- [Yaming] Let's study Bayesian Updating Simulation definition, input data for Bayesian Updating Simulation, steps of Bayesian Updating Simulation, example of Bayesian Updating Simulation, at the end, let's make a summary. Bayesian Updating Simulation is to simulate discrete variables like facies. It provides a straightforward framework for integrating various secondary grid data. For example, seismic data. Indicator kriging is used to generate the conditional probability distribution of each facies from hard data alone, and then Bayesian Updating modifies or updates the probabilities as follows: This is the formula to update the computing of probabilities where i star star or k equal to one to K are the updated probabilities for simulation. I star or k equal to one to K are the probabilities for simulation. P is the seismic derived probability of facies k at the location u. Pk is overall proportion of facies k. C is the normalization constant to ensure that the sum of the final probability is one. The factor p divided by pk operates to increase or decrease the probability depending on the difference of calibrating facies proportion from the global proportion. It uses input data and simulated data when computing a value at an un-simulated grid cell. It generates many equally probable realizations which can be post-processed to quantify and assess uncertainty. It preserves the variance observed in the data, instead of just as the mean value, as in interpolation. Input data are conditional data, global facies proportion table, secondary grid data, indicator variogram parameters, search parameters, seed, and number of realizations. Steps are, one, transform original data into facies. Two, assign transformed data into simulation grid, Three, prepare secondary grid data. Four, prepare global facies proportion table. Five, generate a random path through the grid cells. Six, apply binary indicator transform to each facies. Seven, prepare variograms. Eight, define search region. Nine, visit a node in the random path and use indicator kriging to estimate a mean for each indicator variable which are updated by secondary grid data and used to build local conditional probability distribution. 10, select at random a value from the local conditional probability distribution based on indicator threshold and set the node value to that number. 11, include the newly simulated value as part of the conditional data. 12, repeat steps nine to 11 until all grid cells have values. It records the input data at discrete indicator variables. In this example, original data, from net to gross. If the threshold is zero point three net to gross which is equal to or less than zero point three where we transformed that to shale that is zero. Please say zero point zero eight, zero point one two, and zero point one one are transformed into zero. A net to gross, which is a larger than zero point three, will be transformed to sand that is one. Please say zero point eight, and zero point six eight are transformed into one. This is an example of two indicators, the procedures are similar for more than two indicators. Assign transformed data into simulation grid which will be used as conditioning data. While conditioning data are assigned linear to the grid is their centers. Prepare the secondary grid data. Here are Probability Map of Sand, under, Probability Map of Shale. Input data table is copied here. This is a global facies proportion table. The facies cause of input data are computed and are shown in the table. They are three zeros, they are two ones the facies proportion values of input data are computed and shown in the table. Three divided by five is zero point six. Two divided by five is zero point four. Remember these two numbers, we are going to use them later. The facies proportion values are editable, which can be modified based on the secondary grid data. They are used as global facies proportions in simulation when the variogram correlation fingers are short and the control data are scaled, the final results might reach the input global facies proportions. A random path is used to avoid the artifacts induced by wasting grid cells in a regular facie. The random path must go through each cell while under in random order. This is an example of a random path, for n x equal to eight, n y equal to seven grid. The order is from first unto second it's staggered until 56. Take a look at the first grid cell, i equal to two, j equal to four. Apply binary indicator transform to each facies, i is equal to 100 percent if facies k presents at location u alpha, otherwise, i is equal to zero if facies k does not present at location u alpha. This is to apply binary indicator transform to sand. The top is the assigned data map, one is sand, zero is shale. Wherever shale presents, its probability is 100 percent, see two gray eclipses. Otherwise if shale its probability is zero percent, see two blue eclipses. The bottom is probability map of sand to apply binary indicator transform to shale. The top is the assigned data map, one is sand and zero is shale. Wherever shale presents its probability is 100 percent, see two blue eclipses. Otherwise it's sand, it's probability is zero percent, see two gray eclipses. The bottom is probability map of shale. Prepare indicator variogram for sand, which is gamma one. Prepare indicator variogram for shale, that's gamma zero. First search region in 2D, it is eclipse. That was the first un-simulated cell, that is i equal to two, j equal to four. Applying search region, is to put the eclipse center to the selected cell center on probability map of sand and put the eclipse center to the selected cell center on probability map of shale. Obtain data in search region on probaility map of sand, under, obtain data in search region on probability map of shale. Let's work on that. Run indicator kriging on probability map of sand, under, run indicator kriging on probability map of shale. Indicator kriging gives us an estimate of mean for each facies. What is the probability of this facies? Using conditional data indicator variogram parameters, on the search parameters the estimator of standard deviation is not in use. We get the mean of sand is 18 percent, the mean of shale is 67 percent. Review Bayesian Updating Equation. Get probabilities of secondary data at the first un-simulated grid cell. Probability of sand is equal to 21 percent, probability of shale is equal to 79 percent. Let's update 18 percent using p equal to 21 percent and pk equal to 40 percent, we get nine point five percent. It's similar, we get 88 percent for shale. Secondary greater facies proportion for the cell are input global facies proportions that are assigned to the cell where they are now conditional data obtained in search region. Sort and rescale computed means zero point five percent, that is zero point zero nine five and 88 percent, that is zero point eight eight. Sorting mu sand zero point zero nine five and mu shale zero point eight eight from shale to sand. That is the first mu shale zero point eight eight, the second is mu sand zero point zero nine five. Rescale them into zero to one cumulative probability space by zero point eight eight divided by zero point eight eight plus zero point zero nine five equal to zero point nine zero, that is 90 percent, this is rescaled mu shale. Next zero point zero nine five divided by zero point eight eight plus zero point zero nine five equal to zero point one, that is 10 percent. This is rescaled mu sand. Update to the rescaled results use mu equal to zero point nine zero and mu equal to zero point one zero. Construct a cumulative probability distribution. Run uniform random number generator which presents a probability narrow. We get probability zero point seven one. Follow the arrowed gray lines, we get indicator value zero. Include the simulated value x equal to 35, y equal to 55, facies zero. In the set of conditional data this suggests the closely spaced values have the crack to show scale corrolation. The second node marked as two is not yet simulated the similar procedures as the first node will be conducted, one difference is the simulated value zero, shown in the purple circle, can be used as the conditional data if it is in the search region. In order to preserve the proper covariance structure between simulated values. Repeat steps nine to 11 until all grid cells have simulated values. Let's make a summary. Bayesian updating simulation method is to simulate facies distribution using facies probabilities computed by indicator kriging and modified by secondary grid data. It is more simple than collocated cokriging and very practical when accounting for seismic data. For details please refer to SPE paper, "Bayesian Updating Simulation of Channel Sands From 3-D Seismic Data in the Oseberg Field, Norway North Sea" by Doyen. This is the end of the lecture.