The transition from the Earth's solid inner core to liquid outer core is the location where the inner core grows and from which compositional convection in the outer core originates. Most seismological models of the Earth describe the inner-core boundary as sharp and simple, although experimental data requiring the presence of a thin transition layer at the bottom of the outer core have been reported. The density jump at the inner-core boundary--an important parameter determining gravitational energy release and constraining the compositional difference between the inner and outer core-is also not well known. Estimates of this density jump obtained using free-oscillation eigenfrequencies give low values of 0.25-1.0 g cm(-3), whereas a method using the amplitude ratio of core-reflected phases yielded values of 0.6-1.8 g cm(-3) (refs 14, 15, 16-17). Here we analyse properties of waves precritically reflected from the Earth's inner core (PKiKP phases) that show significant variability in amplitude, consistent high-frequency content and stable travel times with respect to a standard Earth model. We infer that the data are best explained by a mosaic structure of the inner core's surface. Such a mosaic may be composed of patches in which the transition from solid inner to liquid outer core includes a thin partially liquid layer interspersed with patches containing a sharp transition.

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We combine classical concepts from different disciplines -those of α-hull and α-shape from computational geometry, splitting data into training and test sets from artificial intelligence, density-based spatial clustering from data mining, and moving average from time series analysis -to develop a robust algorithm for reconstructing the shape of a curve from noisy samples.The novelty of our approach is two-fold. First, we introduce the notion of k-order α-hull and α-shape -generalizations of α-hull and α-shape. Second, we use white noise to "train" our k-order α-shaper, i.e., to choose the right values of α and k.The difference of the k-order α-hull and α-shape from the α-hull and α-shape is also two-fold. First, k-order α-hull and α-shape provide a robust estimate of the shape by ignoring outliers. Second, it reconstructs the "inner" shape, with the amount of "digging" into the data controlled by k. Index Terms: I.5.2 [Pattern Recognition]: Design MethodologyPattern analysis; J.2 [Physical Sciences and Engineering]: Earth and atmospheric sciences- α -HULL AND α -SHAPELet P ⊂ R 2 be a finite planar set. Classical tools, defining "shape" of the set, are those of the α-hull and α-shape of P [1]. Adapting the notions from [1] to make our exposition simpler, we make the following definitions (cf. Definitions 2, 3, 4 in [1]):Definition 1. Let α > 0. An α-ball is an (open) disc of radius α. An empty α-ball is an α-ball containing no points of P. The α-hull of P is the complement of the union of all empty α-balls. Two points p, q ∈ P are α-neighbors if there exists an empty α-ball having both p and q on its boundary. The α-shape of P is the straightline graph, whose vertices are points in P and whose edges connect α-neighbors.For an example of α-shape see Figure 1, (a) and (b). HANDLING OUTLIERS WITH k-ORDER α -HULL AND α -SHAPEIt is easy to imagine situations in which several outliers will severely harm the shape produced by the α-hull or α-shape (see Fig. 1, (a) and (b)). To reduce the impact of outliers on shape reconstruction, we introduce a generalization of the α-hull and α-shape, the k-order α-hull and k-order α-shape:Definition 2. Let α > 0, k ≥ 0. An α-ball is an (open) disc of radius α. An empty α-ball is an α-ball containing exactly k points of P. The k-order α-hull of P is the complement of the union of all * empty α-balls. Two points p, q ∈ P are called α-neighbors if there exists an empty α-ball having both p and q on its boundary. The k-order α-shape of P is the straight-line graph, whose vertices are points in P and whose edges connect a-neighbors.Of course, α-hull and α-shape are just 0-order α-hull and α-shape. With k ≥ 1 we allow the k-order α-hull and α-shape to ignore some amount of outliers (Fig. 1). RECONSTRUCTING INNER SHAPE WITH k-ORDER α -HULL AND α -SHAPEα-hull and α-shape, being generalizations of the convex hull, capture the "outer" shape of P. At the same time, in many applications it is of interest to infer the "inner" shape of the set. (For a polygonal domain its inner shape may be represe...

We present a new method for estimation of seismic coda shape. It falls into the same class of methods as non-parametric shape reconstruction with the use of neural network techniques where data are split into a training and validation data sets. We particularly pursue the wellknown problem of image reconstruction formulated in this case as shape isolation in the presence of a broadly defined noise. This combined approach is enabled by the intrinsic feature of seismogram which can be divided objectively into a pre-signal seismic noise with lack of the target shape, and the remainder that contains scattered waveforms compounding the coda shape. In short, we separately apply shape restoration procedure to pre-signal seismic noise and the event record, which provides successful delineation of the coda shape in the form of a smooth almost non-oscillating function of time.The new algorithm uses a recently developed generalization of classical computationalgeometry tool of α-shape. The generalization essentially yields robust shape estimation by ignoring locally a number of points treated as extreme values, noise or non-relevant data. Our algorithm is conceptually simple and enables the desired or pre-determined level of shape detail, constrainable by an arbitrary data fit criteria.The proposed tool for coda shape delineation provides an alternative to moving averaging and/or other smoothing techniques frequently used for this purpose.The new algorithm is illustrated with an application to the problem of estimating the coda duration after a local event. The obtained relation coefficient between coda duration and epicentral distance is consistent with the earlier findings in the region of interest.