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In this section we summarize the results concerning the dense media radiative transfer equation as derived for active remote sensing. The incident electromagnetic wave is in the direction (1T - Boi , i)' The incident electric field is given by (7.2.21), with E ih and E ie being the amplitudes of TM and TE wave, respectively.

Step 1: Effective Propagation Constants, Extinction Rates, and Coherent Transmitted Wave As shown in Section 3, the lower half-space of region 1 can be represented by an effective propagation constant K, where K obeys the equation

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succumbed and fell over the next year to $4.50, losing about half its peak value. As you learn about other drawn objects (such as channels, retracements, horizontal lines, and so on), you will see how to combine them to perform more complex analyses. Figure 7.8 shows Cleveland Cliffs (CLF) dipping beneath its trendline. In retrospect, when it did this, it was about halfway through the job of creating what is known as a head and shoulders pattern, highlighted here by the use of a horizontal line. Now that you have the general idea as to what constitutes a trendline and why it is useful, it s time to learn how to make them in ProphetCharts.

K 2 ~ K,2

+ i2K'K"

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3 . aslYs, (Ko) + ~ YS j(Ko)a 3 n sj 8rr 3 H 8jS / (p [

= 0)] }

(7.4.1)

sJ-1

D(Ko) = 1 -

8j=1

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to assume reasonable design values to be used in the analysis. Certainly, in historic structures, species and density play a most signi cant role in determining the strength of a particular timber. Of course, the timber s history is critical also. Wood that appears to be of a superior grade may have been subjected to overstressing, cyclic loading, elevated temperatures, or other environmental conditions. The true capacity of a historic timber is often less than one would expect based on a visual inspection. The testing of representative samples to destruction is the surest way to determine reasonable design values. Results from such tests can be used to establish appropriate design values in a simpli ed manner or through statistical analysis of the results. Certainly, the old-growth, dense, clear timber found in many existing structures built before World War II, in North America, should be judged on the quality of the wood and not on published design values assigned for timber cut today. In spite of the insistence of the forest products industry that today s timber is in every way equal to previously cut old-growth timber, I have observed a consistent difference in the density of today s timber used for construction and that of the past.

fsjYs j(Ko)

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(7.4.3)

(Ko )

= 3K; ~j (k; _ k2)

(7.4.4)

To calculate numerical results of the nonlinear equation of Kin (7.4.1), first solve (7.4.2) for K o . Equation (7.4.2) with (7.4.3)-(7.4.4) is a polynomial equation for K o , and only one solution has the correct physical meaning of effective propagation constant. That solution is then substituted back in (7.4.1) to calculate K. The extinction rate is K,e = 2K", where K" = Im(K). Snell's law can then be applied so that the transmitted angle of the coherent wave is 0i = sin- 1 (ksinO o dK'), where K' = Re(K). The coherent transmission wave amplitudes for TM and TE waves are given, respectively, by (7.2.24) and (7.2.25) with k iz = k cos (}oi and K iz = (K2 - k 2 sin 2 (}oi)1/2. Step 2: Dense Media Radiative Transfer Equation for the Incoherent Wave By using the definitions of Stokes parameters of (7.3.53)-(7.3.56) in (7.3.48)-(7.3.49), it follows after some algebraic manipulations that the dense media radiative transfer equation for the incoherent 4 x 1 Stokes vector

21Kol 4 = K;eI D (Ko )12

~ fS1YSl (Ko )

FIGURE 1-3 Cutting through the oor joists at Market Hall in Charleston, South Carolina, revealed the material to be dense southern pine.

.{a;,Y:, (Ko) + ~ n"a;,Y:, (Ko)81f H"" (ji = 0) }

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