Computation of epistemic uncertainty due to limited data samples in small field dosimetry using fuzzy set theory

Objective: To estimate the epistemic (or fuzzy) uncertainty, arising due to limited data samples in the measurement of the output factors (OF) of the small fields using Fuzzy Set Theory (FST). Methods: EBT3 film samples of size 50 × 50 mm were used for the measurement of the OF of stereotactic radiosurgery (SRS) cones of size 4, 6, 7.5, 10, 12.5 and 15 mm diameter, normalized with respect to the output of 100 × 100 mm open field size. Three measurements were done per cone/field size. Red color channel was chosen for the dosimetry purpose, net optical density (NOD) was converted to the dose using non-linear relation. To estimate the epistemic uncertainty associated with the measured OF due to limited number of data samples, a triangular fuzzy number (TFN) was assumed as the fuzziness in the dose delivered by the individual SRS cone/field. Uncertainty in the OF was estimated by applying the Fuzzy Vertex Method (FVM). The membership functions of the OF were constructed for each cone size and the nature of the uncertainty in the OF of the cones was expressed in the terms of its fuzziness. For the sake of completeness of the study, the statistical uncertainty involved in the procedure has also been calculated. Results: The statistical and fuzzy uncertainties in the measurement of OF of cones range from 3.28 to 6.25% and 2.58 to 5.44% respectively. The smallest cone of 4 mm has the largest values of statistical and fuzzy uncertainties. The membership functions of the OF for the studied cones were triangular in nature. Conclusions: The epistemic uncertainty arising due to limited number of data samples holds a significant fraction of the prescribed dose, and therefore, should not be ignored in the total uncertainty estimation. Advances in knowledge: This study highlights the significance of epistemic component of measurement uncertainty arising out due to the insufficient/limited number of measurements of a quantity.

Imaging active magmatic systems at Oldoinyo Lengai volcano (Tanzania) via earthquake distribution and seismic scattering and absorption mapping

<p>Oldoinyo Lengai volcano, located in the Natron Basin (Tanzania), is the only active natrocarbonatite volcano world-wide. As such, it presents an important endmember magmatic system, which occurs in a young rift segment (~3 Ma) of the East African Rift System. At this volcano, effusive episodes of long-duration are interrupted by short-duration explosive eruptions. At the end of February 2019, we installed a dense seismic network and four infrasound stations as part of the SEISVOL - Seismic and Infrasound Networks to Study the Volcano Oldoinyo Lengai - project. The seismic network spans an area of 30 x 30 km and encompasses Oldoinyo Lengai volcano, the extinct 1 Ma-old Gelai shield volcano, the active Naibor Soito monogenetic cone field and surrounding fault population. Here, we present temporal earthquake distributions combined with 2D absorption and scattering imaging.</p><p>On average, we report up to 34 earthquakes per day within and in the vicinity of our network. Given the dense station spacing, we are able to lower the detection threshold to -1.0 M<sub>L</sub> with a M<sub>C</sub> of -0.3. During the first months of data acquisition, the seismicity is clustered in distinct areas as background seismicity and in intermittent seismic swarms:</p><ol><li>Most of the events are located beneath the eastern and southern flank of Gelai shield volcano. These events are shallow and close to the dike intrusion that preceded the last explosive eruption of Oldoinyo Lengai in 2007-2008.</li> <li>In April 2019, a seismic swarm of ~262 earthquakes in three days forms a pipe-like structure beneath the north western flank of Gelai.</li> <li>Deeper events cluster beneath the monogenetic cone field located just NE of Oldoinyo Lengai. A distinct gap in seismicity can be traced down to 10 km depth between the monogenetic cone field and Gelai volcano.</li> <li>While there seems to be little seismicity directly beneath Oldoinyo Lengai in the upper 5 km of the crust, we observe a number of different, recurring seismic and infrasound signals at the crater, which are indicative of magmatic activity.</li> </ol><p>To image the magmatic plumbing system, we map scattering and absorption of the seismic dataset using the MuRAT (Multi-Resolution Attenuation Tomography) code. Our preliminary results show two well-resolved high-absorption and high-scattering anomalies below Oldoinyo Lengai and the Gelai intrusion in 2007 at all frequencies. With decreasing frequency (increasing depth) the anomalies converge, suggesting a link of the plumbing systems at depth.</p>

Volume lemmas and large deviations for partially hyperbolic endomorphisms

We consider partially hyperbolic attractors for non-singular endomorphisms admitting an invariant stable bundle and a positively invariant cone field with non-uniform cone expansion at a positive Lebesgue measure set of points. We prove volume lemmas for both Lebesgue measure on the topological basin of the attractor and the SRB measure supported on the attractor. As a consequence, under a mild assumption we prove exponential large-deviation bounds for the convergence of Birkhoff averages associated to continuous observables with respect to the SRB measure.

SRB measures for partially hyperbolic attractors of local diffeomorphisms

We contribute to the thermodynamic formalism of partially hyperbolic attractors for local diffeomorphisms admitting an invariant stable bundle and a positively invariant cone field with non-uniform cone expansion at a positive Lebesgue measure set of points. These include the case of attractors for Axiom A endomorphisms and partially hyperbolic endomorphisms derived from Anosov. We prove these attractors have finitely many SRB measures, that these are hyperbolic, and that the SRB measure is unique provided the dynamics is transitive. Moreover, we show that the SRB measures are statistically stable (in the weak$^{\ast }$ topology) and that their entropy varies continuously with respect to the local diffeomorphism.