{"id":356,"date":"2013-10-16T07:25:34","date_gmt":"2013-10-16T07:25:34","guid":{"rendered":"http:\/\/ahay.org\/blog\/?p=356"},"modified":"2015-08-04T23:51:30","modified_gmt":"2015-08-04T23:51:30","slug":"dsr-eikonal-tomography","status":"publish","type":"post","link":"https:\/\/ahay.org\/blog\/2013\/10\/16\/dsr-eikonal-tomography\/","title":{"rendered":"DSR eikonal tomography"},"content":{"rendered":"<p>A new paper is added to the <a href=\"\/wiki\/Reproducible_Documents\">collection of reproducible documents<\/a>:<br \/>\n<a href=\"\/RSF\/book\/tccs\/dsrtomo\/paper_html\/\">First-break traveltime tomography with the double-square-root eikonal equation<\/a><br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"\/RSF\/book\/tccs\/dsrtomo\/marm\/Fig\/ntomo.png\" alt=\"\" width=\"430\" height=\"360\" \/><\/p>\n<blockquote><p>First-break traveltime tomography is based on the eikonal equation. Since the eikonal equation is solved at fixed shot positions and only receiver positions can move along the ray-path, the adjoint-state tomography relies on inversion to resolve possible contradicting information between independent shots. The double-square-root eikonal equation allows not only the receivers but also the shots to change position, and thus describes the prestack survey as a whole. Consequently, its linearized tomographic operator naturally handles all shots together, in contrast with the shot-wise approach in the traditional eikonal-based framework. The double-square-root eikonal equation is singular for the horizontal waves, which require special handling. Although it is possible to recover all branches of the solution through post-processing, our current forward modeling and tomography focus on the diving wave branch only. We consider two upwind discretizations of the double-square-root eikonal equation and show that the explicit scheme is only conditionally convergent and relies on non-physical stability conditions. We then prove that an implicit upwind discretization is unconditionally convergent and monotonically causal. The latter property makes it possible to introduce a modified fast marching method thus obtaining first-break traveltimes both efficiently and accurately. To compare the new double-square-root eikonal-based tomography and traditional eikonal-based tomography, we perform linearizations and apply the same adjoint-state formulation and upwind finite-differences implementation to both approaches. Synthetic model examples justify that the proposed approach converges faster and is more robust than the traditional one.<\/p><\/blockquote>\n","protected":false},"excerpt":{"rendered":"<p>A new paper is added to the collection of reproducible documents: First-break traveltime tomography with the double-square-root eikonal equation First-break traveltime tomography is based on the eikonal equation. Since the eikonal equation is solved at fixed shot positions and only receiver positions can move along the ray-path, the adjoint-state tomography relies on inversion to resolve [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_import_markdown_pro_load_document_selector":0,"_import_markdown_pro_submit_text_textarea":"","activitypub_content_warning":"","activitypub_content_visibility":"","activitypub_max_image_attachments":4,"activitypub_interaction_policy_quote":"anyone","activitypub_status":"","footnotes":""},"categories":[5],"tags":[],"class_list":["post-356","post","type-post","status-publish","format-standard","hentry","category-documentation"],"_links":{"self":[{"href":"https:\/\/ahay.org\/blog\/wp-json\/wp\/v2\/posts\/356","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ahay.org\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/ahay.org\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/ahay.org\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/ahay.org\/blog\/wp-json\/wp\/v2\/comments?post=356"}],"version-history":[{"count":1,"href":"https:\/\/ahay.org\/blog\/wp-json\/wp\/v2\/posts\/356\/revisions"}],"predecessor-version":[{"id":559,"href":"https:\/\/ahay.org\/blog\/wp-json\/wp\/v2\/posts\/356\/revisions\/559"}],"wp:attachment":[{"href":"https:\/\/ahay.org\/blog\/wp-json\/wp\/v2\/media?parent=356"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/ahay.org\/blog\/wp-json\/wp\/v2\/categories?post=356"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/ahay.org\/blog\/wp-json\/wp\/v2\/tags?post=356"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}