Determination of the angle in a Fisher distribution which will be exceeded with a given probability

Summary. The angle, θ _p , in a Fisher distribution which will be exceeded with probability p is shown to be given by The solution does not involve any estimation of the population precision and as such is the small sample solution. Fisher derived the analogue of 'Student's' test for mean directions and the test derived here is the equivalent test for individual observations.     

Comment on 'Systematic determination of earthquake rupture directivity and fault planes from analysis of long-period P-wave spectra' by L. M. Warren and P. M. Shearer

Warren and Shearer describe a method of estimating the duration of P pulses radiated by earthquakes, on the assumption that the source is a unilateral fracture. The estimates are made in the frequency domain. The estimates obtained by Warren and Shearer for seven of the earthquakes are compared to durations estimated here in the time domain; the time-domain measurements being made on broad-band seismograms (∼0.1–4.0 Hz) derived by filtering from short-period recordings. Overall, the time-domain method indicates that the pulse duration of the earthquakes studied here range from 2.0 to 7.6 s, whereas the estimates from the results of Warren and Shearer range from 7.1 to 9.8 s. This suggests that the method of Warren and Shearer cannot resolve pulse lengths less than about 7 s. The suggestion is supported by the estimates of the duration of rupture—fault length/speed of rupture—implied by the results of Warren and Shearer. For, although the estimated fault-length ranges from 0.8 km to over 40 km most rupture durations are around 8 s.     

Reply to comment by J. L. Mateos, O. Novaro, T. H. Seligman and J. Flores on 'Can horizontal P waves be trapped and resonate in a shallow sedimentary basin?'

Using a complete mathematical formulation, we show that the trapping of horizontal P waves in a very soft shallow alluvial layer is a minor effect. These waves do not have a stable way of propagation since in order to exist they require an incident wave and are therefore incapable of resonating in the lateral direction when confined in a basin of limited extent.     

Comment on: 'Can horizontal P waves be trapped and resonate in a shallow sedimentary basin?' by F. J. Sácnhez-Sesma and F. LuzÓ

We show that most of the arguments in the above paper are either incorrect or irrelevant to the point the authors are trying to make. We show that their results have no bearing on the model proposed by our group, as they claim. They discuss the seismic response of a valley with a 2-D trapezoidal cross-section in a vertical plane, whereas we dealt with a closed basin with a 2-D cross-section but of arbitrary geometry and in the horizontal plane. Even more significantly, the width of the valley they use is much smaller than the wavelength of the horizontal P waves that can resonate, thereby precluding any possibility of them being trapped. Therefore, their arguments do not clarify the issue posed in the title of their article.     

Comment on 'Consequences of progressive eclogitization on crustal exhumation, a mechanical study' by H. Raimbourg, L. Jolivet and Y. Leroy

Two simple end-member models of a subduction channel have been proposed in the literature: (i) the 'pressure-imposed' model for which the pressure within the channel is assumed to be lithostatic, the channel walls have negligible strength with respect to lateral pressure gradients, and the channel geometry therefore varies with time and (ii) the 'geometry-imposed' model of constant channel geometry, rigid walls and resultant lateral variation in pressure. Neither of these models is realistic, but they provide lower and upper bounds to potential pressure distributions in natural subduction zones. The critical parameter is the relative strength of the confining plates, reflected in the effective viscosity ratio between the channel fill and the walls. The assertion that the 'geometry-imposed' model is internally inconsistent is incorrect—it merely represents one bound to possible behaviour and a bound that may be approached for realistic values of the effective viscosity for weak channel fill (e.g. unconsolidated ocean-floor sediments) and relatively cold and strong subducting and overriding lithospheric plates.     

Comment on Non‐unique stratal geometries: implications for sequence stratigraphic interpretations,by: P.M. Burgess and G.D. Prince,Basin Research (2015) 27, 351–365

The non‐unique variability highlighted by Burgess & Prince (Basin Res. 2015, 27 , 351) (i.e. the origin and timing of maximum flooding surfaces, maximum regressive surfaces and subaerial unconformities; the process of topset aggradation in relation with the various types of shoreline trajectory; and the multiple controls that may affect the progradation and retrogradation of a shoreline) is irrelevant to the workflow of sequence stratigraphy. What is relevant is the observation of the unique stratal geometries that are diagnostic to the definition of all units and surfaces of sequence stratigraphy. In downstream‐controlled settings, these unique stratal stacking patterns relate to the forced regressive, normal regressive and transgressive shoreline trajectories. Multiple controls interplay during the formation of each type of stacking pattern, including accommodation, sediment supply and the energy of the sediment‐transport agents. This interplay explains the non‐unique variability, but does not change the unique criteria that afford a consistent application of sequence stratigraphy. Failure to rationalize the non‐unique variability within the context of unique stratal geometries is counterproductive, and obscures the simple workflow of sequence stratigraphy.