Reliability-constrained spatial b-value mapping via kernel-weighted kNN with the magnitude-difference estimator and effective sample size

Abstract We present a framework for spatial Gutenberg–Richter (GR) b -value mapping in which three criteria are explicitly and jointly quantified and enforced: spatial locality (each local estimate uses only events within a bounded distance of the evaluation point), statistical support (each estimate is required to rest on a sufficient number of complete-magnitude events), and estimator uncertainty (a confidence measure is computed and reported for every estimate). The method couples (i) a kernel-smoothed map of event density with (ii) a local b -value estimator that, at each grid point, uses the k nearest earthquakes and computes b from the positive differences between their magnitudes following the method of Lippiello and Petrillo (in J Geophys Res Solid Earth 129(2):e2023JB027849, 2024. https://doi.org/10.1029/2023JB027849 ), which is robust to catalog incompleteness, and (iii) an effective sample size $$N_{\textrm{eff}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>N</mml:mi> <mml:mtext>eff</mml:mtext> </mml:msub> </mml:math> that measures how many events effectively contribute to each local estimate once the distance weighting is taken into account. Locality is imposed by requiring the radius $$r_k$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>r</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:math> enclosing the k nearest events to stay below a chosen bound $$r_{\max }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>r</mml:mi> <mml:mo>max</mml:mo> </mml:msub> </mml:math> ; statistical support is imposed by minimum values of $$N_{\textrm{eff}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>N</mml:mi> <mml:mtext>eff</mml:mtext> </mml:msub> </mml:math> and of the local complete-magnitude event count $$n_{\textrm{tail}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>n</mml:mi> <mml:mtext>tail</mml:mtext> </mml:msub> </mml:math> ; uncertainty is obtained by a weighted bootstrap that yields a map of the standard error (SE) of b , together with percentile-based $$95\%$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>95</mml:mn> <mml:mo>%</mml:mo> </mml:mrow> </mml:math> confidence intervals. The pair $$(k,r_{\max })$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>k</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>r</mml:mi> <mml:mo>max</mml:mo> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> is chosen by a single numerical score that combines spatial coverage, locality, support, and uncertainty. We test the estimator on a synthetic catalog whose true b ( x , y ) field is known by construction, and then apply the framework to a catalog of $$N=42{,}007$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mn>42</mml:mn> <mml:mo>,</mml:mo> <mml:mn>007</mml:mn> </mml:mrow> </mml:math> events recorded across Turkiye between 18 January 2023 and 30 September 2025 by the Kandilli Observatory and Earthquake Research Institute (KOERI). The resulting maps reveal interpretable spatial variations in b while suppressing estimates in regions where the data do not provide adequate information.