Data Extrapolation Methods

If we subtract the second equation from the first, the In i terms drop out, which yields the following  

It is now possible to calculate by substitution of this value of n into either of the preceding equations. Using the first one, 

finally, we solve for e at o- = 10 MPa by incorporation of these values of n and into Equation 8.24  

Several theoretical mechanisms have been proposed to explain the creep behavior for various materials these mechanisms involve stress-induced vacancy diffusion, grain boundary diffusion, dislocation motion, and grain boimdary sliding. Each leads to a different value of the stress exponent n in Equations 8.24 and 8.25. It has been possible to elucidate the creep mechanism for a particular material by comparing its experimental n value with values predicted for the various mechanisms. In addition, correlations have been made between the activation energy for creep (Qc) and the activation energy for diffusion (g Equation 5.8). 

Creep data of this nature are represented pictorially for some well-studied systems in the form of stress-temperature diagrams, which are termed deformation mechanism maps. These maps indicate stress-temperature regimes (or areas) over which various mechanisms operate. Constant-strain-rate contours are often also included. Thus, for some creep situation, given the appropriate deformation mechanism map and any two of the three parameters—temperature, stress level, and creep strain rate—the third parameter may be determined. 

---

While the arbitrary extrapolation methods used to evaluate v( and f° for a supercritical component are partly compensated by evaluating dt from data for binary mixtures, such compensation cannot apply generally to mixtures containing supercritical components i.e., for a supercritical component, 5, found from data for solutions of i in one solvent may be quite different from that found from data for the same component i in another solvent. 

The mathematical methods used for interpolation and extrapolation of the data obtained from accelerated tests, as described in Chapters 8 and 9, include both the mechanistic and the empirical. Arrhenius formula, based on chemical rate kinetics and relating the rate of degradation to temperature, is used very widely. Where there are sufficient data, statistical methods can be applied and probabilities and confidence limits calculated. For many applications a high level of precision is unnecessary. The practitioners of accelerated weathering are only too keen to tell you of its quirks and inaccuracies, but this obscures... 

Recently Hoover 29> compared various extrapolation methods for obtaining true solution resistances concentrated aqueous salt solutions were used for the comparisons. Two Jones-type cells were employed, one with untreated electrodes and the other with palladium-blacked electrodes. The data were fitted to three theoretical and four empirical extrapolation functions by means of computer programs. It was found that the empirical equations yielded extrapolated resistances for cells with untreated electrodes which were 0.02 to 0.15 % lower than those for palladium-blacked electrodes. Equations based on Grahame s model of a conductance cell 30-7> produced values which agreed to within 0.01 %. It was proposed that a simplified equation based on this model be used for extrapolations. Similar studies of this kind are needed for dilute nonaqueous solutions. 

Dynamic extractions of the organic flavonr and fragrance componnds from dried lavender flowers and rosemary leaves nsing SCCO2 were carried ont. The data from the lavender and rosemary extractions were fitted to a model to prodnce the characteristic extraction cnrve. Using data obtained from rosemary extractions, an extrapolation method derived from the model was used with data from shorter extractions to show that the model provided qnantitative analytical information (Walker et al., 1994). 

Kinetic Light-Scattering Method. Isochronous Interpolation. When high-activity samples of endocellulase are used, the reaction proceeds so quickly that, since the measurements of scattered intensity at different angles are not performed at the same extent of reaction, the extrapolations to zero angle and the subsequent calculations are erroneous. For this reason Kratochvil et al. (27) have proposed an isochronous interpolation method, whereby the Kc/Re values are plotted against sin21 (0/2) + kft. As in the double-extrapolation method of Zimm, the value of k may be chosen arbitrarily in order to space the experimental data. 

Finally, many extrapolation methods are limited by the availability of suitable databases. Although these databases are generally largest for chemical stressors and aquatic species, even in these cases data do not exist for all taxa or effects. Chemical effects databases for mammals, amphibians, or reptiles are extremely limited, and there is even less information on most biological and physical stressors. Extrapolations and models are only as useful as the data on which they are based and should recognize the great uncertainties associated with extrapolations that lack an adequate empirical or process-based rationale. 

The yield stress values given in Table 3 demonstrate that the yield stresses determined with the Herschel-Bulkley model were lower than the yield stresses determined with all the other methods at equal concentrations. The yield stress predicted by direct data extrapolation and by the Herschel-Bulkley model was similar for each concentration of corn stover. 

The development of extrapolation methods for all possible steps strongly relies on both theory and the quality and number of available data. Studies can be conducted... 

In single-species assessments, the interpretations of mixture assessments tend to be mostly absolute. Hence, risk assessors often focus on particular species and particular compound groups (e.g., risks of PCB mixtures for birds), allowing them to interpret and explain their experimental data to the best of their abilities. On the other hand, many risk assessors apply mixture extrapolation methods to address risks for communities. The applications of SSD-based methods for this evolved fast and now cover a wide set of approaches, ranging from ecological multiple-stress analyses to overall approaches such as life-cycle assessment. Especially in the latter set of approaches, the risk assessor can often allow the method to only yield relative... 

Ecological risk assessments cannot be done without applying extrapolation methods. Sufficient data to execute such risk assessments are usually lacking. For example, there may be no toxicity data for the suspect substance, the tested species may differ from the species in the assessed ecosystem, exposure is to single substances in test systems but mixtures occur in the field, or risks are to be assessed for communities rather than for species. The lack of data is a consequence of practical and ethical considerations. 

Extrapolation methods are used for various types of risk assessment. Methods may be used in the process of deriving environmental quality objectives, in the registration of new substances, and in the process of site-specific risk assessment. Suter (1993) called these approaches prospective (the former 2) and retrospective (the latter) risk assessments. The specific process in which extrapolation methods are used has implications for the concepts to be applied and the data to be used as input in extrapolation. Strictly described approaches are in place for the derivation of environmental quality criteria (EQCs) and the registration of pesticides and newly developed substances. The prescribed approaches for deriving EQCs can differ between jurisdictions. The approaches for retrospective investigations have more degrees of freedom. A characteristic of the latter approach is that the methods can make use of measured local exposure levels and can estimate local risk with known precision (or known uncertainty ). The latter is uncommon for EQCs. 

When the scope is known and the extrapolation methods are defined, the data for the risk assessment steps (and extrapolations) can be collected. For tiered systems, this implies that choices need to be made on the manner in which uncertainties are addressed and what to do when these are only addressed by simple methods. A distinction was already made between prospective use and retrospective use of extrapolation methods. In both cases, extrapolations are being applied, but the way in which existing methods are selected for an assessment problem can differ. 

It is noteworthy that comparisons of existing assessment schemes reveal dissimilarities in the use of extrapolation methods and their input data between different jurisdictions and between prospective and retrospective assessment schemes. This is clearly apparent from, for example, a set of scientific comparisons of 5% hazardous concentration (HC5) values for different substances. Absolute HC5 values and their lower confidence values were different among the different statistical models that can be used to describe a species sensitivity distribution (SSD Wheeler et al. 2002a). As different countries have made different choices in the prescribed modeling by SSDs (regarding data quality, preferred model, etc.), it is clear that different jurisdictions may have different environmental quality criteria for the same substance. Considering the science, the absolute values could be the same in view of the fact that the assessment problem, the available extrapolation methods, and the possible set of input data are (scientifically) similar across jurisdictions. When it is possible, however, to look at the confidence intervals, the numerical differences resulting from different details in method choice become smaller because confidence intervals show overlap. 

Step 5 Choosing a consistent set of extrapolation methods Step 6 Collecting and judging the data that are needed Step 7 Working with the set of extrapolation methods Step 8 Interpreting assessment results... 

The list of extrapolation items can be subjected to a test in terms of pragmatic issues. Choosing amongst all possible extrapolation methods may be needed due to resource limitations (pragmatism) or preferred methods and/or estimated lack of data or methods (science). This stage consists of a preliminary sensitivity analysis to identify... 

A key step in any risk assessment is the collection of data to feed into the extrapolations. The use of an extrapolation method will be possible only with appropriate input data. The data that are needed can be literature data (e.g., laboratory toxicity data) or field data (measured or predicted environmental concentrations), and the extrapolation is then to move these toward common currencies, that is, the estimation of bioavailable concentrations, derived from total concentrations, in both test systems and the field, or species-to-community extrapolation when the concern is the effect of exposure on biodiversity. Step 6 will often be made together with Step 7, because a method without data is as useless as data without a method. 

---