Survey

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Survey

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SIO 226: Introduction to Marine Geophysics Gravity LeRoy Dorman Scripps Institution of Oceanography Winter, 2013 Goals: To leave you with some tools to help see the connection between observed gravity (1)density variations in the earth and (2)The earth's shape(including topography), which is the basis for estimation of bathymetry from satellite altimetry. Much of the analysis of gravity is based on a phenomenon called “upward continuation”. We start with the terminology, the types of gravity anomalies, and what they show us. Gravity Reductions The first task is to remove the earth's main field, which is much larger than the anomalies we study. 2 The formula of 1967 4 is g ϕ= 978031.846(1+ 0.005278895sin ϕ+ 0.000023462sin ϕ) mGal Where phi is latitude. This is based on a rotating ellipsoidal earth. To correct for altitude difference from the ellipsoid, we use the derivative, -0.3086 mGal/meter. The result of these two corrections is the Free Air anomaly. To correct for the material between the point and observation, we make the Bouguer plate correction of 2 Gh, which is 0.1119h mGal/m, for a density of 2670kg/m^3. The result of making this additional correction is the Bouguer anomaly. can be adjusted to optimize topographic suppression, thus estimating ρ. Eliminate Things We Know Elevation: Free Air reduction add 0.3088 mGal/m Topography: Bouguer and terrain reductions subtract 0.1967 mGal/m Net: +0.1121 mGal/m Isostasy: (later) The horizontal scale of the previous figure was ~1 km. The one below is ~1000 km and provides a very different picture. Things which worked on a local scale leave us with large anomalies when applied to a larger Continent-scale Data data set. Mantle Bouguer Anomaly At sea, the plate correction (evaluated for field at the sea surface for topographic height directly beneath the ship) is a poor approximation to the field from the actual topography, so real bathymetric data are used. To reveal features of the mantle, Kuo and Forsyth (1988) suggested removal of the gravity field from the crust, using whatever data are available- from a seismic survey or assuming a simple constant-thickness crust. The usefulness of this, of course depends on the accuracy of the estimate of the crustal field, which, in turn, depends on the survey data used and other assumptions. Basics from Physics I Gravity is a vector force per unit mass caused by the presence of other masses. For two masses, this force is, from Newton's law, m1 m2 F= G . l2 The theory which describes gravity is called potential theory, since this force is the gradient (derivative) of a scalar potential, Gm U= l and scalars are simpler to deal with. Since taking a derivative of both sides of an equation leaves us with a valid equation, many transformations can be used on the potential apply as well to components of gravity. Basics from Physics II F⋅ ds= − 4 πG M Gauss's Theorem∫ is∫ that F perp ( x)= − 2 πG σ( x) which implies that This means that we can choose a σ( x) which produces any value of F perp ( x) we desire. Thus we have no hope of extracting the earth's density structure uniquely from gravity data alone. Laplace's Equation ∇ 2U = 0 In mass-free space, Laplace's equation applies, and this provides a strong constraint on the potential field U. Writing this out, we get ∂ 2x U + ∂ 2x U + ∂ 2x U = 0 1 2 3 and after Fourier transformation, − k 21 U (k )− k 22 U (k )− k 23 U (k )= 0 or, k 12+ k 22+ k 23= 0 Fourier Transform and derivatives In one dimension, the ∞ Fourier Transform of a function is F (k )= ∫ f ( x)e− 2 πi k x dx. −∞ ∞ The FT of a deriv is F x [ f ' ( x)](k )= ∫ f ' ( x)e− 2 πi k x dx −∞ Integration by parts, ∫ v du= [uv ]− ∫ u dv with du= f ' ( x)dx and v= e− 2 πi k x so u= f ( x) ,and dv= 2 πi k e− 2 πi k x dx ∞ F x [ f ' ( x)](k )= [ f ( x)e− 2 πk x ]− ∫ f ( x)(− 2 πi k e− 2 πi kx dx) −∞ so multiplication by 2 πi k corresponds to differentiation, since we can omit the 2nd term. Fourier Transformation in 2D The Fourier Transformation is where the real part of i k⋅ x the F (k )=is∫plotted ∫ f ( x)e 1 dxfor 2 kernel at the dx right one value of k. The direction of k is normal to the wave “crests” and the magnitude of k is 2π̸λ, where is the wavelength. Upward Continuation In geophysical surveys, we normally gather data in two dimensions, xand x, 1which 1 k2 transform into and . The equation k1 2 2 2 tells us that if we know any two, we khowever, + k + k 1 2 3 = 0, can calculate the third! Thus if we have the gravity field at some level in the earth (such as from a lamina), we can calculate the field at the surface. Upward decay of anomaly amplitude If we take a 2D Fourier transform of a spatial grid of data, we end up with a 2D array of amplitudes evaluated at two wavenumbers k = − (k + k ) where k i = 2 π/ λ i When we reconstruct the gravity field, thek i enter the calculation as the exponential kernel of the Fourier transform. A real value of k produces a sinusoidal variation but an imaginary k produces an exponential decay. So long wavelengths decay slowly and short wavelengths decay rapidly. 2 3 2 1 2 2 Simple modeling We can calculate the gravity field of a geological model in 4 steps. (1) Approximate the structure using one or more lamina using the equivalent stratum. (2) Make a Fourier expansion of the field from the model. (3) “Upward continue” the Fourier components. (4) Synthesize the anomaly from the Fourier components. Seamount Example Seamount radius: 10km Water depth: 4km Ridge Example Water Depth: 4km Is this a realistic depth? A Simple Isostatic Model Consider a simple model in which the topography is mirrored by a structure of like density, but opposite sign, at some (initially one) depth. This model is consistent with the the data, both local and continental, which we have just seen. Implications of this model We represent topography as a lamina with density σ(x)= h(x) and the Compensation is a lamina with density -(x). Take one harmonic component at a time, h ( x)= h ei k ⋅ x a sinusoid with wavelength k =2π/λ, whose gravity field is 1 ik x 1 gThe field from hthe ( x)= 2 πρG e 1compensating B lamina is and the BA is ik x − k z e gDividing by h e produces Q (k )= − 2 πρG a log scale as a line whose slope is 1 g BA (k )= − 2 πρG h e g BA gB 1 1 1 which plots on c ik 1 x 1 − k 1 z c e Q(k 1 )= − e − zc − k 1 zc Isostatic Response Data Surveyor data is oceanic from off Oregon, McNutt, 1978. USA is from Lewis and Dorman, 1970. Australia is Mcnutt, 1978. Plotted on a log scale Shows straight lines, more or less. Slope of oceanic is shallower, since compensation depth is smaller. Circles are oceanic, squares and triangles are US and Australia. 0 -1 Idealized Models In terms of out simple model, the classical Pratt and Airy models look like: Mass Anomalies and Geodesy A mass anomaly not only changes the gravity, it changes the shape of the earth's geoid as well. Fortunately, the changes in the geoid are for > 400 km. In The Space Domain Convolving the gravity anomaly field with the Stokes functions produces the geoid height. Since these functions are not localized around the computation point, gravity at distant parts of the earth is important. Principle of SIO's Bell Gravimeter which is a forcebalance accelerometer. Its accuracy depends on a very accurate digital voltmeter. Its principle of operation is shown at right. Bell Gravimeter (reality) In operational reality, its appearance is a little different. Summary: Gravity measurements allow us to: measure the density of the surficial material, measure the shape of the earth Show that mountains “float”