Akaike Bayesian Information Criterion Research Articles

Smooth 2-D Inversion for Magnetotelluric Data Based on Statistical Criterion ABIC.

We often apply a smoothness constraint to a two-dimensional (2-D) linearized least-squares inversion of magnetotelluric (MT) data to achieve a stable result. A substantial problem with this scheme lies in the choice of the optimum smoothness, and in this paper, a statistical approach which is very versatile for this purpose is presented. It uses a statistical criterion called ABIC (Akaike's Bayesian Information Criterion), which was derived by introducing the entropy-maximization theorem into the Bayes statistics. On applying the Bayesian procedure to 2-D MT inversion, we seek simultaneous minimization of data misfit and model roughness. ABIC works as a number that represents goodness, or an entropy, of a model in a sense of this simultaneous minimization. Tests with both synthetic and real field data have revealed the effectiveness of this method for non-linear inversion problems. Regardless of the magnitude of the observation error, the method objectively adjusts the tradeoff between the misfit and the roughness, and stable convergence is attained.

赤池のベイズ型情報量規準を用いた種雄牛評価モデルにおける母数効果の選択

赤池のベイズ型情報量規準を用いた種雄牛評価モデルにおける母数効果の選択

Inversion of COPROD2 Magnetotelluric Data by Use of ABIC Minimization Method.

A smooth two-dimensional least-squares inversion was applied to the COPROD2 magnetotelluric (MT) data provided by the Geological Survey of Canada. Selection of the optimum smoothness at each iteration was based on a statistical criterion ABIC (Akaike's Bayesian Information Criterion), which was derived from the Bayes statistics and the entropy-maximization theorem. Six frequencies, below 2 Hz, were picked for the inversion. Both TM and TE mode data at 35 sites were used for the weighted inversion in which the weights were calculated from the observation errors listed in the data file. The MT responses and their partial derivatives with respect to model parameters were calculated with a finite-element method. The inverted model basically shows a three-layer earth: conductive, resistive, and conductive from the surface. Two major conductive anomalies are recognized in the resistive layer of 100 km thickness. One is interpreted as the North American Central Plains (NACP) anomaly, and the other corresponds to the Thompson Belt (TOBE) anomaly.

Three-dimensional topography of the sediment/basement interface in the Tokyo metropolitan area, central Japan

Abstract An extensive seismic refraction experiment was conducted in the Tokyo metropolitan area, central Japan, to reveal the three-dimensional basin structure in this area. Sixty-three explosions were fired over 14 years and more than 2000 travel times were picked for several phases. For our inversion, the basement is divided into two homogeneous regions where the sediment is assumed to be of uniform velocity. The interface separating the sediment and basement is then parameterized by bi-cubic B-splines and a smoothness constraint is imposed to avoid oscillatory artifacts in the solution. The relative weight of the constraint to data residuals is determined by minimizing Akaike's Bayesian information criterion. About 1500 travel times of refracted P waves are inverted tomographically, and the interface shape determined agrees well with the results of other geophysical surveys such as borehole measurements and gravimetric observations. The basement velocities in the northern and southern regions are 5.7 and 5.4 km/sec, respectively. Previous two-dimensional analyses suggested there is a 4.5 km/sec layer in the uppermost part of the southern basement, but we ignore it for simplicity so that estimated depths are likely underestimated by 1 ∼ 2 km in the southern region. Nevertheless, this study provides the first detailed image of the basin structure in the area, which will help strong motion and earthquake hazard studies as additional constraints.

Geodetic data inversion using a Bayesian information criterion for spatial distribution of fault slip

SUMMARY We developed a new inversion method to reconstruct static images of seismic sources from geodetic data, using Akaike’s Bayesian Information Criterion (ABIC). Coseismic surface displacements are generally related with a slip distribution on a fault surface by linear integral equations. Parametric expansion of the fault slip distribution by a finite number of known basis functions yields a set of observation equations expressed in a simple vector form. Incorporating prior constraints on the smoothness of slip distribution with the observation equations, we construct a Bayesian model with unknown hyperparameters. The optimal values of the hyperparameters, which control the structure of the Bayesian model, are objectively determined from observed data by using ABIC. Once the values of hyperparameters are determined, we can use the maximum likelihood method to find the optimal distribution of fault slip. We examined the validity of this method through a numerical experiment using theoretical data with random noise. We analysed geodetic data associated with the 1946 Nankaido earthquake (Ms = 8.2) by using this method. The result shows that the fault slip distribution of this earthquake has two main peaks of 4 and 6 m, located off Kii Peninsula and Muroto Promontory. These two high-slip areas are clearly separated by a low-slip zone extending along Kii Strait. Such a slip distribution corresponds with the fact that the rupture process of this earthquake in the western part is notably different from that in the eastern part.

Changes in the magnitude—frequency b-value prior to large ( M ⩾ 6.0) earthquakes in Japan

On the basis of Akaike's Bayesian information criterion (ABIC), a new method of estimating temporal and spatial variations in the magnitude-frequency relation ( b-value) is developed. The space-time volume studied is divided into a number of segments having equal volumes, for which b-values are estimated. The smoothness of the estimated b-value is guaranteed by the introduction of a prior distribution. The parameters of the prior distribution are chosen in such a way as to minimize the ABIC. This method is employed to study the temporal and spatial variations of b-values for microearthquake activity in the Kanto, Tokai and Tottori areas, where seven earthquakes of M ⩾ 6.0 have recently occurred. The temporal variation of b-values for the source region of the western Nagano earthquake (Sep. 1984, M 6.8) in the Tokai area shows a decrease in b-values about 2 years before the occurrence of the main shock. Similar findings before the southwestern Ibaraki (Feb. 1983, M 6.0), the southern Ibaraki (Oct. 1985, M 6.0) and the Chiba (Dec. 1987, M 6.7) earthquakes in the Kanto area, and the Tottori earthquake (Oct. 1984, M 6.2) in the Tottori area were obtained. These findings are confirmed by a precursor analysis (PA), in which effects of aftershocks are eliminated. Using a quantitative definition, temporal variations of b-values are classified into three groups: increase, decrease and no change. The space-time volume of decrease accounts for approximately 10% of the total volume. For a period before the occurrence of earthquakes, the decrease appears more frequently than might be expected from random chance. The results suggest that the decrease in b-values is a promising candidate to act as a precursory phenomena.

An application of Bayesian (ABIC) smoothing methods to estimating space and time variations in the magnitude distributions of earthquakes

On the basis of Akaike's Bayesian information criterion (ABIC), a new method of estimating temporal and spacial variations in the magnitude-frequency relation (b value) is developed. The space-time volume studied is divided into a number of segments having equal volume, for which b values are estimated. The smoothness of the estimated b value is guaranteed by the introduction of a prior distribution. The parameters of the prior distribution are chosen in such a way as to minimize the ABIC. This method is employed to study the temporal and spatial variations of b values for microearthquake activity in the Kanto, Tokai and Tottori areas, where seven earthquakes having magnitudes of M6.0 and larger have recently taken place. States of temporal variations of b values are classified into three groups; increase, decrease and nochange. The space-time volume of decrease accounts for approximately 10% of the total volume. For a period before the occurrence of earthquakes, the decrease appears more frequently than might be expected. It is concluded that the decrease in the b value is a promising candidate to act as a precursory phenomena.

On the Optimum Design of a Smoothing Filter for Geophysical Tomography

An investigation is made on the optimum design of a smoothing filter that may be applied in order to reconstruct a subsurface image based on the computer tomography technique (or so-called geotomography). Such a smoothing filter is especially necessary in geotechnical applications where the angle of projection is usually severely restricted. However, hitherto, the most intractable problem encountered when applying a smoothing filter is how to adjust the relative power of the smoothing filter to the observed data. The Akaike-Bayesian Information 'Criterion (ABIC), which was proposed by Akaike (1980), is applied in order to solve this problem. Some numerical experiments are carried out to examine the effectiveness of the proposed method. It is concluded that the proposed method is very effective in the process of selecting the optimum smoothing filter.

Simultaneous inversion of waveform and geodetic data for the rupture process of the 1984 Naganoken-Seibu, Japan, earthquake

SUMMARY A simultaneous inversion method of strong motion seismograms and geodetic data for inferring earthquake rupture process was developed. We assumed a source model composed of many point-sources of which slip vectors and rupture times are unknown parameters. In addition to observed data, we introduced smoothing constraints to determine many parameters without instability. Since the leastsquares solution depends on the relative weights of the smoothing constraints to a data misfit function, we determined their values by minimizing Akaike’s Bayesian information criterion (ABIC). By using ABIC, we can objectively choose a unique model. This inversion method was applied to the Naganoken-Seibu, Japan, earthquake of 1984. Dividing a 15 x 10 km fault into 150 subfaults, we obtained the minimum ABIC model with a seismic moment of 2.7 X los dyne cm. This model has a slip peak in the eastern and shallow area of the assumed fault. The high slip area is just under the areas where thrown-out boulders were found by Umeda et al. (1987) and Iio & Yoshioka (1988). Comparing the aftershock distribution with the estimated slip distribution, we found that the high slip area had few aftershocks. This suggests that aftershocks are caused by the stress generated in the marginal region around the main shock slip. The inversion analysis also revealed that there was a slight change in slip angle during the earthquake. The high slip area has some amount of dip-slip component while the slips in the other areas are almost purely right-lateral.

Waveform inversion using ABIC for the rupture process of the 1983 Hindu Kush earthquake

Abstract An inversion method was developed to infer earthquake rupture process from far-field velocity waveforms. We devise a source model that has variable slip and variable rupture velocity. The fault plane is divided into many subfaults. Slip amplitude is assumed to be dependent on the subfaults, and slip velocity at a point is assumed to be a triangle function of time. Rupture time is expressed by an interpolation equation using the rupture times at the corners of the subfaults. The slip distribution and the rupture-front motion are estimated by constrained least-squares inversion. We incorporate smoothing constraints on slip distribution and on rupture propagation into the observed data. The degree of the constraints is controlled by hyperparameters. The optimal values of the hyperparameters are determined by minimizing Akaike's Bayesian information criterion (ABIC). Under the assumption of ABIC, we can obtain a unique solution with reasonable resolution. We apply this inversion method to the Hindu Kush earthquake of 30 December 1983 with a depth of 212 km. We found that the rupture process of this earthquake was composed of two subevents. The first subevent occurs near the initiation point of the rupture. The centre of the second subevent region is approximately 20 km to the west of the first subevent. The released moment of the second subevent is about seven times larger than that of the first subevent.

A Bayesian approach to the detection of temporal changes in P wave velocity

On the basis of Akaike's Bayesian information criterion (ABIC), a new method is proposed for detecting a temporal change in a seismic velocity in a source region. The method of joint hypocenter determination was modified in order to determine a seismic velocity in a source layer as a function of time together with hypocenters and station corrections. Arrival times of initial waves of shallow earthquakes in a small area are analyzed in this method. The smoothness of the estimated temporal variation in the velocity is guaranteed by the introduction of a prior distribution of the parameter. The hyperparameter of the prior distribution of the velocity, the reading error of arrival times, and the initial velocity in the source layer are chosen to minimize ABIC. This procedure was applied to the 1983 eastern Yamanashi M = 6.0 earthquake in central Japan. We analyzed P arrival times of 374 earthquakes observed at 12 stations in the network of the National Research Center for Disaster Prevention by dividing the whole period (from October 1981 to May 1987) into 12 six‐month subperiods. Calculating ABICs for different combinations of the three parameters above, we searched for the minimum value of ABIC and found two minima. The first one corresponds to a model of a constant velocity in time, and the other corresponds to a model of a variable velocity with 5% velocity change at maximum. However, since ABIC in the former is 10 smaller than that in the latter, the former constant velocity model is statistically more suitable than the latter. Furthermore, generating artificial data with the same reading errors as the actual data, we used computer simulation to examine the lower limit of the velocity change detectable for this data set. In conclusion, the velocity in the source region is 6.24±0.18 km/s, and the velocity change exceeding 6–7% at maximum did not exist during the 6 years before and after the M = 6.0 earthquake.

A bayesian approach to nonparametric test problems

We propose an alternative approach to the classical “nonparametric” test problems, such as the goodness of fit test and the two-sample “nonparametric” test. In this approach, those problems are reviewed from the viewpoint of the estimation of the underlying population distributions and are formulated as the problem of model selection between Bayesian models which were recently proposed by the present authors. The model selection can be easily realized by choosing a model with the smallest ABIC, Akaike Bayesian information criterion. The approach provides the estimates of the density of the underlying population distribution(s) of any shape as well as the evaluation of the goodness of fit or the check of homogeneity of distributions. The practical utility of the present procedure is demonstrated by numerical examples. The difference in behavior between the present procedure and a density estimator GALTHY proposed by Akaike and Arahata is also briefly discussed.

An approach to the nonstationary process analysis

A Bayesian approach to nonstationary process analysis is proposed. Given a set of data, it is divided into several blocks with the same length, and in each block an autoregressive model is fitted to the data. A constraint on the autoregressive coefficients of the successive blocks is considered. This constraint controls the smoothness of the temporal change of spectrum as shown in Section 2. A smoothness parameter, which is called a hyper parameter in this article, is determined with the aid of the minimum ABIC (Akaike Bayesian Information Criterion) procedure. Numerical examples of our procedure are also given.

A Bayesian method for estimating earthquake magnitude distribution and changes in the distribution with time and space in New Zealand

Abstract A Bayesian procedure for examining whether the time-space variation of the magnitudefrequency relation conforms with the exponential distribution or not is proposed. The relation in each time-space segment is obtained as a histogram. The smoothness of the estimated histogram is guaranteed by the introduction of prior distributions of the parameters in time, space, and magnitude. The parameters of the prior distribution are chosen to minimise Akaike's Bayesian Information Criterion (ABIC). This procedure is applied to three earthquake catalogues in New Zealand: those for the Wellington Seismograph Network, the Pukaki Network, and the New Zealand Network respectively. For both the Wellington and Pukaki Networks, departure from the exponential distribution is the main effect and the time variation is secondary. For the New Zealand Network, where the magnitude- frequency relation appears to be an exponential distribution, the variation of b over time and space suggests a relation between periods of active seismicity and periods of high b-value, not locally, but overall in the main seismic region.

A Bayesian modeling of seismicity in time and space.

Basing on the Bayesian statistics, a new method to quantify seismicity is proposed. Earthquake occurrences are assumed to follow a non-stationary Poisson process of which the parameter (Poisson rate) changes both in time and in space Time-space for study is divided into hundreds of segments, and earthquakes are sorted into these segments. The Poisson rate is assumed to be constant in each segment, and the time-space variation of the rate is estimated by using Akaike's Bayesian Information Criterion (ABIC). To demonstrate the effectiveness of this method, the time-space variation in seismicity was studied for the area of the Nemuro-oki earthquake of June 17, 1973(MJMA=7.4). As a result of the analysis, seismicity in the source area is found to be rather constant for the period of 1927-1972. The precursory seismic quiescence pointed out in earlier works should be attributed to the high seismic activities in the surrounding areas due to aftershocks of the 1952, 1968, and 1969 earthquakes.

A Bayesian approach to the detection of changes in the magnitude-frequency relation of earthquakes.

Based on Akaike's Bayesian Information Criterion (ABIC), a new method is proposed for detecting changes in the magnitude-frequency relation. Earthquakes are classified into a few magnitude classes and the ratio between the numbers of shocks in these classes is estimated by using ABIC. The time variation of the magnitude-frequency relation is investigated by this method for the source areas of three events with magnitude 6.0 and above in central Japan; the Southern Ibaraki Earthquake of 1983, the Eastern Yamanashi Earthquake of 1983 and the Western Nagano Earthquake of 1983. In.the southern Ibaraki and the Eastern Yamanashi regions, the frequency of larger shocks in comparison with smaller shocks increased about 1 year preceding the main shocks. In the Western Nagano region, the number of large shocks may have increased 2 years before the main shock. The time variations of the former cases are statistically significant.

A bayesian approach to the probability density estimation

A Bayesian procedure for the probability density estimation is proposed. The procedure is based on the multinomial logit transformations of the parameters of a finely segmented histogram model. The smoothness of the estimated density is guaranteed by the introduction of a prior distribution of the parameters. The estimates of the parameters are defined as the mode of the posterior distribution. The prior distribution has several adjustable parameters (hyper-parameters), whose values are chosen so that ABIC (Akaike's Bayesian Information Criterion) is minimized.