A Bayesian approach to inverse modelling of stratigraphy, part 1: method

The inference of ancient environmental conditions from their preserved response in the sedimentary record still remains an outstanding issue in stratigraphy. Since the 1970s, conceptual stratigraphic models (e.g. sequence stratigraphy) based on the underlying assumption that accommodation space is the critical control on stratigraphic architecture have been widely used. Although these methods considered more recently other possible parameters such as sediment supply and transport efficiency, they still lack in taking into account the full range of possible parameters, processes, and their complex interactions that control stratigraphic architecture. In this contribution, we present a new quantitative method for the inference of key environmental parameters (specifically sediment supply and relative sea level) that control stratigraphy. The approach combines a fully non‐linear inversion scheme with a ‘process–response’ forward model of stratigraphy. We formulate the inverse problem using a Bayesian framework in order to sample the full range of possible solutions and explicitly build in prior geological knowledge. Our methodology combines Reversible Jump Markov chain Monte Carlo and Simulated Tempering algorithms which are able to deal with variable‐dimensional inverse problems and multi‐modal posterior probability distributions, respectively. The inverse scheme has been linked to a forward stratigraphic model, BARSIM (developed by Joep Storms, University of Delft), which simulates shallow‐marine wave/storm‐dominated systems over geological timescales. This link requires the construction of a likelihood function to quantify the agreement between simulated and observed data of different types (e.g. sediment age and thickness, grain size distributions). The technique has been tested and validated with synthetic data, in which all the parameters are specified to produce a ‘perfect’ simulation, although we add noise to these synthetic data for subsequent testing of the inverse modelling approach. These tests addressed convergence and computational‐overhead issues, and highlight the robustness of the inverse scheme, which is able to assess the full range of uncertainties on the inferred environmental parameters and facies distributions.     

A transformational approach to geophysical inverse problems

A set of coordinate transformations is used to linearize a general geophysical inverse problem. Statistical and analytic techniques are employed to estimate the parameters of such linearization transformations. In the transformed space, techniques from linear inverse theory may be utilized. Consequently, important concepts, such as model parameter covariance, model parameter resolution and averaging kernels, may be carried over to non-linear inverse problems. I apply the approach to a set of seismic cross-borehole traveltimes gathered at the Conoco Borehole Test Facility. the seismic survey was conducted within the Fort Riley formation, a limestone with thin interbedded shales. Between the boreholes, the velocity structure of the Fort Riley formation consists of a high-velocity region overlying a section of lower velocity. It is found that model parameter resolution is poorest and spatial averaging lengths are greatest in the underlying low-velocity region.     

A GIS approach to modelling CO2 emissions associated with the pupil-school commute

Concerns have been raised in numerous countries over declining rates of active transport to school. In a UK context, the pupil-school commute is estimated to contribute around 658 kilotonnes of CO₂ per year; however, tackling this issue effectively requires an improved understanding of how emissions can be modelled and mapped over a variety of scales. This paper implements a new estimation technique for the modelling of CO₂ emissions linked with the school commute that integrates both transport network-level routing and geographically disaggregate vehicle emissions data. The model is then applied to a national cohort of pupils in England. Areas demonstrating the highest emissions were typically more rural and/or comprising more affluent resident populations. Emissions were also shown to increase with school year, with larger step changes between educational stages reflecting the different geography of school locations. Furthermore, where secondary school entry policies were selective or based on a religious domination, average emissions were typically higher than in non-selective schools.     

A continued fraction approach to the inverse problem of electrical conductivity

Summary. The problem of determination of the electrical conductivity of the Earth from the geomagnetic induction data is formulated as that of finding the coefficients of the continued fraction expansion of a certain rational fraction representation of the total response (or impedance) of the medium at the surface when this quantity is given for N frequencies. The coefficients of expansion are related to the conductivities of N layers of constant attenuation, where, within each layer the conductivity is assumed to be constant. Thus in this approach the conductivity profile resulting from the inversion of the response function is given as a series of step functions.     

A Bayesian approach to estimating tectonic stress from seismological data

Earthquakes are conspicuous manifestations of tectonic stress, but the non-linear relationships between the stresses acting on a fault plane, its frictional slip, and the ensuing seismic radiation are such that a single earthquake by itself provides little information about the ambient state of stress. Moreover, observational uncertainties and inherent ambiguities in the nodal planes of earthquake focal mechanisms preclude straightforward inferences about stress being drawn on the basis of individual focal mechanism observations. However, by assuming that each earthquake in a small volume of the crust represents a single, uniform state of stress, the combined constraints imposed on that stress by a suite of focal mechanism observations can be estimated. Here, we outline a probabilistic (Bayesian) technique for estimating tectonic stress directions from primary seismological observations. The Bayesian formulation combines a geologically motivated prior model of the state of stress with an observation model that implements the physical relationship between the stresses acting on a fault and the resultant seismological observation. We show our Bayesian formulation to be equivalent to a well-known analytical solution for a single, errorless focal mechanism observation. The new approach has the distinct advantage, however, of including (1) multiple earthquakes, (2) fault plane ambiguities, (3) observational errors and (4) any prior knowledge of the stress field. Our approach, while computationally demanding in some cases, is intended to yield reliable tectonic stress estimates that can be confidently compared with other tectonic parameters, such as seismic anisotropy and geodetic strain rate observations, and used to investigate spatial and temporal variations in stress associated with major faults and coseismic stress perturbations.     

Multistate supernetwork approach to modelling multi-activity,multimodal trip chains

Least-costs path finding methods belong to the standard toolkit of geographical information science and allow one to assign trips with known origin and destination to a transport network. An extension of the network representation is proposed that allows one to model complete trip chains (tours) that may involve multiple transport modes and multiple activities, as a least-cost path through the network. The extension is based on the notion that link costs may change as a consequence of activity and vehicle choices during the tour. The extended network or supernetwork contains as many copies of the network as there are possible activity–vehicle states and additional state transition links to interconnect them. In this way, not only the generalized link cost function, but also the start and end points of a path can be defined state dependently. A least-cost path through a multistate network defines an optimal sequence of travelling, transferring, parking, conducting activities and dropping off products of activities. An application shows how a multistate network can be constructed and used to assess the impacts of network design variables on path choice.     

Shallow-marine sequences as the building blocks of stratigraphy: insights from numerical modelling

Two types of depositional sequences can be defined within the sequence stratigraphic framework: the parasequence and the high‐frequency sequence. Both sequences consist of stacked regressive and transgressive deposits. However, a parasequence forms under conditions of overall sea‐level rise, whereas a high‐frequency sequence forms as the sea level oscillates which results in typical forced regressive deposits during sea‐level fall. Both depositional sequences may develop over comparable temporal (10–100 kyr) and spatial (1–20 km wide and 1–40 m thick) scales. Numerical modelling is used to compare the architecture, preservation potential, internal volumes, bounding surfaces, condensed and expanded sections and facies assemblages of parasequences and high‐frequency sequences. Deposits originating from transgression are less pronounced than their regressive counterparts and consist of either preserved backbarrier deposits or shelf deposits. Shoreface deposits are not preserved during transgression. The second half of the paper evaluates in detail the preservation potential of backbarrier deposits and proposes a mechanism that explains the occurrence of both continuous and discontinuous barrier retreat in terms of varying rates of sea‐level rise and sediment supply. The key to this mechanism is the maximum washover capacity, which plays a part in both barrier shoreline retreat and backbarrier‐lagoonal shoreline retreat. If these two shorelines are not balanced, then the retreat of the coastal system as a whole is discontinuous and in time barrier overstep may take place.     

A three-dimensional approach to fault seal analysis: fault-block juxtaposition & argillaceous smear modelling

Three‐dimensional (3D) numerical modelling of fault displacement enables the building of geological models to represent the complex 3D geometry and geological properties of faulted sedimentary basins. Using these models, cross‐fault juxtaposition relationships are predicted in 3D space and through time, based on the geometries of strata that are cut by faults. Forward modelling of fault development allows a 3D prediction of fault‐zone argillaceous smear using a 3D application of the Shale Gouge Ratio. Numerical models of the Artemis Field, Southern North Sea, UK and the Moab Fault, Utah, USA are used to demonstrate the developed techniques and compare them to traditional one‐ and two‐dimensional solutions. These examples demonstrate that a 3D analysis leads to significant improvements in the prediction of fault seal, the analysis of the interaction of the sealing properties of multiple faults, and the interpretation of fault seal within the context of sedimentary basin geometry.     

农户土地规模与生产率的关系及其解释的印证——以吉林省为例

依据原中共中央政策研究室和农业部的农户固定观察点数据,利用数理模型对吉林省农户土地规模-生产率之间的关系进行了验证。结果表明:(1)农户土地规模-土地生产率之间的关系并不是简单的直线关系,当农户经营的土地面积小于30亩时,负向关系并不明显,但农户经营土地的规模超过30亩后,两者之间则呈现出明显的负向关系;(2)吉林省大规模农户土地产出率较低的主要影响因素为化肥投入,劳动投入因素的影响位列次席,这与前人的研究有所差别;(3)农户劳动力的影子工资率与农户土地规模存在正向关系。鉴于目前吉林省大部分农户的经营规模在20亩以下,因此笔者建议鼓励土地流转,进一步推动耕地的适度经营规模;大规模农户可适当地增加化肥和劳动投入量。     

A LiDAR-based flood modelling approach for mapping rice cultivation areas in Apalit,Pampanga

Majority of rice cultivation areas in the Philippines are susceptible to excessive flooding owing to intense rainfall events. The study introduces the use of fine scale flood inundation modelling to map cultivation areas in Apalit, a rice-producing municipality located in the province of Pampanga in the Philippines. The study used a LiDAR-based digital elevation model (DEM), river discharge and rainfall data to generate flood inundation maps using LISFLOOD-FP. By applying spatial analysis, rice cultivation zone maps were derived and four cultivation zones are proposed. In areas where both depth and duration exceed threshold values set in this study, varieties tolerant to stagnant flooding and submergence are highly recommended in Zone 1, where flood conditions are least favorable for any existing traditional lowland irrigation varieties. The study emphasizes that a decline in yield is likely as increasing flood extents and longer submergence periods may cause cultivation areas for traditional irrigated lowland varieties to decrease over time. This decrease in yield may be prevented by using varieties most suitable to the flooding conditions as prescribed in the rice zone classification. The method introduced in this study could facilitate appropriate rice cultivation in flood-prone areas.     

Simple equations of sedimentation: applications to sequence stratigraphy

Abstract An equation to relate the thickness of sediment deposited (ΔSed), eustatic sea-level change (ΔE), and subsidence (ΔSub), to changes in depth of water (ΔD) is: ΔSub +ΔE-ΔSed =ΔD.
Using existing sea-level curves, the equation shows that some transgressive-regressive sequences in a foreland basin and a composite seismic facies sequence on a passive margin cannot result solely from eustatic variation. In each case, the space created by subsidence is greater than that provided by eustatic rise. However, eustatic variation could have triggered sequence development if superimposed on a basin with relatively constant values of the other parameters. Short-period sea-level fluctuations with high rates of change, exceeding 70–100 m Myr^-1 for periods less than 2–3 Myr, affect the stratigraphy and sedimentology more than longer period, higher amplitude variations.
Clinoforms are generated because of lateral variations in sedimentation rate compared to the rate of creation of accommodation space. These variations may result from differing sedimentation rates, subsidence rates, or rates of eustatic change, superimposed on a basin with lateral sediment supply. Clinoform slopes and curvatures are interpre table in terms of these variables as well as the type of sediment supplied and the energy distribution in the basin.
These equations put some well-known geological principles on a simple quantitative basis. They force precision in definition of variables, and may lead to further development of quantitative techniques in stratigraphy and sedimentology.     

Tectonic modelling of the Middle Jurassic synrift stratigraphy in the Oseberg–Brage area, northern Viking Graben

A finite difference model, allowing for episodic movements along different faults, is used to examine the effect of tectonics on the stratigraphic signature in the Oseberg–Brage area in the northern Viking Graben. Constraints are provided by local exploration and production well data and 3-D seismic coverage, and a regional depth-converted seismic line.
In the modelling, we focus on the influence of varying rates of fault movement on stratigraphic signatures such as upflank unconformities and changes in layer thickness. We couple the basinwide features of the northern Viking Graben with the fault-block-scale features of the Oseberg–Brage area by using parameter constraints derived by large-scale modelling as input for the local-scale model. In addition, subsidence patterns resulting from the basinwide model were used as background subsidence for the fault block model of the Oseberg–Brage area.
The model results indicate that the alternating activation of different faults with varying extension rates can cause stratigraphic features such as unconformities, condensation and onlap/offlap patterns. Onlap occurs during periods of low extension rates. An increase in extension rate along a fault causes footwall uplift, resulting in condensation or upflank erosion yielding unconformities. This influence can also affect sub-basins further away from the fault. Downdip layer thickening reflects the local tilting of fault blocks.
The coupling of the local and regional scales turns out to be essential in explaining the stratigraphy of the Oseberg–Brage area: basinward and, notably, updip layer thickening as observed on some of the fault blocks can only be explained by activity of the boundary fault on the opposing, western margin of the northern Viking Graben.
     

A whole-basin, mass-balance approach to paleolimnology

Lake sediments record the flux of materials (nutrients, pollutants, particulates) through a lake system both qualitatively, as changes in the composition of geochemical and biological tracers, as well as quantitatively, through changes in their rate of burial. Burial rates provide a direct link to contemporary (neo-) limnological studies as well as management efforts aimed at load reductions, but are difficult to reconstruct accurately from single cores owing to the spatial and temporal variability of sediment deposition in most lakes. The accurate determination of whole-lake burial rates from analysis of multiple cores, though requiring more effort per lake, can help resolve such problems and improve our understanding of sediment heterogeneity at multiple scales. Partial solutions to these problems also include focusing corrections based on ²¹⁰Pb flux, co-evaluation of concentration profiles, trend analysis using multiple lakes, and trend replication based on a small number of cores from the same lake. Recent multi-core studies demonstrate that no single core site faithfully records the whole-lake time-resolved input of materials, but that as few as five well-placed cores can provide a reliable record of whole-lake sediment flux for morphometrically simple basins. Lake-wide sediment fluxes can be coupled with reconstructed outflow losses to calculate historical changes in watershed and atmospheric loading of nutrients, metals, and other constituents. The ability of paleolimnology to accurately assess the sedimentary flux and extend the period of reference into the distant past represents an important contribution to the understanding of biogeochemical processes and their response to human and natural disturbance.     

Simulation of soil erosion and deposition in a changing land use: A modelling approach to implement the support practice factor

Using the USPED (Unit Stream Power Erosion Deposition) model, three land use scenarios were analysed for an Italian small catchment (15 km²) of high landscape value. The upper Orme stream catchment, located in the Chianti area, 30 km south of Florence, has a long historical agriculture record. Information on land use and soil conservation practices date back to 1821, hence offering an opportunity to model impacts of land use change on erosion and deposition. For this study, a procedure that takes into account soil conservation practices and potential sediment storage is proposed. The approach was to calculate and model the flow accumulation considering rural and logging roads, location of urban areas, drainage ditches, streams, gullies and permanent sediment sinks. This calculation attempts to assess the spatial variability, especially the impact of support practices (P factor). Weather data from 1980–2003 were taken into account to calculate the R factor. However, to consider the intense pluviometric conditions in terms of the erosivity factor, the 0.75th quantile was used, while the lowest erosivity was modelled using the 0.25th quantile. Results of the USPED model simulation show that in 1821 the mean annual net erosion for the watershed was 2.8 Mg ha^− 1 y^− 1; in 1954 it was 4.2 Mg ha^− 1 y^− 1; and in 2004 it was 5.3 Mg ha^− 1 y^− 1. Conservation practices can reduce erosion processes by ≥ 20 Mg ha^− 1 y^− 1 when the 1821 practices are introduced in the present management. On the other hand, if the support practices are not considered in the model, soil erosion risk is overestimated. Field observation for the present-day simulation confirmed that erosion and associated sediment deposition predicted by the model depend, as expected, on geomorphology and land use. The model shows limitations that are mainly due to the input data. A high resolution DEM is essential for the delineation of reliable topographic potential to predict erosion and deposition especially in vineyards.