the Bouguer correction, we take the actual rock density intoaccount. The Bouguer correction terms can be calculated from a infinite-longslab of height h and average density ρ:

δgB = 2 π G ρ h (EQ 1.15a)G is again the gravitation constant.

For marine surveys, the Bouguer correction is given by:δgB = 2 π G (ρr – ρw) hw, (EQ 1.15b )

Where ρr is density of rock, ρw density of seawater, and hw is water depth.





Isostatic correctionIf there were no lateral density variations in theEarth’s crust, the fully reduced gravity datawould everywhere be the same. However, thereare lateral variations, a gravity anomaly knownas the Bouguer anomaly. The average Bougueranomaly in oceanic areas is positive, whereas itis negative over mountains.Two hypotheses were developed in the 19thcentury describing the large-scale systematicvariation in gravity. The geodesist G.B. Airyproposed in 1855 that mountains may havedeep roots, but that density changes across themountain are small. J.H. Pratt, however,proposed in 1859 that all mountains arecompensated at a constant depth, but densitychanges strongly across the mountain range.





Airy isostatic compensation modelAiry suggested that the density of theEarth’s crust does not change stronglyenough to give rise to observed Bougueranomalies, rather the thickness of the crust(roots) changes remarkably. The lowdensity crust is floating in the high densitymantle. The higher the mountain, the biggerthe crustal root beneath.The elevation-thickness relationship isdetermined with respect to the sealevel:HWρW + HCρC +HMρM = HρC,Where ρW, ρC, and ρM are densities of thewater, crust and mantle.





Pratt isostatic compensation model

Pratt suggested that the density of the rocksforming the mountains is less than thoseforming the lowlands, such that the totalmass of a column of mountain to a givendepth (called depth of compensation) isequal to the total mass of a column in thelowland to the same depth.In this model the Earth’s crust isapproximated as blocks of equal massesfloating on the mantle. The elevation-densityrelationship is given as:ρA HA = ρB HBUsing the diagram to the right as anexample.





Airy’s model is preferred geologically and seismically, whereas Pratt’s model iseasier to use to calculate the isostatic correction, but the results are similar.The aim of the isostatic correction is that effects of the large-scale changes indensity should be removed, thereby isolating the Bouguer anomaly.

Free-air and Bouguer correction are commonly combined, referred to aselevation correction (δgE) to simplify data handling.δgE = δgF – δgBδgE = 3.086 h – 2 π G ρ h = (3.086 – 0.4192 ρ ) h [g.u.] (EQ 1.16)[ 2 π G = 0.4192 in g.u.]





Eötvös-correctionFor a gravimeter mounted on a vehicle (ship or plane) the measured gravitationalacceleration is affected by the vertical component of the Coriolis force, which is afunction of the speed and the direction in which the vehicle is traveling. Tocompensate for this, gravity data are adjusted by applying the Eötvös correction,named after the Hungarian geophysicist Baron von Eötvös who described thiseffect in the late 1880s.There are two components to the correction. The first is the outward-actingcentrifugal acceleration associated with the movement of the vehicle over thesurface of the Earth, and the second is the change in centrifugal accelerationresulting from the movement of the vehicle relative to the Earth’s axis of rotation.In the second case, a stationary object is traveling with the speed of the Earth’ssurface at that particular location. If the same object is then moved by x km/htowards the east, its speed relative to the rotational velocity is increased by thatsame amount (or reduced if it travels westward).





Eötvös-correctionAny movement of a gravimeter which involves an east-west component will have asignificant effect on the measurement of gravity. For ship-mounted gravimeters theEötvös-correction can be as large as 350 g.u.. For airborne gravimeters, wherespeeds over 90 km/h are common, the correction can be as high as 4000 g.u..

The Eötvös correction is given by:

δgEC = 75.08 V cos φ sin α + 0.0416 V2 [g.u.]

or δgEC = 40.40 V’ cos φ sin α + 0.01211 V’2 [g.u.] [EQ 1.17]

Where V and V’ are the speed of the vehicle in knots and km/h, respectively, φ isthe geographic latitude, and α is the azimuth of the vehicle movement relative tothe Earth rotational axis.





The final result: The Bouguer anomaly

The main end product of gravity data reduction is the Bouguer anomaly, whichshould only correlate with lateral variations in density of the upper crust.The Bouguer anomaly is the difference between the observed gravity value (gobs)adjusted by the algebraic sum of all necessary corrections (Σ(corr)) and that of abase station (gbase).The Bouguer anomaly ΔgB is thus given by:

ΔgB = gobs + Σ(corr) – gbase, [EQ 1.18]

With Σ(corr) = δgL +(δgF-δgB) + δgTC ± δgEC ± δgIC – δgDWhere L = latitude, F = free-air, B = Bouguer, TC = terrain correction, EC = Eötvös correction,IC = isostatic correction, and D = drift and tide correction





Mathematical derivation of the Eötvös-correctionIn general, the centrifugal acceleration (a1) is given by (velocity)2 / d. The total eastwest speed of a vehicle is the sum of the linear speed of the rotation of the Earth (v) and the east-west component of the vehicle’s movements (VE). Thus the total centrifugal acceleration is:a1 = (v + VE)2 / dCentrifugal acceleration a2 along the radius vector, due to the north-component ofthe vehicle’s movement (VN) is given by:a2 = VN

2 / RHowever, we have to subtract the centrifugal acceleration of a static object a3, as itis only the change in acceleration that is required here, thus:a3 = v2 / d

In total we get: δgEC = a1 cos φ + a2 – a3 cos φ [EQ1.19]





Mathematical derivation of the Eötvös-correctionNote that:d = R cos φv = ω R cos φV = (VN

2 + VE2)1/2

VE = V sin αSubstituting these expression into the previous formula (EQ1.19) yields:

[EQ 1.20]


Mathematical derivation of the Eötvös-correction

Given the following values we can rewrite equation EQ 1.20:ω = 7.2921·10-5 radian/sR = 6371 km1 knot = 51.479 cm/s1 Gal = 104 g.u.

δgEC = 2·(7.2921·10-5 · 51.479·104) V cos φ sin α + (51.479 V)2·104 / 6.371·108

Which is equivalent to:δgEC = 75.08 V cos φ sin α + 0.0416 V2 [g.u.] [EQ 1.17]





eps – 435 applied geophysics potential field methods - gravity · 2016-03-03 · eps – 435...

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