Extend the four-cell resolution test of this chapter to a six-cell mantle: two rows…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Earth, Climate & Energy Physics · Seismology, tectonics and geodynamics

The Solid Earth, Ch Surface waves, free oscillations and seismic tomography

Problem

Extend the four-cell resolution test of this chapter to a six-cell mantle: two rows and three columns of cells, each of unit size, crossed by two horizontal rays (one along each row) and three vertical rays (one down each column), every ray of unit length inside every cell it crosses. Number the cells

m_1,\dots,m_6

reading left to right along row 1 then row 2, so row 1 is

(m_1,m_2,m_3)

and row 2 is

(m_4,m_5,m_6)

. ((a)) Write the

5\times6

sensitivity matrix

\mathbf{G}

(rows: the two horizontal rays then the three vertical rays; columns:

m_1,\dots,m_6

). ((b)) Show that the five rows of

\mathbf{G}

satisfy exactly one linear relation, so that

\mathrm{rank}(\mathbf{G})=4

and the model-space null space has dimension

2

. Exhibit one nontrivial pattern of the six cells that produces zero residual on every one of the five rays. ((c)) Suppose the true model has a single fast cell,

m_1=1

and every other cell zero, so the noise-free data on the five rays are

\mathbf{d}=(1,0,1,0,0)

. Solve the damped normal equations

(\mathbf{G}^{\mathrm T}\mathbf{G}+\epsilon^2\mathbf{I})\hat{\mathbf{m}}=\mathbf{G}^{\mathrm T}\mathbf{d}

with

\epsilon=1

(in the same units as the ray lengths), and report

\hat{\mathbf{m}}

. ((d)) Compare the recovered value in cell

1

with the true value, and describe how the rest of the anomaly is distributed among the other cells.

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