Estimating Te using admittance and coherence analyses

Loads on the Earth. When a load is imposed on the Earth's surface it must be supported in some way. If the Earth's lithosphere (the outer shell of the Earth though which heat is conducted) is strong the Earth's surface does not bend much (load 1 in the cartoon). The load therefore is an excess of mass on the Earth's surface, and thus the gravity field over the Earth at that point is greater than where there is no mass excess. This produces a positive free-air gravity anomaly (gf in the cartoon; note that the cartoon is just that — the actual shape of the anomaly is different due to edge effects). The amplitude of gf is dependent on the difference in the density contrast between the load and the surrounding medium (usually air, but it could be water or rock). The admittance is the ratio between gf and the topography of the load in the frequency domain. Therefore, the admittance, Z, in the region of load 1 is high: there is a significant gravity anomaly and a significant topographic signal.

Load 2 is emplaced on a weak lithosphere (T_e is small). The load is therefore not supported elastically and causes the Earth's surface to bend. The principle of isostasy states that at some depth, the pressure will be equal everywhere (in the cartoon this is represented by the dashed line). This means that density contrasts higher up in the lithosphere, such as the boundary between densities 2 and 3, which might be the Moho or the base of the lithosphere, are deflected by the load. Hence load 2 'floats' in the lithosphere. The mass deficit caused by the 'root' of load 2 is made up for by the extra mass above the surface. If load 2 has the same density as layer 2, the root of load 2 causes the density contrast between layers 2 and 3 to be deflected downwards. This is possible because the lithosphere is weak and can flex over short distances. In this case, load 2 is compensated, that is, the pressure at some depth is equal everywhere and thus there is no mass excess. In this case, gf is zero across the load. (This isn't actually the case because the positive gf associated with the topography of load 2 is not quite compensated for by the negative gf of the root of load 2 because this density contrast is further away from the surface.) Hence the admittance in the region of load 2 is negligible because the gravity anomaly is very small.

For a lithosphere of given strength, there is a size of load at which the admittance will transition from low to high. People, for example, are a load on the Earth's surface, but do not produce little roots in the Moho. Mountain ranges, on the other hand, are very large and will have a low-density crustal root, relative to the surrounding mantle. The wavelength of load (large loads = long wavelengths) at which this transition occurs can be used to estimate T_e. T_e is estimated by measuring the admittance of a region, i.e. dividing the free-air gravity anomaly by the topography in the frequency domain. This produces a curve which generally has low values of Z at long wavelengths (the large topographic features are compensated) and high values of Z at short wavelengths (little features are elastically supported). This curve can be modelled using a formula which relates Z to T_e by the flexural rigidity. In the examples presented here this modelling is done using an inverse method to find the best solution. The methods used here have been pioneered by McKenzie & Fairhead (1997) and McKenzie (2003). Watts (2001) provides a comprehensive explanation of how all this works.

An alternative method of estimating T_e was established by Forsyth (1985). He used the coherence between the topography and the Bouguer gravity anomaly, gb. gb is the gravity anomaly with rock between the surface and sea level removed. So in the above cartoon, gb for load 1 would be zero (the mass excess has been removed), and for load 2 would be negative (there is mass deficit below the surface). Coherence is basically how well gb and the topography correlate. Uncompensated loads (load 1) have no coherence; gb is negligible whilst there is significant topography. Compensated loads (load 2) have negative coherence; gb and topography are correlated, but in opposite directions. The wavelength of load at which the coherence transitions from high to low enables T_e to be estimated as with the admittance. The problem is, significant amounts of topography are removed over time by erosion and sedimentation. This reduces the topographic signal and decreases the coherence between short wavelength loads which are compensated, but have no topographic expression (i.e. there are loads at depth, but they have no surface expression). Hence the coherence method is sensitive not only to T_e but also to erosion and sedimentation, i.e. topographic maturity. Therefore the coherence method only provides an upper bound on estimates of T_e, rather than a true estimate.

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