Multiparameter Inversion: Cramer's Rule for Pseudodifferential Operators

<table class="table-group" width="800px"><tr class="fig-image" id="a"><td><img alt="780291.fig.003a" src="https://static.hindawi.com/articles/ijge/volume-2011/780291/figures/780291.fig.003a.jpg"/></td></tr><tr class="fig-caption"><td align="center"><table><tr><td><b>(a) </b><svg height="15.475" id="M163" style="vertical-align:-6.20941pt" version="1.1" viewbox="0 0 39 15.475" width="39" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
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<g transform="matrix(.012,-0,0,-.012,13.475,11.762)"><path d="M797 0h-214v26q46 5 58 17.5t12 63.5v167q0 116 -88 116q-48 0 -112 -52q3 -18 3 -42v-195q0 -46 11 -58t53 -17v-26h-214v26q48 5 59.5 17t11.5 60v172q0 115 -88 115q-47 0 -109 -52v-235q0 -47 10 -59.5t48 -17.5v-26h-211v26q50 5 62.5 17t12.5 60v207q0 48 -9 59.5
t-52 18.5v23q79 14 139 41v-81q3 2 29 20.5t44 29.5q45 28 81 28t66 -22t46 -61q25 20 83 57q39 26 76 26q59 0 93 -40t34 -113v-193q0 -49 10.5 -60.5t54.5 -16.5v-26z" id="x6D"></path></g><g transform="matrix(.012,-0,0,-.012,23.22,11.762)"><path d="M135 536q-20 0 -35 15.5t-15 35.5q0 22 15 37t36 15t35.5 -15t14.5 -37q0 -21 -15 -36t-36 -15zM252 0h-220v26q48 5 59 17t11 63v206q0 47 -9 58t-54 18v24q78 12 142 39v-345q0 -51 11.5 -63t59.5 -17v-26z" id="x69"></path></g><g transform="matrix(.012,-0,0,-.012,26.409,11.762)"><path d="M463 437l5 -8q-21 -38 -45 -51l-57 3q30 -35 30 -88q0 -76 -52.5 -118t-122.5 -42q-24 0 -51 5q-30 -21 -30 -42q0 -16 16 -27t47 -11q19 0 54 0.5t52 0.5q36 0 65.5 -8t53.5 -36t24 -74q0 -77 -75 -137.5t-174 -60.5q-79 0 -126 37.5t-48 84.5q0 32 24 56q21 21 83 70
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<g transform="matrix(.012,-0,0,-.012,32.6,15.412)"><path d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z" id="x31"></path></g>
</svg>, velocity component of the migrated image</td></tr></table></td></tr><tr class="fig-image" id="b"><td><img alt="780291.fig.003b" src="https://static.hindawi.com/articles/ijge/volume-2011/780291/figures/780291.fig.003b.jpg"/></td></tr><tr class="fig-caption"><td align="center"><table><tr><td><b>(b) </b><svg height="15.475" id="M164" style="vertical-align:-6.20941pt" version="1.1" viewbox="0 0 39 15.475" width="39" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,7.675)"><path d="M766 88q-40 -45 -83.5 -72.5t-63.5 -27.5q-39 0 -16 103l49 224q15 68 -10 68q-36 0 -112.5 -78.5t-105.5 -133.5q-27 -101 -38 -165q-35 -4 -77 -18l-6 6q42 153 70 288q12 55 10 78t-19 23q-38 0 -114 -80.5t-101 -131.5q-24 -68 -41 -165q-36 -3 -76 -18l-8 6
q56 202 87 347q7 33 -3 33q-8 0 -31.5 -17.5t-41.5 -35.5l-12 28q42 45 84 72t63 27q44 0 10 -124l-20 -75h2q92 121 193 178q36 21 64 21q62 0 28 -159l-8 -37h2q50 67 103.5 112t94.5 65q39 19 62 19q60 0 23 -156l-45 -189q-9 -39 2 -39q17 0 71 49z" id="x1D45A"></path></g>
<g transform="matrix(.012,-0,0,-.012,13.475,11.762)"><path d="M797 0h-214v26q46 5 58 17.5t12 63.5v167q0 116 -88 116q-48 0 -112 -52q3 -18 3 -42v-195q0 -46 11 -58t53 -17v-26h-214v26q48 5 59.5 17t11.5 60v172q0 115 -88 115q-47 0 -109 -52v-235q0 -47 10 -59.5t48 -17.5v-26h-211v26q50 5 62.5 17t12.5 60v207q0 48 -9 59.5
t-52 18.5v23q79 14 139 41v-81q3 2 29 20.5t44 29.5q45 28 81 28t66 -22t46 -61q25 20 83 57q39 26 76 26q59 0 93 -40t34 -113v-193q0 -49 10.5 -60.5t54.5 -16.5v-26z" id="x6D"></path></g><g transform="matrix(.012,-0,0,-.012,23.22,11.762)"><path d="M135 536q-20 0 -35 15.5t-15 35.5q0 22 15 37t36 15t35.5 -15t14.5 -37q0 -21 -15 -36t-36 -15zM252 0h-220v26q48 5 59 17t11 63v206q0 47 -9 58t-54 18v24q78 12 142 39v-345q0 -51 11.5 -63t59.5 -17v-26z" id="x69"></path></g><g transform="matrix(.012,-0,0,-.012,26.409,11.762)"><path d="M463 437l5 -8q-21 -38 -45 -51l-57 3q30 -35 30 -88q0 -76 -52.5 -118t-122.5 -42q-24 0 -51 5q-30 -21 -30 -42q0 -16 16 -27t47 -11q19 0 54 0.5t52 0.5q36 0 65.5 -8t53.5 -36t24 -74q0 -77 -75 -137.5t-174 -60.5q-79 0 -126 37.5t-48 84.5q0 32 24 56q21 21 83 70
q-49 15 -74 55q-13 20 -11 42q56 21 88 61q-37 16 -62.5 51t-25.5 83q0 78 56.5 122t122.5 44h1q51 0 96 -27q71 4 136 15zM219 418q-34 0 -58 -32t-24 -87q0 -59 26.5 -96.5t67.5 -37.5q35 1 59 32t24 87q0 59 -26 96.5t-69 37.5zM241 -11q-62 0 -87 -21q-46 -39 -46 -78
q0 -41 38 -68.5t94 -27.5q69 0 106.5 31t37.5 80q0 52 -45 71q-33 13 -98 13z" id="x67"></path></g>
<g transform="matrix(.012,-0,0,-.012,32.6,15.412)"><path d="M412 140l28 -9q0 -2 -35 -131h-373v23q112 112 161 170q59 70 92 127t33 115q0 63 -31 98t-86 35q-75 0 -137 -93l-22 20l57 81q55 59 135 59q69 0 118.5 -46.5t49.5 -122.5q0 -62 -29.5 -114t-102.5 -130l-141 -149h186q42 0 58.5 10.5t38.5 56.5z" id="x32"></path></g>
</svg>, density component of the migrated image</td></tr></table></td></tr></table>

<table>Migrated images mixing the contributions from density and velocity, and effecting a phase space scaling.</table>

International Journal of Geophysics

fig3

Figure 3

Figure 3: Multiparameter Inversion: Cramer's Rule for Pseudodifferential Operators 