Hydromagnetic waves in Earth's core and their influence on ...

24 Chapter 2 — Space-time analysisco-ord<strong>in</strong>ate axes, <strong>and</strong> (z, u) are the new co-ord<strong>in</strong>ates axes after rotation clockwise byan angle q. The action of the Radon transform on a TL plot is shown schematically <strong>in</strong>figure 2.4.−Figure 2.4: Radon transform of a time-longitude plot.The Radon transform H R (q, z) of an image H(x, t) takes the projection of the field <strong>in</strong>the direction normal to the l<strong>in</strong>e def<strong>in</strong>ed by an angle q, with z measur<strong>in</strong>g position alongthe projected image (after Cipoll<strong>in</strong>i et al. (2004)).S<strong>in</strong>ce the TL plot consists of an array of discrete values, a discrete version of the Radontransform which f<strong>in</strong>ds an array of transformed values H R (q n , z m ) is employed. Thisimplementation of the Radon transform has the orig<strong>in</strong> of its coord<strong>in</strong>ate axes <strong>in</strong> thecentre of the image be<strong>in</strong>g transformed <strong>and</strong> measures q clockwise from the x axis up to±90 ◦ . The discretisation of q n is <strong>in</strong> steps of 2 degrees from -90 degrees to 90 degrees. Thediscretisation of z m is more complex <strong>and</strong> is arranged such that when q n =0 it matchesthat <strong>in</strong> the x direction <strong>and</strong> when q n = ±90 ◦ it matches that <strong>in</strong> the t direction. Figure2.5 shows an example of the application of this discrete Radon transform to a syntheticmodel of field evolution on a spherical surface consist<strong>in</strong>g of a wave pattern travell<strong>in</strong>gwestward at 17 kmyr −1 at latitude 10 ◦ S. In Figure 2.5, (a) shows a snapshot of the signalon the spherical surface (100 years <strong>in</strong>to the model); (b) shows the example TL plot at10 ◦ S <strong>and</strong> (c) shows the result follow<strong>in</strong>g the application of the discrete Radon transform.Hav<strong>in</strong>g computed H R (q n , z m ) each value <strong>in</strong> the array H R (q n , z m ) is squared, <strong>and</strong> summedover all m (along the z direction) to give a discrete measure of ∫ |H R | 2 dz. It is thennecessary to divide by a normalised correction factor C(q n ) that accounts for the factthat with a rectangular array, different numbers of grid po<strong>in</strong>ts will be summed over fordifferent q m (for further details see the analytic solution for the Radon transform for arectangle region of constant amplitude on p.63 of Deans (1988)). If all po<strong>in</strong>ts <strong>in</strong> a TLplot have the same constant value then tak<strong>in</strong>g the discrete Radon transform, squar<strong>in</strong>g