Wavelet estimation in seismic convolved hidden Markov models
Inversion of seismic AVO-data is an important part of reservoir evalua- tion. These data are convolved but the convolution kernel and the associated error- variances are largely unknown. We aim at estimating these model parameters with- out using calibration observations in wells. This constitu tes the first step in socalled blind deconvolution. We solve the seismic inverse problem in a Bayesian setting and perform the associated model parameter estimation by an approximate marginal maximum likelihood method. A small test study indicate that bell-shaped wavelets with smooth edges are identified well, even for approximatio ns of low orders.
Approximate posterior distributions for convolutional two-level hidden Markov models
A convolutional two-level hidden Markov model is defined and evaluated. The bottom level contains an unobserved categorical Markov chain, and given the variables in this level the middle level contains unobserved conditionally independent Gaussian variables. The top level contains observable variables that are a convolution of the variables in the middle level plus additive Gaussian errors. The objective is to assess the categorical variables in the bottom level given the convolved observations in the top level. The inversion is cast in a Bayesian setting with a Markov chain prior model and convolved Gaussian likelihood model. The associated posterior model cannot be assessed since the normalizing constant is too computer demanding to calculate for realistic problems. Three approximate posterior models based on approximations of the likelihood model on generalized factorial form are defined. These approximations can be exactly assessed by the forward-backward algorithm. Both a synthetic case and a real seismic inversion case are used in an empirical evaluation. It is concluded that reliable and computationally efficient approximate posterior models for convolutional two-level hidden Markov models can be defined.
Equilibrium Cross-section of Meandering and Braided Rivers
THERE are two simple models which describe the equilibrium state of a channel cross-section with constant conditions, one static, the other dynamic. In the first case equilibrium is said to hold when all grains on the bed are just at the threshold of movement1. Because movement is produced by a combination of fluid drag and the downslope gravity component and because the fluid drag is greatest in the centre of the channel, the channel slopes can increase up the sides to compensate for the decrease in drag. The resulting profile is roughly a parabola1 and is very similar to many natural channel forms. This static model is particularly relevant to the design of some canals where discharges are fairly constant and the bed material is normally immobile. Natural rivers transport sediment, however, and the mean channel form develops at high stages when nearly all the bed material is in intermittent motion. When sediment is transported as bed load it must be continually tending to creep downslope into the centre of the channel under the influence of gravity. If, however, the channel maintains a constant profile there must be an equal and opposite tendency at all points for the sediment to move sideways upslope. Apart from buoyant upthrust, the only other force acting on the grains is the fluid drag acting in the direction of the water flow, leading to the second model. When river channels are in dynamic equilibrium the flow along the bed has a sideways component up the channel banks which exactly compensates the tendency of the sediment to move downslope under the influence of gravity. As water flowing sideways along the bed must return to the channel centre along the surface this implies that the river has a double spiral secondary flow (Fig. 1). Although double spiral flow in straight channels has been demonstrated2,3, single spiral flow in meander bends is better known1. As, however, the sense of the spiral changes from one bend to the next, it follows from considerations of continuity and symmetry that in the straight reaches between bends there is a transitional double spiral structure (Fig. 1 C). In short the secondary flow in river channels is a double spiral which is asymmetrically developed in bends so that one vortex momentarily excludes the other at the crest of the meander. The two spirals adjoin where the flow attaches itself to the bed and diverges sideways; this line is also the thalweg or deepest part of the channel. It is no coincidence that the same hypothesis of double spiral flow is commonly used to explain the origin of many sand ridge bed configurations formed parallel to the flow, such as primary current lineation, sand ribbons, tidal current ridges and longitudinal aeolian dunes and draas4,5. River channels are analogous to the troughs between longitudinal sand ridges in certain other respects too: they arise spontaneously during sediment transport, reach a dynamic equilibrium by interaction with the flow and are associated with transverse structures, forming meander bars, that migrate downstream. In short, there is a good case for saying that the entire form of a river channel is a bed configuration sensu stricto.