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Pre-stack or angle stack gathers are inverted to estimate pseudo-logs at every surface location for building reservoir models. Recently several methods have been proposed to increase the resolution of the inverted models. All of these methods, however, require that the total number of model parameters be fixed a priori. Here, we investigate an alternate approach in which we allow the data to choose model parameterization. In other words, in addition to the layer properties, the number of layers is also treated as a variable in our formulation. Such trans-dimensional inverse problems are generally solved by using the Reversible Jump Markov Chain Monte Carlo (RJMCMC), which is an effective tool for model exploration and uncertainty quantification. Here we introduce a new gradient based RJMCMC, called the Hamiltonian Monte Carlo, where the model perturbations are generated according to the birth-death approach. Model updates are computed using gradient information and the Metropolis-Hastings criterion is used for model acceptance. We have applied this technique to pre-stack (angle stack) AVA inversion for estimating acoustic and shear impedance profiles. Our results demonstrate that RJHMC converges rapidly and can be a practical tool for inverting seismic data.
The process of modeling the earth’s interior from the observed data is known as ‘inversion’. We use seismic recording for inversion so as to indirectly infer the subsurface structure and various processes inside the earth. Our observations are deficient of complete information, which leads to non-uniqueness in the solution. Most of the inverse algorithms assume fixed model dimension, which is perhaps the least known of all parameters.
In an inverse problem it is very important to determine exact model parametrization, i.e. the number of model parameters, to be consistent with resolving power of the data. Using too few parameters can lead to under-fitting the data, estimating biased parameters and under-estimating parameter uncertainties. On the other hand, considering too many model parameters can over-fit the data, which leads to estimating under-determined parameters with enormous uncertainty (Dosso et al., 2014). Thus it makes sense to make the number of model parameters itself a parameter to be solved for. Green (1995) introduced a new framework for construction of reversible Markov chain sampler, which can jump between the dimensionality of parameter sub space making it very flexible and entirely constructive. The problem with conventional reversible jump Markov chain sampler is that it is very slow and therefore not practical for routine analysis. Here we have proposed a new method which combines the conventional reversible jump sampler with a gradient based Monte Carlo method named Hamiltonian Monte Carlo.
This paper was prepared for the Northern Plains Section Regional Meeting of the Society of Petroleum Engineers of AIME, to be held in Omaha, Neb., May 18-19, 1972. Permission to copy is restricted to an abstract of not more than 300 words. Illustrations may not be copied. The abstract should contain conspicuous acknowledgment of where and by whom the paper is presented. Publication elsewhere after publication in the JOURNAL OF PETROLEUM TECHNOLOGY or the SOCIETY OF PETROLEUM ENGINEERS JOURNAL is usually granted upon request to the Editor of the appropriate journal provided agreement to give proper credit is made.
Discussion of this paper is invited. Three copies of any discussion should be sent to the Society of Petroleum Engineers Office. Such discussions may be presented at the above meeting and, with the paper, may be considered for publication in one of the two SPE magazines.
This paper describes two illustrative examples demonstrating the application of Monte Carlo techniques to gas process design. In addition, a detailed discussion is presented of the characteristics of the Monte Carlo procedure. In the first example Monte Carlo procedure. In the first example Monte Carlo simulation is used to determine the distribution of required MEA circulation rates to sweeten gas in a proposed installation. A simple problem formulation is used; therefore, it is possible to test the sensitivity of the calculated circulation rate distribution to the number of Monte Carlo events and to the selected probability distribution. In the second example, the probability distribution. In the second example, the optimum size of a lean oil absorption plant is determined. Feed rate and composition together with product prices are selected as random variables. Distributions of present-value profit are generated for several plant sizes. profit are generated for several plant sizes. Optimum plant size is selected to maximize the mean present-value profit.
In designing gas processing facilities, plant size or capacity must be established prior plant size or capacity must be established prior to determining process configuration and operating conditions. Establishing the optimum capacity is complicated in that plant inputs and economic constraints, such as feed volume and composition, product price, and equipment operation, may not be known with certainty. This paper demonstrates use of Monte Carlo techniques in gas plant design problems when system inputs are uncertain.
To use Monte Carlo techniques in analyzing gas processing projects, plant inputs and economic constraints that characterize the project must first be defined. These project project must first be defined. These project parameters fall into two classes: deterministic parameters fall into two classes: deterministic and stochastic variables. Deterministic variables have a known value, but can vary with time (e.g., the gas price in a long-term gas sales contract). Stochastic variables are not known with certainty and are described by a probability distribution spanning a range of probability distribution spanning a range of values. For most variables a triangular distribution adequately relates the value of a stochastic variable to its minimum, maximum and expected value.
Next, appropriate yardsticks for comparing investment decisions must be selected. Present-value profit and investor's interest rate are Present-value profit and investor's interest rate are commonly used economic yardsticks. It may be preferred to use key process variables, plant preferred to use key process variables, plant investment, or product output to characterize performance so that the result may be used in a performance so that the result may be used in a subsequent economic analysis.
Monte Carlo simulation is a process of running a model numerous times with a random selection from the input distributions for each variable. The results of these numerous scenarios can give you a "most likely" case, along with a statistical distribution to understand the risk or uncertainty involved. Computer programs make it easy to run thousands of random samplings quickly. Monte Carlo simulation begins with a model, often built in a spreadsheet, having input distributions and output functions of the inputs. The following description is drawn largely from Murtha.