Interpretation of MUSIC for Location Detecting of Small Inhomogeneities Surrounded by Random Scatterers

<div>Maps of <svg height="11.5564pt" id="M269" style="vertical-align:-2.26807pt" version="1.1" viewbox="-0.0498162 -9.28833 42.2862 11.5564" width="42.2862pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M663 549C663 621 607 680 521 680C283 680 82 444 82 188C82 80 142 -15 258 -15C311 -15 361 -1 408 23C428 -8 451 -38 500 -38C535 -38 584 -16 633 44L618 58C582 23 543 10 516 10C485 10 469 28 454 52C564 133 637 268 637 396C637 502 561 542 497 542C414 542 336 469 336 362C336 331 339 284 371 232L393 242C369 289 366 322 366 347C366 448 430 506 494 506C539 506 586 470 586 396C586 266 508 157 469 118C458 106 446 97 435 86C418 115 396 142 349 142C313 142 284 123 241 84L254 67C283 90 305 96 330 96C359 96 375 77 390 53C350 29 307 15 256 15C179 15 142 78 142 150C142 331 347 645 517 645C590 645 624 603 643 547L663 549Z" id="g198-18"></path><glyph.data ascent="687" descent="-226" horiz-adv-x="692" 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130C70 130 46 114 46 88C46 72 54 65 59 64C93 56 121 33 121 -3C121 -41 93 -68 45 -88L56 -118C117 -99 186 -56 186 22C186 91 131 130 95 130Z" id="g113-60"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="228" vert-adv-y="228"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,25.06,0)"><path d="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z" id="g113-50"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="480" vert-adv-y="480"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,31.3,0)"><path d="M241 635C89 635 35 457 35 312C35 153 89 -12 240 -12C390 -12 443 166 443 312C443 466 390 635 241 635ZM238 602C329 602 354 454 354 312C354 172 330 22 240 22C152 22 124 173 124 313S148 602 238 602Z" id="g113-49"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="480" vert-adv-y="480"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,37.586,0)"><path d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z" id="g113-42"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="346" vert-adv-y="346"></glyph.data></g></svg> with first <svg height="8.55521pt" id="M270" style="vertical-align:-0.2063904pt" version="1.1" viewbox="-0.0498162 -8.34882 6.36303 8.55521" width="6.36303pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M285 378C315 398 338 416 353 432C373 451 384 474 384 503C384 579 325 635 236 635H235C182 635 136 610 108 579L65 516L85 496C110 533 150 575 205 575C258 575 300 543 300 481C300 407 232 369 141 339L147 310C163 315 188 321 211 321C268 321 338 284 338 192C338 94 288 40 217 40C160 40 119 68 93 91C85 98 77 97 69 91C60 84 47 71 46 58C44 46 48 35 62 22C75 10 116 -12 162 -12C234 -12 424 62 424 224C424 297 373 359 285 376V378Z" id="g113-52"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="480" vert-adv-y="480"></glyph.data></g></svg>-singular values (a) and with <svg height="8.55521pt" id="M271" style="vertical-align:-0.2063904pt" version="1.1" viewbox="-0.0498162 -8.34882 15.6015 8.55521" width="15.6015pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M241 635C89 635 35 457 35 312C35 153 89 -12 240 -12C390 -12 443 166 443 312C443 466 390 635 241 635ZM238 602C329 602 354 454 354 312C354 172 330 22 240 22C152 22 124 173 124 313S148 602 238 602Z" id="g113-49"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="480" vert-adv-y="480"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,6.24,0)"><path d="M113 -12C146 -12 170 11 170 46C170 78 146 103 114 103S58 78 58 46C58 11 82 -12 113 -12Z" id="g113-47"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="228" vert-adv-y="228"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,9.204,0)"><path d="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z" id="g113-50"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="480" vert-adv-y="480"></glyph.data></g></svg>-threshold scheme (b).</div>

Mathematical Problems in Engineering

fig7

Figure 7

Figure 7: Interpretation of MUSIC for Location Detecting of Small Inhomogeneities Surrounded by Random Scatterers 