The Probability Perturbation Method: A New Look at Bayesian Inverse Modeling

Building of models in the Earth Sciences often requires the solution of an inverse problem: some unknown model parameters need to be calibrated with actual measurements. In most cases, the set of measurements cannot completely and uniquely determine the model parameters; hence multiple models can describe the same data set. Bayesian inverse theory provides a framework for solving this problem. Bayesian methods rely on the fact that the conditional probability of the model parameters given the data (the posterior) is proportional to the likelihood of observing the data and a prior belief expressed as a prior distribution of the model parameters. In case the prior distribution is not Gaussian and the relation between data and parameters (forward model) is strongly non-linear, one has to resort to iterative samplers, often Markov chain Monte Carlo methods, for generating samples that fit the data likelihood and reflect the prior model statistics. While theoretically sound, such methods can be slow to converge, and are often impractical when the forward model is CPU demanding. In this paper, we propose a new sampling method that allows to sample from a variety of priors and condition model parameters to a variety of data types. The method does not rely on the traditional Bayesian decomposition of posterior into likelihood and prior, instead it uses so-called pre-posterior distributions, i.e. the probability of the model parameters given some subset of the data. The use of pre-posterior allows to decompose the data into so-called, “easy data” (or linear data) and “difficult data” (or nonlinear data). The method relies on fast non-iterative sequential simulation to generate model realizations. The difficult data is matched by perturbing an initial realization using a perturbation mechanism termed “probability perturbation.” The probability perturbation method moves the initial guess closer to matching the difficult data, while maintaining the prior model statistics and the conditioning to the linear data. Several examples are used to illustrate the properties of this method.     

Migration dynamics of clupeoids in the Schelde estuary: A stable isotope approach

Large numbers of young of the year herring (Clupea harengus L.) and sprat (Sprattus sprattus (L.)) typically enter and remain within North Sea estuaries during the winter months. The main purpose of this study was to examine their migration dynamics between the North Sea and the Schelde estuary using C and N stable isotopes. Prior to this, stomach contents were used to verify the isotopic differences between the food sources at the sampling stations. From May 2000 to April 2001 fish were collected monthly in the upper and lower estuary. Muscle tissue and stomach contents were analyzed for δ¹³C and δ¹⁵N using an EA-IRMS. Based on the stomach contents, it was demonstrated that δ¹⁵N could not be used as a tracer for fish migration because the longitudinal estuarine δ¹⁵N gradient had reversed completely during autumn. The δ¹³C gradient, however, was found to be reliable for studying fish movement in the Schelde estuary.Seasonal movements of clupeoids in the Schelde estuary were analyzed by separating the temporal abundance patterns into migration groups based on their muscle isotopic composition. Immigration and emigration seem to occur continuously throughout the year. Most exchange occurred in November. During winter, immigration remained high but gradually decreased. Although the herring and sprat abundance further declined in the estuary during February and March, large seaward emigration was not fully demonstrated. As temporal overlap between immigration and emigration is concluded the results support the hypothesis that migration to estuarine nurseries is individually based.     

Climatic and tectonic controls on the fluvial morphology of the Northeastern Tibetan Plateau (China)

The geomorphological evolution of the Northeastern Tibetan Plateau (NETP) could provide valuable information for reconstructing the tectonic movements of the region. And the considerable uplift and climatic changes at here, provide an opportunity for studying the impact of tectonic and monsoon climate on fluvial morphological development and sedimentary architecture of fluvial deposits. The development of peneplain-like surface and related landscape transition from basin filling to incision indicate an intense uplift event with morphological significance at around 10–17 Ma in the NETP. After that, incision into the peneplain was not continuous but a staircase of terraces, developed as a result of climatic influences. In spite of the generally persisting uplift of the whole region, the neighbouring tectonic blocks had different uplift rates, leading to a complicated fluvial response with accumulation terraces alternating with erosion terraces at a small spatial and temporal scale. The change in fluvial activity as a response to climatic impact is reflected in the general sedimentary sequence on the terraces from high-energy (braided) channel deposits (at full glacial) to lower-energy deposits of small channels (towards the end of the glacial), mostly separated by a rather sharp boundary from overlying flood-loams (at the glacial-interglacial transition) and overall soil formation (interglacial). Pronounced incision took place at the subsequent warm-cold transitions. In addition, it is hypothesized that in some strongly uplifted blocks energy thresholds could be crossed to allow terrace formation as a response to small climatic fluctuations (10³–10⁴ year timescale). Although studies of morpho-tectonic and geomorphological evolution of the NETP, improve understanding on the impacts of tectonic motions and monsoonal climate on fluvial processes, a number of aspects, such as the distribution and correlation of peneplain and the related morphological features, the extent and intensity of tectonic movements influencing the crossing of climatic thresholds, leading to terrace development, need to be studied further.     

3D Modeling of Large-Scale Geological Structures by Linear Combinations of Implicit Functions: Application to a Large Banded Iron Formation

Natural Resources Research - Implicit methods for modeling geological structures such as stratigraphy and faults have been developed for more than a decade, and they have made automatic model...     

A Distance-based Prior Model Parameterization for Constraining Solutions of Spatial Inverse Problems

Spatial inverse problems in the Earth Sciences are often ill-posed, requiring the specification of a prior model to constrain the nature of the inverse solutions. Otherwise, inverted model realizations lack geological realism. In spatial modeling, such prior model determines the spatial variability of the inverse solution, for example as constrained by a variogram, a Boolean model, or a training image-based model. In many cases, particularly in subsurface modeling, one lacks the amount of data to fully determine the nature of the spatial variability. For example, many different training images could be proposed for a given study area. Such alternative training images or scenarios relate to the different possible geological concepts each exhibiting a distinctive geological architecture. Many inverse methods rely on priors that represent a single subjectively chosen geological concept (a single variogram within a multi-Gaussian model or a single training image). This paper proposes a novel and practical parameterization of the prior model allowing several discrete choices of geological architectures within the prior. This method does not attempt to parameterize the possibly complex architectures by a set of model parameters. Instead, a large set of prior model realizations is provided in advance, by means of Monte Carlo simulation, where the training image is randomized. The parameterization is achieved by defining a metric space which accommodates this large set of model realizations. This metric space is equipped with a “similarity distance” function or a distance function that measures the similarity of geometry between any two model realizations relevant to the problem at hand. Through examples, inverse solutions can be efficiently found in this metric space using a simple stochastic search method.     

Assessment of Planetary-Boundary-Layer Schemes in the Weather Research and Forecasting Model Within and Above an Urban Canopy Layer

We tested several planetary-boundary-layer (PBL) schemes available in the Weather Research and Forecasting (WRF) model against measured wind speed and direction, temperature and turbulent kinetic energy (TKE) at three levels (5, 9, 25 m). The Urban Turbulence Project dataset, gathered from the outskirts of Turin, Italy and used for the comparison, provides measurements made by sonic anemometers for more than 1 year. In contrast to other similar studies, which have mainly focused on short-time periods, we considered 2 months of measurements (January and July) representing both the seasonal and the daily variabilities. To understand how the WRF-model PBL schemes perform in an urban environment, often characterized by low wind-speed conditions, we first compared six PBL schemes against observations taken by the highest anemometer located in the inertial sub-layer. The availability of the TKE measurements allows us to directly evaluate the performances of the model; results of the model evaluation are presented in terms of quantile versus quantile plots and statistical indices. Secondly, we considered WRF-model PBL schemes that can be coupled to the urban-surface exchange parametrizations and compared the simulation results with measurements from the two lower anemometers located inside the canopy layer. We find that the PBL schemes accounting for TKE are more accurate and the model representation of the roughness sub-layer improves when the urban model is coupled to each PBL scheme.     

The fluvial system — Research perspectives of its past and present dynamics and controls

During the conference “The fluvial system — past and present dynamics and controls" held at the Department of Geography of Bonn University from 16 to 22 of May 2005 the participants organised in 12 international organisations working in the fluvial environment were asked about their opinions about the main aspects to be considered for sustainable progress in future research projects. The individual comments can be grouped by the following headlines: integration and application of experiences, considering system analytical approaches, considering effects of climate and global change, interdisciplinary work, regarding extreme events and their frequencies and quantification of human impact. Detailed explanations and selected references of previous studies initially considering the mentioned aspects are given as a review.