End-point contributions to synthetic seismograms

Summary . Synthetic seismograms represented by integrals generally display signals associated with the limits of integration. Sometimes these 'end-point' contributions are spurious (e.g. in the WKBJ seismogram) and sometimes they are the main physical interest (e.g. the Kirchhoff integral for an edge). The end-point contributions may be asymptotically approximated using integration by parts or Laplace's method and it may then be possible to reduce them if desired. We describe examples in the WKBJ seism ogram for reflected or transmitted waves in homogeneous layers and for turning waves. We also study signals due to discontinuities in reflection coefficients, by partitioning the real slowness integral so that the discontinuities lie at end points. Examples are the head wave, which is a physically correct signal, and spurious diffractions caused by using plane-wave coefficients for grazing rays in the WKBJ seismogram.     

Application of the Maslov Seismogram Method in Three Dimensions

Asymptotic methods provide an efficient way to compute seismograms in heterogeneous media. However, zeroth-order ray theory, the simplest of the asymptotic methods, often fails because of the presence of caustics. Maslov theory is an extension of zeroth-order ray theory, which gives a uniformly valid expression of the wavefield everywhere, including the caustics. This result is given in terms of an integral of ray data over one or two ray parameters. It is shown in this paper how geometrical arrivals are constructed in the one and two-parameter Maslov integrals.In practice Maslov seismograms have been computed using only one ray parameter. However, in three-dimensional media two parameters are needed to uniquely define a ray. In this paper we present an efficient algorithm to compute two-parameter Maslov integrals. The Maslov integral is evaluated by computing the frequency-to-time Fourier transform prior to integration over the ray parameters. The wavefield is then discretized by smoothing with a boxcar function. The resulting expression, which only requires the results of ordinary kinematic and dynamic ray tracing, cen be computed efficiently and robustly. A numerical example is given that illustrates the use of this algorithm.     

Modulating synoptic scale convective activity and boundary layer dissipation in the IPESD models of the Madden–Julian oscillation

A self-contained derivation of the IPESD models [Majda, A.J., Klein, R., 2003. Systematic multi-scale models for the tropics. J. Atmos. Sci. 60, 393–408] governing synoptic and planetary scale tropical flows is provided. This derivation demonstrates the analytic tractability of the model and the effect of zonally and meridionally tilted synoptic scale heating on the forcing of planetary scale flows through upscale momentum and temperature fluxes. Exploiting the analytic tractability of the models, different aspects of the planetary scale forcing are traced to meridional and vertical tilts in the synoptic scale heating profile. Variants of the archetypal IPESD models for the Madden–Julian oscillation (MJO) presented in Majda and Biello [Majda, A.J., Biello, J.A., 2004. A multi-scale model for tropical intraseasonal oscillations. Proc. Natl. Acad. Sci. 101, 4736–4741; Biello, J.A., Majda, A.J., 2005. A new multi-scale model for the Madden–Julian oscillation. J. Atmos. Sci. 62, 1694–1721] are studied. In addition to vertically tilted synoptic scale heating, the models discussed herein incorporate upscale zonal momentum flux due to meridional flux convergence arising from meridionally tilted heating. The effect of a boundary layer momentum drag at the base of the free troposphere is also systematically incorporated into the IPESD models. Both meridional tilts and lower boundary layer drag are shown to meridionally confine the MJO westerly wind burst and drive a planetary scale barotropic flow. Meridionally tilted heating can also greatly strengthen the wind burst at the base of the troposphere and modify its vertical profile. The competing effects of meridionally tilted, and off-equatorial heating can also significantly weaken the MJO winds. Appendices are provided which discuss generalizations and a solution algorithm for the IPESD models.     

Fixed-Domain Asymptotics for Variograms Using Subsampling

The variogram is a measure of the local variation in space of a random field. For large geostatistical data sets, the traditional empirical variogram may be hard to compute. This article presents, for processes with a fixed domain, the effect of using a subsample of the available data on the performance of the empirical variogram. The motivation of this work, apart from the saving on computation, is to study how dense the observations need to be in the bounded sampling region to obtain most of the information we would get from continuous observations in the fixed domain.     

Global and local spatial autocorrelation in bounded regular tessellations

This paper systematically investigates spatially autocorrelated patterns and the behaviour of their associated test statistic Moran's I in three bounded regular tessellations. These regular tessellations consist of triangles, squares, and hexagons, each of increasing size (n=64; 256; 1024). These tesselations can be downloaded at http://geo-www.sbs.ohio-state.edu/faculty/tiefelsdorf/regspastruc/ in several GIS formats. The selection of squares is particularly motivated by their use in raster based GIS and remote sensing. In contrast, because of topological correspondences, the hexagons serve as excellent proxy tessellations for empirical maps in vector based GIS. For all three tessellations, the distributional characteristics and the feasibility of the normal approximation are examined for global Moran's I, Moran's I ^(k) associated with higher order spatial lags, and local Moran's I _i. A set of eigenvectors can be generated for each tessellation and their spatial patterns can be mapped. These eigenvectors can be used as proxy variables to overcome spatial autocorrelation in regression models. The particularities and similarities in the spatial patterns of these eigenvectors are discussed. The results indicate that [i] the normal approximation for Moran's I is not always feasible; [ii] the three tessellations induce different distributional characteristics of Moran's I, and [iii] different spatial patterns of eigenvectors are associated with the three tessellations. Received: 2 July 1999 / Accepted: 9 November 1999