Global estimation error

For the sake of completeness, of conrse analytical resnlts also inclnde the total analytical error (TAE) in addition to all sampling errors. The snm of these two categories is termed the global estimation error (GEE), which thus includes all possible errors which could ever be of interest in PAT  

This chapter will not deal with TAE, as analytical errors are amply dealt with elsewhere in the specific chapters on the individual analytical techniques as well as in the voluminous general pertinent analytical chemistry literature. 

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This value characterizes the upper level of relative scattering of estimated elements concentration in all the considered snow samples, which is associated both with estimation errors in groups 1-3 and natural variation of the elements abundance in samples. Thus, the results of successive testing of ypotheses Ht -H3 allow us to conclude that the basic hypothesis Hb is true and only global source of chemical contamination exists on the territory of Karabash. 

Theorem 5.1 If the rate constants are bounded as in (2.32) and (2.33), then, there exists a set of observer gains such that the state estimation error x is globally uniformly convergent lo 0 as / -> oc and the parameter estimation error 90 is bounded for every t. 

This guarantees the boundedness of all error signals. By invoking Barbalat s lemma [3], it can be recognized that V -> 0, which implies the global convergence to 0 of both x and e, while the parameters estimation errors 60 and 6C are only guaranteed to be uniformly bounded. ... 

Figure 1.12 Global plots of objectively analyzed N (pmol kg" ) on isopycnal surfaces. (A) N on the as = 26.50 surface. This surface is characteristic of subtropical mode waters, such as the 18° Water in the North Atlantic, (B) N on the as = 27.10 surface representing Subpolar Mode Waters. Solid circles denote the stations stippling denotes areas where the waters of this potential density are not present in winter time. Cross-hatched areas are regions where the estimated error in objectively analyzed JV is greater than 0.7 pmol kg". Data are from the WOCE program augmented by selected pre-WOCE cruises and were adjusted by the GLODAP project.

In practice, we do not know the exact solution of the problem, and thus we cannot detennine the magnitude of tlie error involved in the numerical method. Knowing that the global discretization error is proportional to the step size is not much help either since there i.s no easy way of determining the value of the proportionality constant. Besides, the global discretization error alone is meaningless without a true estimate of the round-off error. Therefore, we recommend (he following practical procedures to assess the accuracy of the results obtained by a numerical method. 

The last line above shows the estimated error to the solution obtained with the smallest step size as being simply (jj -y ) i. In terms of Figure 10.18 the first order estimate of the solution error is then simply 1/3 of the difference between the dotted curve and the solid curve. This is the error estimate of the most accurate solution with the largest number of data points. This technique of numerically estimating the global error of a solution will be referred to as the h-2h error estimation technique. 

One approach to an adaptive step size is simply to first assume some best guess at a good step distribution and then calculate the solution and obtain an estimated error as has been done in the previous section. Based upon the resulting global error a new step size distribution could be developed and the calculation could be repeated, hopefully until a given accuracy had been achieved. This would proba-... 

Fig. 4.3. (A) Composite multispecies benthic foraminiferal Mg/Ca records from three deep-sea sites DSDP Site 573, ODP Site 926, and ODP Site 689. (B) Species-adjusted Mg/Ca data. Error bars represent standard deviations of the means where more than one species was present in a sample. The smoothed curve through the data represents a 15% weighted average. (C) Mg temperature record obtained by applying a Mg calibration to the record in (B). Broken line indicates temperatures calculated from the record assuming an ice-free world. Blue areas indicate periods of substantial ice-sheet growth determined from the S 0 record in conjunction with the Mg temperature. (D) Cenozoic composite benthic foraminiferal S 0 record based on Atlantic cores and normalized to Cibicidoides spp. Vertical dashed line indicates probable existence of ice sheets as estimated by (2). 3w, seawater S 0. (E) Estimated variation in 8 0 composition of seawater, a measure of global ice volume, calculated by substituting Mg temperatures and benthic 8 0 data into the 8 0 paleotemperature equation (Lear et al., 2000).

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