In weighted least-squares refinement the quantity minimized is where X is one of the quantities |F|, or I, and ΔX is |X(rel)-X(cal)|. The weight, w(X), reflects the accuracy of the measurement of the intensity of a reflection. A properly estimated weight will lead to an accurate crystal structure. The correct value of the weight is the reciprocal of the variance, X. Usually the contribution of counting statistics to σX is easily calculated, but the contribution from various other sources of error is not.

REGWT provides information to assist in the modification of weights to account for other sources of error. It is equivalent to WTANAL and WTLSQ in XRAY76. REGWT contains an additional feature which allows the calculation of coefficients for the weight modification expression in order to modify the weights. The program does not differentiate between random and systematic error. The procedure is described in detail elsewhere, (Wang and Robertson, 1985). In this program description, w refers to the weight rather than the square root of the weight as it does in some other program descriptions in this manual.

REGWT is used to examine existing weights based on the distribution of w(X)Δ X. The data is divided into blocks bounded by both intervals of X(rel) and s. By choosing either X(rel) or s the analysis becomes one-dimensional with respect to X(obs) or s. For each grid point, the number of data, the average value of and weighted R index are given.

The weighting schemes used here are based on the program WTLSSQ of the XRAY76 system. Nine weight functions and an opportunity for manipulation of the location of the weights in the bdf are specified on the normal line. The weight functions are specified by a weighting scheme number (0-9) and the coefficients (A to I) corresponding to those functions. The default values of these coefficients are all zero. The full weighting schemes are described in detail below.

Scheme 0: Retrieve the old least-squares weight from one of the least-squares weight locations n901, n902, n903 and place it in the least-squares weight location n900, which is used in least-squares refinement in the logical record lrrefl: when the bdf weights are specified.

Scheme 1: W = 1 / (A + B X + C/(oldweight) + D*X(rel) + E*X(rel) + G*X(rel) + I*sin )

Scheme 2: W = 1 / (A + B*X(rel) + C*X(rel) /oldweight + D*X(rel) + E*sin )

Scheme 3: W = X * Y If A*X(rel) is greater than |X(cal)| then weight is 0.000000001. Let B = sin limit, then if sin is greater than B, X = 1, else X = sin /B B should not be zero. Let C = X(rel) limit, then if X(rel) is less than C, Y = 1, else Y = C/X(rel).

Scheme 4: If X(rel) is less than or equal to A, then W = (X(rel)/A) , else W = (A/X(rel)) . Reflections with X(rel) = A are given maximum weight.

Scheme 5: If X(rel) is less than or equal to A, then W = 1, else W = (A/X(rel)) . Small reflections are given constant weight. In Hughes' original treatment, A = 4 X(min).

Scheme 6: W = 1 /(1 + ((X(rel) - B)/A) ) Reflections with X(rel) = B are given maximum weight, but if A is very large , the weights are constant.

Scheme 7: W = 1 /( A + X(rel) + B*X(rel) + C*X(rel) ) Cruickshank suggests that A = 2 X(min), B = 2 / X(max), and C = 0 are useful. A larger value of C, say C = 5 / X(max) , may help down-weight large X(rel) subject to extinction. These values may prove useful in schemes 1 and 2.

Scheme 8: W = A / max( X(rel), (B*X(rel) + C), (D*G + E) )where G is X(rel)(max). For a complete description of this (Univ. of Washington) scheme, see the general section for the program DATRDN of the XRAY76 system.

Scheme 9: W = A This weighting scheme produces constant weights

If the variances of the structure factor amplitudes are correctly estimated, their average value should correspond to X. (The variance will be assumed to represent error in both the experiment and the model; i.e. in |X(rel)| and |X(cal)|.) The process of least squares allows some adjustment of the model to errors in the experiment, so that

rather than unity. N is the number of reflections used for least-squares refinement and M is the number of least-squares variables. <A> is the average value of A. The quantity (N-M)/N will be referred to in what follows as the "freedom factor". The calculated variance, X(cal), is the variance obtained from a knowledge of the experimental conditions and is usually based entirely on counting statistics. A quantity X(mod), the modified variance, may be added to X(cal) to give an improved value of X or 1/w(X). The correction, X(mod), commonly called the "ignorance factor", may be entered by the user as some function such as

(see Scheme 1, normal weighting schemes) or it may be estimated automatically. An approximate expression for is

(4.83)

where VC is a correction term including variance (VAR) and covariance (COV) terms.

This term results from replacing by , and the user may choose to not use it. Also, the "freedom factor" may be replaced by unity. The independent variables are normalized by dividing by and . The new independent variables are:

The correction, , is estimated by least-squares fitting of the following expression to equation ???

(4.84)

If p=0, a constant term is determined.

If p=1, the coefficients in the expression of the form

A(0,0) + A(1,0)*V(F) + A(0,1)*V(S)

are determined; etc. The coefficients A(p,q) can then be used to calculate for an individual reflection. The weight for a reflection is then calculated as:

(4.85)

The structure is refined by normal least squares using the modified weights and new values of are created. If the individual reflection option is chosen, the variance and covariance correction is not used or calculated. The process is iterated until the coefficients do not change. The change in the standard deviations of least-squares variables on the first iteration will probably be between 0 and 40% of their initial value with non-modified weights. The improvement on the second iteration will typically be 10% of the improvement of the previous one. A third iteration would seldom seem justified. The quantity is required for calculating the variance and covariance correction (VC), but is not well known until after first iteration. Therefore must be used instead of in the first iteration, if the VC correction is applied.

In the REGWT calculation, the value of in (4.85) above is obtained from IDN 1900 in logical record lrrefl: on the bdf. If IDN 1900 is empty, the value of X(cal) is then obtained from IDN 130n (n=1, 3 or 5 for I, or F, respectively). After the modified weight w(X) is calculated, its value will be stored in IDN 1900, replacing the old value of X(cal).

The square root of the number of reflections in a grid point is used to weight the grid points when fitting equation (4.84) to equation ???. Not surprisingly, grid points with high V(X) and high V(S) may be empty and others may have few reflections. The option also exits to not average but treat every reflection as a separate grid point. Since averaging to form grid points involves first order differences, the influence of reflections whose contribution to the average in equation ??? deviate far from the mean will be enhanced if individual reflections are used to find the A(p,q). If the individual reflection option is chosen, the variance and covariance correction is not used or calculated.

REGWT lst wta       
maxhkl 12 12 12 0.0 0.824
fgrid 2.3 4.3 7.6 12.4       
fgrid 13.4 17.5 20.5 25.5 30.5 35.5       
fgrid 45.0 55.5 65. 80. 100. 99999.       
sgrid .1 .15 .2 .25 .3 .35 .4 .45 .5 .55 .6 .65       
sgrid .7 .75 .80 .85

In this example, the weight-analysis process is called to analyze the weights which have been assigned for each reflection. The analysis maps are specified as 16x16 by the fgrid and sgrid lines. The program does not update the archive bdf. The reflection data will be printed.

REGWT nolist wfc 10.0 rel cnt 99.0
regina pow 2 ind fac 1. avc       
archiv 1902 -1903       
fgrid 3.0 4.0 5.0 6.0 7.0 8.0
fgrid 9.50 12.0 17.0 21.0 27.0 30.0 50.0       
fgrid 70.0 999.0       
sgrid 0.150 0.200 0.250 0.300 0.350 0.400 0.450       
sgrid 0.500 0.515 0.530 0.550 1.000

The weight modification function with 2 as the highest power is specified. The estimation of the weights is based on the individual reflection mode. The variance and covariance correction is applied and the least-squares freedom factor is 1. The weight in location 1903 will be deleted from the binary data file and the old weight will be stored in the weight location 1902. Only the observed reflections are used excluding the reflections for which the value of ?(F) is greater then 99.0. The output analysis map (15x12) will be scaled by the factor 10.

REGWT nor       
normal 0 b 2       
archiv -1903

A normal weighting scheme is indicated in the REGWT line. The scheme number is specified as 0, which transfers the weight saved at location 1902 to location 1900 where it will be used for weighted least-squares refinement. The old weights at location 1903, if they exist, will be erased from the bdf.

References