This article gives the details and results of an investigation into the properties of the temperature and relative humidity errors from the Navy Operational Global Atmospheric Prediction System for a 4-month period from March to June 1998. The spatial covariance data for temperature errors and for relative humidity errors were fit using eight different approximation functions/weighting methods. From these, two were chosen as giving good estimates of the parameters and variances of the prediction and observation errors and were used in further investigations. The vertical correlation between temperature errors at different levels and relative humidity errors at different levels was approximated using a combination of functional fitting and transformation of the pressure levels. The cross covariance between temperature and relative humidity errors at various pressure levels were approximated in two ways: 1) by directly computing and approximating the cross-covariance data, and 2) by approximating variance of the difference of normalized data. The latter led to more consistent results. Figures illustrating the results are included.
The Navy Atmospheric Variational Data Assimilation System (NAVDAS; Daley and Barker 2001) computes an optimal estimation of the atmospheric state by blending observations of the atmosphere with the background, which is provided by a short-range forecast. The success of this approach depends on the accuracy in the specification of the errors of both the background and observations. McNally’s (2000) estimates on the effect of background temperature error correlations on the direct assimilation of radiance data suggest that optimal methods may actually produce an analysis that is less accurate than the background when the error structures are poorly specified. In NAVDAS the innovation statistics method is used, where the innovation vectors, which are the observation minus background values, are statistically analyzed to derive the error structures of the background and observations.
It is not known how relative humidity background errors are correlated with the temperature background errors. There are physical reasons to expect that they are, especially in forecasts of the thickness of a well-mixed boundary layer under a dry and warm stable layer. Clearly, errors in the location of the temperature inversion will result in errors in the location of the boundary layer top, and consequently the depth of the moisture near the surface will also be in error. Similarly in the upper atmosphere where tropopause folds are deep intrusions of the dry warm stratospheric air into the troposphere, position errors of these folds will result in errors in temperature and moisture, and the errors will be correlated. If the structure of these correlations could be captured in the innovation statistics, then optimal methods like NAVDAS could use this information to couple the moisture and temperature analysis in a manner similar to the geostrophic coupling between temperature and winds currently being done. In previous work on this topic, Steinle and Seaman (1995) used the “NMC method” (Parrish and Derber 1992) to estimate the cross correlation between temperature and relative humidity. The NMC (National Meteorological Center, now known as the National Centers for Environmental Prediction) method computes prediction error by differencing two different forecasts verifying at the same time; for example, the 48-h forecast minus the 24-h forecast would be an estimate of the 24-h prediction error. The method’s popularity has come from the emergence of three-dimensional variational analysis methods where prediction error needs to be described in spectral space to simplify the solution. The method is not as accurate as the innovation statistics method because data-sparse regions, where prediction error is underestimated, are included in the computation. Neither method can do much to resolve prediction error over the data-void regions. NAVDAS works in observation space and therefore can continue to use the innovation statistics method. Courtier et al. (1993) did find a close agreement between this method and the innovation statistics method for vertical structures of temperature background error.
This study tests procedures for computing the background error structures of temperature and relative humidity, and explores whether the cross covariance between temperature and relative humidity has a structure that might be useful for optimal estimation methods. We used the method of simultaneous fit and transformation of the vertical coordinate, which allows one to express the vertical correlation function in some kind of isotropic form as it guarantees that the function is positive definite while allowing considerable nonisotropy in either log pressure or pressure coordinates (Franke 1999b). For purposes of this investigation, the usual assumptions were made: the prediction errors were assumed to be homogeneous (stationary) and isotropic, with independent observation errors at different locations, but with identical variances. The assumptions of homogeneous and isotropic innovations are certainly not true, but given the limitations in the amount of data, the assumptions are ones that are commonly made. The procedures employed here are very much like those used previously by one of the investigators (see Franke 1999b) and others. The temperature and relative humidity innovation data were averaged at each level for each station over the 122 days. The mean values were then subtracted from the innovation data. Then the raw covariance data were formed and summed into bins of size 0.01 rad, keeping a count of the number of terms. True radian distance on the sphere was used. Use of great circle distance poses a potential difficulty with positive definiteness of the spatial covariance function approximation, but it is thought to not pose a problem over regions of the size contemplated. Experiments were carried out using several different forms of the covariance function approximation and different weights for the squared residuals of the binned data. Complete details of all results are not given here, but sufficient data to indicate the nature of the results and reasons for the choices made are given.
The observations are quantities measured by radiosondes, temperature, and relative humidity (OFCM 1997). The background fields are short predictions supplied from the Naval Operational Global Atmospheric Prediction System (NOGAPS), pressure p, temperature Tp, and specific humidity Qp. In the data file along with the NOGAPS data is the observed temperature To and observed dewpoint depression Do (which is actually reported), along with the quality control flags. Using the formulas given by OFCM (1997), the observed value of relative humidity is retrieved as follows: dewpoint temperature Td = To − Do, observed vapor pressure e = c exp[aTd/(b + Td)], saturation vapor pressure es = c exp[aTo/(b + To)], and finally observed relative humidity uo = e/es. Here a, b, and c are constants, a = 17.502, b = 240.97 K, and c = 6.1121 mb. These equations reverse those that obtained Do from uo. This is important, because otherwise the relative humidity may be contaminated by the observation error in To. Similar formulas allow the calculation of the predicted value of relative humidity from the predicted values of temperature, specific humidity, and pressure p: mixing ratio r = Qp/(1 − Qp), predicted vapor pressure e = rp/(r + 0.622), saturation vapor pressure at the predicted temperature es = c exp[aTp/(b + Tp)], and predicted relative humidity up = e/es, where a, b, and c are as before.
As we will see later, the independence of the observation errors for temperature and relative humidity is questionable. Whether the values were cross contaminated from the above operations, or whether the relative humidity observation error is somehow dependent on temperature measurement errors is unknown.
This article is a condensation of a technical report that gives additional details and many more illustrative figures and tables. Here the concentration will be on the most successful approaches and will deal almost exclusively with the 0000 UTC data. There are some differences at 1200 UTC, and more extensive results using different fitting functions in the report, and the interested reader is referred to Franke (1999a).
2. Covariance functions for temperature and relative humidity
Samples of the binned covariance data for temperature and relative humidity errors are similar to those for pressure height errors that have been published (e.g., Franke 1999b) and are not given here. For curve fitting purposes, only data out to a maximum distance of 0.40 rad were used.
There is no clear-cut choice of functions to fit data such as this. It is important that the function be positive definite and that it embody enough parameters to give a good estimate of the intercept value for the dependent variable using only data for positive distances since the zero distance covariance is a sum of the observation and prediction error variances. It is assumed that the observation errors are spatially uncorrelated since different instruments (of the same type) are used at different stations. Thus we seek to fit the prediction error spatial covariance function, with the intercept then being the prediction error variance and the difference between the empirical innovation variance and the prediction error variance being the observation error variance. In previous studies a number of different functions and weights have been used (e.g., Franke 1986, 1999b). For this study we have used some that have been used previously, and some that we have not used before. The three basic functions used here are the special second-order autoregressive function,
a convex combination (0 ⩽ c ⩽ 1) of two second-order autoregressive functions,
and the full third-order autoregressive function,
Function (1) has been used extensively by the investigator and others. Note that function (2) was inspired by Mitchell et al. (1990), where a sum of special third-order autoregressive functions were used (but with specified weights and relation between constants in the exponentials). Function (3) was chosen because it has more parameters and embodies two different exponential decay rates, giving considerably more flexibility than function (1), and with a different connection between the exponential decay terms than that of function (2). All nonlinear least squares fits were computed using the standard minimization function fmins in Matlab,1 which uses a Nelder–Mead simplex algorithm. All nonlinear minimization routines are sensitive to the initial guess, and some effort was expended in trying different initial guesses.
Table 1 gives a list of the functions and weights we have used to fit the temperature and relative humidity innovations.
The standard deviations of the temperature and relative humidity prediction errors (i.e., the square roots of the intercepts) obtained from the eight approximations indicated in Table 1 are shown graphically for various levels at time 0000 UTC in Fig. 1. Two things are striking: the general consistency of the results for the special second-order autoregressive fits (especially for temperature) and to a lesser extent the overall consistency when some nonequal weighting is applied to the binned data.
The difficulties in obtaining appropriate approximations to the intercept (and hence the variance of the prediction error) is a long-standing problem. In some sense the data for short distances are most critical to defining that value, but on the other hand the amount of data is very much less. Of course, this goes hand in hand with a suitable assumption for the local behavior of the spatial covariance function for short distances. As in previous work (Franke 1999b), it is felt here that some form of nonequal weighting for the binned data is appropriate. The choice is not clear-cut, and somewhat arbitrarily we have decided to use the weighting indicated in Table 1 by reference 1, weighting of the squared residuals by the number of the data collected in the bin. We note that this gives each observation pair equal weighting. In addition, averaging empirical interstation covariances may give unreliable pairs of stations with small amounts of data greater influence in the binned averages. We have decided to pursue further investigations using only two fitting functions with that weighting: the special second-order autoregressive function and the full third-order autoregressive function. These will be referred to hereafter as SAR2 and FAR3, respectively. It is presently unknown what restrictions on the parameters will guarantee that the full third-order autoregressive function is positive definite in two dimensions, but the additional parameters and flexibility available will allow some different behavior by the approximating function for small distances. The FAR3 will also be used for the vertical correlation approximations.
The standard deviations of the error data implied by the two choices for fitting the spatial covariance functions are presented in Fig. 2. The upper subplots show the standard deviations of the temperature observation and prediction errors for the various levels for each of the two approximations, while the lower two subplots show the corresponding data for relative humidity.
The correlation distance parameter is of interest. For SAR2 this is the reciprocal of the parameter a, while for FAR3 there are two exponential parameters, b and c. The correlation distance is generally a measure of the distance at which the function decays by a certain amount,2 and for FAR3 the term is taken to mean the larger of the reciprocals of b and c. Plots of the correlation distance parameters are given in Fig. 3. It is noted that the correlation distances arising from the SAR2 approximations are generally better behaved across various levels than those arising from the FAR3 fits.
To estimate the vertical correlation between temperature errors and between relative humidity errors, the same strategy as has been used previously for pressure level height errors was used. That is, the spatial covariance functions for the differences in the errors between all possible pairs of (different) levels were approximated. Letting Vj represent the error of the quantity in question (temperature or relative humidity), we estimate var(Vj − Vi). Then, since var(Vj − Vi) = var(Vj) − 2 cov(Vj, Vi) + var(Vi), we obtain
Having estimated each of the quantities on the right side of Eq. (4) by approximating the spatial covariance function for the quantity, we are able to obtain the vertical covariance. The vertical correlation is then cor(Vj, Vi) = cov(Vj, Vi)/[var(Vj) var(Vi)]1/2.
In order to assure a positive definite vertical correlation matrix, it is desirable that the vertical correlation functions be expressed in a stationary isotropic form. In an attempt to achieve this, the simultaneous fit and transformation of the vertical coordinate that was used by Franke (1999b) was applied to these eight cases for both prediction and observation error. Because there is appreciable negative correlation at short distances, the full autoregressive function of order three given by Eq. (3) was used. After translation of the values to the new coordinate system, the resulting approximations, along with the correlation points, are shown in Figs. 4 and 5. The transformations generated for each of the approximations are shown in Figs. 6 and 7. The previous application (to height error correlations) generated considerable improvement in terms of reducing the scatter and making the data look reasonably coherent when viewed as isotropic in the transformed coordinate; however, we see less of this in Figs. 4 and 5. At short distances the temperature prediction errors (Fig. 4) are fit well, but the temperature observation errors are fit less well. The situation is somewhat reversed with the relative humidity errors, where the fit is generally better for the observation error (Fig. 5) than that for the prediction error. The use of the full third-order autoregressive function for the vertical correlation approximation induces some negative lobes in the approximation for the temperature prediction error and the relative humidity observation error. Generally the approximating correlation functions for temperature observation errors appear to be quite narrow, indicating that the vertical errors are nearly uncorrelated. It is noted that the transformation associated with some fits (Fig. 6, lower left) shows extreme deformation between certain points. It is also noted that some of the correlation values obtained through the FAR3 fits are larger than one (in the temperature observation error, and this also occurs at 1200 UTC), although these points are off the plotting window. There is no such occurrence with SAR2.
Some interesting things can be seen by looking at Figs. 4–7, but the plot of all correlation data as a scatterplot does not allow one to see the fit to the correlation points on any one level nor how the translation affects the correlation curve approximation. For this purpose Figs. 8–11 show the correlation curves mapped back to the log pressure coordinate. The 16 subplots of Fig. 8 show the vertical correlation of temperature prediction error at the 16 levels. The label indicates the correlation curve for the error at that level with other levels. Also shown in each subplot is a dotted line showing the empirical correlation. It is noted that for purposes of plotting the correlation curves the transformation between levels was obtained by piecewise linear interpolation. The figures can be perused at length and yield considerable information about the approximate behavior of the vertical correlation functions for temperature and relative humidity prediction and observation error. It is seen in Fig. 8 that the calculated vertical correlation of prediction error oscillates a larger number of times and with larger amplitude than can be captured by FAR3. The vertical correlation of observation error is quite narrow, and also oscillates widely, as seen in Fig. 9. The use of the transformations does allow considerable nonisotropy when viewed in log pressure coordinates. Figures 10 and 11 are the corresponding plots for relative humidity prediction error and relative humidity observation error, respectively.
3. Estimation of cross correlation of temperature and relative humidity errors by direct computation
The cross correlation between temperature and relative humidity errors has not been extensively studied. The only work known to the investigators is that mentioned in the introduction, by Steinle and Seaman (1995) using the NMC method. The more important part of this work was to attempt to compute from the innovation data the behavior of the cross correlation between temperature and relative humidity prediction errors.
First it is noted that the variance of the temperature prediction errors and the relative humidity prediction errors must be known (estimates, anyway) before the cross correlation of the two quantities can be estimated from the innovation data. It is believed that reasonable estimates are available from the work reported in the previous section.
The first attempt was to directly compute the cross-covariance data from the innovation data, bin it, and then approximate the spatial cross-covariance function. In principle the observation errors should be independent, and only the zero distance empirical cross covariance would need to be computed. In practice it was quickly discovered that this was not the case. In what follows, the innovations will be referred to as errors for simplicity. Figure 12 shows the binned cross-covariance data for relative humidity errors at 850 mb and the temperature errors at 925, 850, 700, and 500 mb at 0000 UTC. The fit using the SAR2 function is also shown for each case. The data for 925-mb temperature innovations are rather scattered, but the 850- and 500-mb data appear well behaved and are fit reasonably well by the approximating function. The 700-mb temperature innovation data seem to have an anomalous fit, perhaps partly because the function cannot change sign. Despite the fact that the function approximates the data fairly well, there is no good reason to believe that such functions would approximate well for other pairs of levels since cross-covariance functions need to satisfy fewer constraints than covariance functions. Note that the intercept of the approximating function misses the value at zero distance, in each case, by a nontrivial amount. Perusal of other pairs of levels also exhibits apparent additional correlated error at zero distance. Whether this error is indeed due to correlated observation error, or whether it is contamination of the innovation values for relative humidity when calculated from predicted specific humidity and observed dewpoint depression is unknown. In any case, it cannot be ignored.
Most of the software was readily available, so it was decided to approximate the directly computed cross-covariance data using both the SAR2 fit and the FAR3 fit. Using the intercepts of these approximations, the cross-covariance matrix for zero distances between the 16 temperature levels and the 7 relative humidity levels was computed, and then the vertical cross-correlation matrix for temperature and relative humidity was computed. The seven panels showing the correlation between relative humidity error and temperature errors at various heights at 0000 UTC are shown in Fig. 13, as derived from the SAR2 fits. Most notable are the strong negative correlation between temperature errors and relative humidity errors at the same level up to 850 mb, and lesser negative correlation at higher levels. Smaller amounts of data suggest one might be suspicious about the correlations between relative humidity error at 1000 mb and temperature error at the upper levels. Perusal of the fits to the cross-covariance data for relative humidity error at 1000 mb and temperature error at 200 mb and above shows that in most cases the fits do not seem unreasonable; however, the fit to the data for relative humidity errors at 700 mb and temperature errors at 20 mb is seen to be unrealistic. There is a similar poor fit at 1200 UTC for the relative humidity error at 850 mb and temperature error at 70 mb.
The fits to the cross-correlation data using the FAR3 approximation do not yield any comparable anomalous points, although at 0000 UTC the computed correlation between relative humidity error at 925 mb and temperature error at 925 mb is slightly less than −1. The FAR3 fits also show slightly better consistency between the two times than the SAR2 fits.
Because of the unknown properties of spatial cross-covariance functions, it seemed desirable to attempt to derive the cross-correlation properties by fitting only spatial covariance data. The analogy on which this idea is based is that of estimating vertical covariances between different levels of the height error (see Franke 1999b) by treating the thickness error for all combinations of levels, just as the temperature and relative humidity data were treated in the previous section. In this case, unfortunately, we are dealing with quantities that have different units (degrees, and none), and various authors (e.g., Cressie 1993) have warned against working with the difference of such quantities. In the next section an approach and the results will be presented.
4. Estimation of cross correlation of temperature and relative humidity errors using a differencing approach
One problem with computing the spatial covariance of the difference of two quantities is assigning a meaning to it when the two quantities have different units. Thus, while one can plunge ahead and compute things such as var(Vi − Wj) when Vi and Wj have different units, exactly what that means physically is questionable and troubling. Cressie (1993) suggests that the quantities need to be normalized in some way. Because it is necessary to compute the variances of the prediction errors for both temperature and relative humidity, it seems natural to normalize by the standard deviations of the error. To lend some consistency, the standard deviations derived from the same fitting function as is used to fit the spatial covariance of the difference will be used.
At this point the details of the equations for the case of temperature innovations and relative humidity innovations will be completely spelled out. Let σi and μj represent the standard deviations of the temperature prediction error at level i and the relative humidity prediction error at level j, respectively. Recall that the innovations are the differences between the observed value and the predicted value and are thus equal to the difference between the observation error and the prediction error. Having previously estimated the variances of the prediction errors for both temperature and relative humidity, we now nondimensionalize the innovation for temperature at level i and relative humidity at level j by dividing by the appropriate standard deviation, σi for temperature and μj for relative humidity. Now, letting δtoi − δtpi represent the normalized temperature innovation at level i and, correspondingly, δuoj − δupj represent the normalized relative humidity innovation at level j, we now consider the variance of the difference. We have
if we assume that the predicted values and the observed values are independent. Now consider Eq. (4) in a more general sense as describing spatial covariance of the difference on the left side. Because the observation error is independent for different stations, at distances greater than zero the right side becomes (here interpreting the quantities as spatial covariances)
Thus when the left side is approximated by the same techniques as used for temperature and relative humidity errors and extrapolated to zero distance, the intercept is an approximation to the quantity in Eq. (5) for zero distance. The actual empirical value at zero distance also includes the terms arising from observation errors, that is,
It may be possible that there are other terms involving error that are correlated with observation error (such problems are assumed away here). In such a case the values of the covariance of observation errors would be impossible to separate from the other correlated errors.
Noting that δtpi and δupj represent the normalized values of the predicted temperature and relative humidity errors, respectively, we see that the two variances have the value one. Further, the covariance on the left side is then seen to be the correlation between the two quantities, and thus we have
One of the key advantages of the difference approach is that all function approximations are to spatial covariances. Using the two approximations SAR2 and FAR3, the approximation of the correlation by the difference method was carried out. To illustrate some of the data involved, Fig. 14 shows the binned covariance data for four different cases. These are the covariances between relative humidity error at 850 mb and temperature error at 925, 850, 700, and 500 mb. When the intercept values obtained from fitting all differences are used in Eq. (6) the correlation matrix for the relative humidity and temperature errors are then available. The results were computed for both the SAR2 and FAR3 fits.
The results obtained from the SAR2 fits at 0000 UTC are shown in Fig. 15. Figure 15 should be compared with Fig. 13. Depending on the level, some the values compare rather well, while there are significant differences in other cases. There are no “out of bounds” points resulting from the difference method, in contrast to the results from the SAR2 fits using the direct method. The difference method tends to show a significant drift toward positive correlations at the higher pressure levels (generally above 200 mb, but lower in a few cases) that are not shown in the direct method calculations. No explanation for this has come to mind, although the investigators would urge skepticism concerning the reality of such results.
To exhibit some properties of the FAR3 approximation, the data and fit corresponding to Fig. 14 are shown in Fig. 16. Note the closer fit at small distances resulting in larger intercept values, the sharp transition from zero slope for the 925-mb curve, and the slight waviness of the 850- and 500-mb curves. In contrast to the SAR2 fits, the out of bounds points with FAR3 are now obtained using the difference method rather than the direct method. The cross-correlation curves derived from FAR3 fits to temperature and relative humidity errors using the difference method are shown in Fig. 17. The strong negative correlation between relative humidity errors and temperature errors at the same level are somewhat suppressed, and in some cases the correlation is positive. However, there are three out of bounds points, and a generally greater variation of the correlation across pressure levels occurs with the FAR3 fits. This variation and the close fits seen at small distances may indicate that the approximation embodies too much flexibility and that the actual behavior at small distances is not modeled properly. The consistent drift to positive correlations at the higher levels seen in Fig. 15 does not occur in Fig. 17, however.
5. Summary and conclusions
This study has attempted to determine some of the properties of the spatial covariance and vertical correlation of temperature and relative humidity prediction errors, and their vertical cross correlation. Use of different spatial approximations has led to consistent results in some respects such as the variance of prediction errors and correlation distances for both temperature and relative humidity. The attempt to approximate the vertical correlation functions with an isotropic function on a transformed domain led to somewhat inconsistent results when comparing those based on spatial fits using the SAR2 function with those based on FAR3 spatial fits. Although neither of the two approximations gave entirely satisfactory results, the SAR2 fits had the fewest real serious problems, such as “correlation” values between levels that are greater than one in value.
In attempting to fit the cross correlation, the more pleasing results were obtained by fitting the spatial cross-covariance data directly, even though the difference method is based entirely on fitting spatial covariance functions (of differences), making it eassier to choose an appropriate fitting function. With the direct method the additional flexibility of the FAR3 function leads to better vertical cross-correlation curves between the two variables at different levels.
Fitting the normalized difference of the two variables in the spatial domain using SAR2 fits yields somewhat more pleasing results for the vertical cross correlation than that using the FAR3 fits. There is, however, a disconcerting increase in the correlation between relative humidity errors and temperature errors as a function of height of the temperature errors for the results from the SAR2 fits. In the case of the difference scheme there are no “bad points” such as were noted in Fig. 13 using the SAR2 fits directly on the spatial cross covariance.
It is noted that the vertical correlation of temperature prediction errors and of temperature observation errors may not be isotropic in any transformed region. That is, for certain levels the negative lobe below the given level may not be matched by a negative lobe at upper levels. See Fig. 8 for the 16 plots of correlation of temperature prediction errors at 0000 UTC derived from SAR2 at a given level with those at other levels, and note especially the curves for 300 and 70 mb, and, to a lesser extent, 200 mb. Further it can be noted that the empirical curves are different at different heights, indicating inhomogeneity, at least to some extent. The corresponding curves for the associated temperature observation errors are shown in Fig. 9 and exhibit some of the same behavior. Whether such vertical correlation functions can be modeled using positive definite functions that have appropriate properties is not known to the investigators. As a matter of interest, it is noted that the empirical vertical correlation matrix for the temperature prediction errors shown in Fig. 8 is not positive definite, although that for the temperature observation errors shown in Fig. 9 is. The empirical vertical correlation matrices for the relative humidity prediction and observation errors (shown in Figs. 10 and 11, respectively) are both positive definite.
In summary, even though the attempt to approximate the vertical correlation functions with an isotropic function on a transformed domain did not perform as hoped, the structures computed for temperature are similar to those derived from geopotential correlations using the hydrostatic matrix (see Daley and Barker 2001) and the vertical correlation lengths are smaller than those derived by McNally (2000) using the NMC method. The cross correlations between temperature and relative humidity did indicate a signature (see Figs. 12 and 13), but only the horizontal structure was evident. The vertical structure appears to have been too small to be defined with the data analyzed. The structure in the cross correlations between temperature and relative humidity does exist, and as model resolutions increase we should be able to describe its vertical structure more accurately and begin using this information in data assimilation.
This work was supported by the Naval Research Laboratory/Monterey under Program Element 0602435N. Discussions of the work and help in obtaining the data were provided by Andy van Tuyl, Roger Daley, and Nancy Baker, and are greatly appreciated.
Corresponding author address: Prof. Richard Franke, Department of Mathematics, Naval Postgraduate School, 1411 Cunningham Rd., Room 341, Monterey, CA 93943-5216.
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In this and previous papers the investigator uses the distance at which the argument of the exponential is −1, but others have used decay to 0.5 of the maximum value.