Restoratives

If a situation arises whereby formation fluid or gas enters the bore bole the driller will notice an increase in the total volume of mud. Other indications such as a sudden increase in penetration rate and a decrease in pump pressure may also indicate an influx. Much depends on a quick response of the driller to close in the well before substantial volumes of formation fluid have entered the borehole. Onoe the BOP is closed, the new mud gradient required to restore balance to the system can be calculated. The heavier mud is then circulated in through the kill line and the lighter mud and influx is circulated out through the choke line. Once overbalance is restored, the BOP can be opened again and drilling operations continue. 

The Supplement C ( reference) is especially valuable it contains procedure to restore the regression dependencies with reference to empirical data. 

The classical computer tomography (CT), including the medical one, has already been demonstrated its efficiency in many practical applications. At the same time, the request of the all-round survey of the object, which is usually unattainable, makes it important to find alternative approaches with less rigid restrictions to the number of projections and accessible views for observation. In the last time, it was understood that one effective way to withstand the extreme lack of data is to introduce a priori knowledge based upon classical inverse theory (including Maximum Entropy Method (MEM)) of the solution of ill-posed problems [1-6]. As shown in [6] for objects with binary structure, the necessary number of projections to get the quality of image restoration compared to that of CT using multistep reconstruction (MSR) method did not exceed seven and eould be reduced even further. 

By imposing the prior information on the solution one implies, that this solution is suited to the only class of problems which corresponds to the information that is involved. But it should be noticed that these classes can be so wide that real constraints to the restored image are reduced to minimum. 

The starting point of this approach is that the 3D restoration is implemented by the solution of the variational problem for the trade-off functional M , which favors in a weighted manner measured data (functional A) and a priori knowledge (functional B) ... 

The main point of our elaboration is, that the Gibbs measure (4) of the potential lattice under interest ctin be considered as a nontrivial prior in the Bayes formula for the conditional probability, applied to the problem of image restoration ... 

In the Maximum Entropy Method (MEM) which proceeds the maximization of the conditional probability P(fl p ) (6) yielding the most probable solution, the probability P(p) introducing the a priory knowledge is issued from so called ergodic situations in many applications for image restoration [1]. That means, that the a priori probabilities of all microscopic configurations p are all the same. It yields to the well known form of the functional 5(/2 ) [9] ... 

Fig. 1. Two spatial views of the restored image of the object using 4 projections and CSA...

Often the a priori knowledge about the structure of the object under restoration consists of the knowledge that it contains two or more different materials or phases of one material. Then, the problem of phase division having measured data is quite actual. To explain the mathematical formulation of this information let us consider the matrix material with binary structure and consider the following potentials ... 

The importance of distinct a priori knowledge account becomes more perceptible if noisy data are under restoration. The noise / ( shifts the solution of (1) from the Maximum Likelihood (ML) to the so called Default Model for which the function of the image constraint becomes more significant. 

To investigate the influence of the noise on the quality of image restoration a specific procedure was implemented, which included ... 

To estimate the quality of restoration having both, simulated and reconstructed images, using some kind of quality criterion For this last purpose the following mean square measure was applied ... 

Fig. 3. One simulated projection of the simulated object with different level of noise (left column) a) 0.0, b) 0.05, c) 0.1, d) 0.15, and the corresponding restored images (right column) using 5 projections.

In this figure the next definitions are used A - projection operator, B - pseudo-inverse operator for the image parameters a,( ), C - empirical posterior restoration of the FDD function w(a, ), E - optimal estimator. The projection operator A is non-observable due to the Kalman criteria [10] which is the main singularity for this problem. This leads to use the two step estimation procedure. First, the pseudo-inverse operator B has to be found among the regularization techniques in the class of linear filters. In the second step the optimal estimation d (n) for the pseudo-inverse image parameters d,(n) has to be done in the presence of transformed noise j(n). 

The adaptive estimation of the pseudo-inverse parameters a n) consists of the blocks C and E (Fig. 1) if the transformed noise ( ) has unknown properties. Bloek C performes the restoration of the posterior PDD function w a,n) from the data a (n) + (n). It includes methods and algorithms for the PDD function restoration from empirical data [8] which are based on empirical averaging. Beeause the noise is assumed to be a stationary process with zero mean value and the image parameters are constant, the PDD function w(a,n) converges, at least, to the real distribution. The posterior PDD funetion is used to built a back loop to block B and as a direct input for the estimator E. For the given estimation criteria f(a,d) an optimal estimation a (n) can be found from the expression... 

Therefore, the adaptive properties of the above system include the PDD restoration algorithms from empirieal data. 

V. N. Vapnic, Function Restoration from the Empirical Data (In Russian), Nauka, Moscow, 1979... 

Trofimov O.E. To the problem of restoration of functin from three variables by its integrals along the lines, crossing given curve., Avtometriya, N5, 1991, p..29-33. 

In sections, where exists high probability of presence of defects, on the base formed in the binary type of projections acoustical tomographic images of only defective structure elements of sections is restored. IT of restoring stipulates such operations ... 

Restored parameters for the evaluation of PDSM, may be different PMF of material tensor of stresses or its invariants, spatial gradients of elastic features (in particular. Young s modulus E and shear modulus G), strong, technological ( hardness HRC, plasticity ), physical (density) and others. 

Tasks of the USCT IT are restoring of SD of the certain acoustic parameter of material in product volume on the base of measured parameters of US signals, but afterwards determination of necessary PMF. It is defined strong factors basically on the grounds of their correlation with acoustic features. 

Restoring of SD of parameters of stress field is based on the effect of acoustoelasticity. Its fundamental problem is determination of relationship between US wave parameters and components of stresses. To use in practice acoustoelasticity for SDS diagnosing, it is designed matrix theory [Bobrenco, 1991]. For the description of the elastic waves spreading in the medium it uses matrices of velocity v of US waves spreading, absolute A and relative... 

Realization of the USCT method for restoring of SD of PMF of material in thick-sheet products with plane parallel surfaces under the unilateral access required special designs of US blocks, in which taken into account particularities of the NDTO, of the method and of the... 

Note some particularities of new USCT method. At first, data collection and search of areas with anomalous (inhomogeneous)SD of acoustic parameters (velocities of spreading of US waves) is joined. As a sought image, on which anomalies is revealed, it is offered total image B (r), which practically is the low frequency copy of restored fimction g(f). As PMF SD of... 

For restoring of three-dimensional SD is used stated above approach. Under restoring of tomographic images for the base undertakes a function of three-dimensional total image, which after double differentiation and inverse projecting describes sought SD 8 (1) ... 

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