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The exchange explains why standard Tikhonov parameter selection methods (L-curve, GCV) and variance component estimation (VCE) can fail under heterogeneous measurements, especially when prior information is deterministic. It summarizes proposed strategies combining VCE-derived relative weights with L-curve-based parameter selection, compares performance via RMS error and correlation, and notes revisions to sections and discussion to clarify limitations and significance. Additional future work is described for real data applications.",{"@graph":69,"@context":123},[70,84,106],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/response-to-interactive-comment-on-regularization-methods-for-combination-of-heterogeneous-observations-using-spherical-radial-basis-functions-reviewer-response/315983/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/response-to-interactive-comment-on-regularization-methods-for-combination-of-heterogeneous-observations-using-spherical-radial-basis-functions-reviewer-response/315983.png","ImageObject",300,407,{"name":92,"@type":93},"Asher","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-31","2026-09-20",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",11,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"Why are conventional regularization parameter choice methods like L-curve and GCV considered unsuitable for heterogeneous observations?","Question",{"text":113,"@type":114},"They cannot determine the relative weighting between different observation types. That limitation affects the reliability of parameter selection when multiple heterogeneous datasets are combined.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"What role does VCE play in the proposed approach, and why can VCE be unreliable in this context?",{"text":118,"@type":114},"VCE can estimate variance components and the regularization parameter, but it may fail when the “prior information” is deterministic. In regional gravity field modeling, background models are typically deterministic, while VCE requires stochastic prior information.",{"name":120,"@type":111,"acceptedAnswer":121},"What evidence supports that the proposed “VCE + L-curve method” improves results?",{"text":122,"@type":114},"The response states that the original L-curve and VCE cannot always provide satisfying outcomes, while the combined “VCE + L-curve method” yields the smallest RMS error and largest correlation across the study cases, with further comparisons based on validation data and tables in the revised manuscript.","https://schema.org",{"og:url":83,"og:type":125,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":127,"canonical":83},"index,follow",{"doc_id":129,"site_id":62},315983,1790819919,{"code":4,"msg":5,"data":132},{"doc_id":129,"user_id":133,"nickname":92,"user_avatar":134,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":105,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":24,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":144,"read_time":145,"preview_page_count":4},687197207639,"https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd","A response to the interactive comment on “Regularization methods for the combination of heterogeneous observations using spherical radial basis functions”by Qing Liu et al.  \nAnonymous Referee \\#1  \nDear Reviewer,  \nThank you very much for taking the time to review the manuscript and give comments that help us to improve it. Please find below our point-by-point response to your comments. The original comments are in black, and our answers are in blue. The revised manuscript with tracked changes is also attached.  \nThe manuscript is a good scientific writing, with clearly defined question, i.e. how to choose the optimal Tikhonov regularization parameter in the combination of heterogeneous observation for regional gravity field modeling. Numerical experiments are properly designed to investigate this problem and the results are fully discussed to get a reasonable conclusion. However, I couldn ’t see much scientific significance from the manuscript.  \nAs you said, this manuscript is trying to answer the question “how to choose the optimal Tikhonov regularization parameter in the combination of heterogeneous observation for regional gravity field modeling.” Xu et al. (2006) pointed out that when heterogeneous measurements are involved, the conventional regularization parameter choice method, including the L-curve method and GCV may be not proper to apply. The reason is that the L-curve method or GCV cannot determine the relative weighting between different observation types. On the other hand, VCE can estimate the variance components of different observation techniques as well as the regularization parameter (interpreted asthe ratio between the variance factors of the observations and the prior information) simultaneously. However, Naeimi (2013), Liang (2017) and Lieb (2017) all showed that VCE sometimes could not provide a reliable regularization result by incorporating prior information. Because in VCE, the prior information is interpreted as an additional observation technique, which is required to be stochastic. In regional gravity field modeling, usually a background model serves as the prior information, which is deterministic.  \nBased on these facts, the three commonly used regularization parameter choice methods seem all have pitfalls when heterogeneous observations are combined. Thus, we proposed two methods which combine VCE for determining the relative weights between the observation types and the L-curve method for determining the regularization parameter. The idea of combining VCE for weighting different data sets and a method for determining the regularization parameter was introduced in the Section ‘future work’ of both Naeimi (2013, p.121) and Liang (2017, p.134) . The study  \nin this manuscript is also inspired by Wang et al. (2018), who combine two methods successively for determining the regularization parameter and relative weights for GPSand InSAR. However, to the best of our knowledge, there are still no publications applying this idea for combining heterogeneous observations in regional gravity field modeling.  \nOur experiment results prove that the original L-curve method or the VCE cannot always give a satisfying result (please refer to Table 6 in the revised manuscript) , and our proposed ‘VCE + L-curve method’ provides the best results which means the smallest RMS error and the largest correlation for all our study cases. A further work based on the proposed ‘VCE + L-curve method’ for combining real observation data to model the regional gravity field in Colorado, USA, is under investigation. Moreover, this new approach could be used not only for regional gravity field modeling but also in many other fields such as atmospheric science.  \nWe have rewritten parts of Section 1 and added two paragraphs explaining the limitations of the existing methods as well as the reasons for proposing the new methods. We have also highlighted the scientific significance of the new methods in the discussion part as well a","cbCaikkWdGYUAMKJ","https://ap.wps.com/l/cbCaikkWdGYUAMKJ","pdf",2901240,"English","# Point-by-Point Response\n## Scientific significance and motivation\n## Limitations of existing parameter choice methods\n## Proposed VCE + L-curve strategy\n## Numerical results and comparisons","[{\"question\":\"Why are conventional regularization parameter choice methods like L-curve and GCV considered unsuitable for heterogeneous observations?\",\"answer\":\"They cannot determine the relative weighting between different observation types. That limitation affects the reliability of parameter selection when multiple heterogeneous datasets are combined.\"},{\"question\":\"What role does VCE play in the proposed approach, and why can VCE be unreliable in this context?\",\"answer\":\"VCE can estimate variance components and the regularization parameter, but it may fail when the “prior information” is deterministic. In regional gravity field modeling, background models are typically deterministic, while VCE requires stochastic prior information.\"},{\"question\":\"What evidence supports that the proposed “VCE + L-curve method” improves results?\",\"answer\":\"The response states that the original L-curve and VCE cannot always provide satisfying outcomes, while the combined “VCE + L-curve method” yields the smallest RMS error and largest correlation across the study cases, with further comparisons based on validation data and tables in the revised manuscript.\"}]","Response to Interactive Comment on Regularization Methods for Combination of Heterogeneous Observations Using Spherical Radial Basis Functions - Reviewer Response | PDF",1789908750,13]