Appendix B. Details of the definition of scale.
Practical implementation of the definitions of scale involves a number of details and caveats, which are discussed below.
1. The Peak Scale
Practical considerations and caveats regarding calculation of the spectrum itself have been covered in Appendix A. The fraction of the overall variance associated with each dominant spectral peak was calculated by integrating the area under the peak (the limits of which were judged by eye) and dividing that area by the integral of the entire spectrum.
Note that the definition of the peak scale requires that we be able to discern discrete peaks in the spectrum. When examining our spectra for the existence of discrete peaks, the following criteria were applied. If, on a log-log plot of the spectrum, the lower 95% confidence limit for a spectral estimate (see Appendix A) fell substantially above a regression line through the data, a peak was suspected, and if two adjacent spectral estimates met this criterion, the case for the existence of a peak was strengthened. In a few cases, the lower 95% confidence limit for a single estimate fell at or just marginally above the regression line, and in these cases we have taken a conservative approach and assume that these "peaks" are not sufficiently dominant to warrant the calculation of a peak scale. For each vetted peak we note the fraction of the overall variance contained within that peak. Only if a substantial fraction of the overall variance is contained within a peak (> approximately 25%), does the scale associated with that peak likely have much ecological significance.
2. The Frequency Scale
The frequency scale defined using Eq. 2 is affected by issues of practical measurement. The upper limit of integration is constrained by the highest frequency at which we can examine a phenomenon. This limiting frequency (fg, the Nyquist frequency, see Appendix A) corresponds to twice the smallest time or distance over which we make measurements. The Nyquist time or distance is in turn a measure of the grain of measurement. The lower limit of integration, fmax, is constrained by the largest time or distance over which we can measure a phenomenon (the extent). Given these practical limits, the average frequency of the measurable variation is
![]() |
(B.1)
|
What are the effects of these limits
of integration on the average frequency? The answer depends on the shape of
the spectrum. If the spectral density is low in the vicinity of both the maximum
and minimum frequencies, the presence of these constraints has negligible effect.
In contrast, if the spectral density is substantial near either limit, the magnitude
of the limit affects the calculated scale. For example, by inserting Eq. 8 into
Eqs. 2 and 3, we find that the frequency scale of a 1/f-noise process
(0 <
<1) is:
![]() |
(B.2)
|
An increase in either the grain (fg) or the extent (fmin) of measurement results in an increase in the calculated frequency scale.
As fmin approaches 0 (that is, as the extent of the measurement gets very large), this expression reduces to:
![]() |
(B.3)
|
In this case, the f-scale
is independent of the extent of measurement, and the grain has the strongest
influence on the calculated f-scale: the smaller the grain, the higher
fg is, and the smaller the scale. Note that when
> 1, the integral in the numerator of Eq. 2 is improper, and it cannot be
used to calculate a scale for a measurement of infinite extent.
In light of these complications, the frequency scale of 1/f-noise processes must be treated with caution. In particular, one would expect that for any practical set of measurements the calculated scale will increase with an increase in either the grain or the extent of the measurements. The f-scale values subject to this cautionary note are indicated in Tables 1 and 2 with an asterisk.
3. The Wavelength Scale
As with the frequency scale, practical
application of the definition of the wavelength scale (Eq. 4) is constrained
by the grain and extent of measurement. The upper limit of integration is constrained
by the extent, max,
and the lower limit by the grain,
g:
![]() |
(B.4)
|
If the spectral density is substantial
in the vicinity of these limits, the magnitude of the limits can affect the
calculated scale. For example, if the spectrum is 1/f noise (0 <
<1), Lw is
![]() |
(B.5)
|
An increase in either the extent or grain of measurement results in an increase in the calculated wavelength scale.
If the grain of the measurement is very small compared to the extent, this expression reduces to
![]() |
(B.6)
|
Clearly, in this case the extent of the measurement is the strongest influence on the calculated wavelength scale.
4. The Integral Scale
The integral scale has been widely
used in the study of turbulence as a measure of the characteristic scale of
velocity fluctuations (Tennekes and Lumley 1972). In this
application, it is generally assumed that the autocorrelation function asymptotically
decays to 0 at infinite lag, and the parameter a in Eq. 5 is thus commonly
cited as . In contrast,
ecological systems often exhibit autocorrelation functions that wander negative,
leading us to the arbitrary selection of a used here.
As with the frequency-, and wavelength scales, the integral scale can be directly affected by the grain and extent of measurement. For the 1/f-noise spatial data typical of this study, the grain of measurement has little effect, whereas the calculated i-scale is sensitive to the measurement’s extent: the larger the extent, the larger the i-scale. This effect is illustrated in Fig. B1. Here, the autocovariance for the 1/f-noise process has been estimated by numerically solving:
![]() |
(B.7)
|
This is the inverse Fourier transform of the spectrum, which, by the Wiener-Kinchine relationships is equal to the autocovariance (Bendat and Piersol 1986).
5. The Derivative Scale
The concept of the derivative scale
is challenged by our ability to calculate the derivative for the type of discrete
data that are available for ecological measurements. The standard finite-difference
technique for estimating the derivative at a point is illustrated in Fig. B2.
The value yi of a process is measured at a series of points,
xi, where the spacing between measurements is a constant x.
For three adjacent points, the derivative at the middle point, xi,
is estimated as a three-point average:
![]() |
|
(B.8)
|
This formula works well as long
as the scale of variation is large relative to 2x.
However, when the measured quantity varies rapidly, Eq. B.8 can lead to a gross
underestimation of the average derivative of the process. No fully satisfactory
solution is apparent for this problem. As a partial solution, we have chosen
to calculate the derivative as a two-point average:
![]() |
(B.9)
|
Use of this two-point derivative reduces the tendency to overestimate the derivative scale. The magnitude of the residual tendency is shown in Fig. B3. In plotting this figure, Eq. B.9 has been used to calculate the derivative scale of a fixed wavelength monochromatic sine wave as the spacing between measurements is varied. The ratio of the sampled vs. the continuous scale is almost always greater than 1.0 (that is, the calculated derivative scale is too large), but the deviation from 1.0 is relatively small (on average, only 5%). This curve has been truncated at a measurement interval equal to half the wavelength, a spacing that places our measurements at the Nyquist frequency (see Appendix A).
6. The Variance Scale
In practice, the scheme described in the text is carried out through the use of spectral analysis. As described in Appendix A, the spectrum was calculated for a particular variable, and the standard deviation at a measurement scale a is defined as the square root of the sum of the variances at all smaller scales (higher frequencies):
![]() |
(B.10)
|
Characterization of the inflection point, ai, in this curve was then carried out as described in the text. This procedure works well for our data, but there is no guarantee that it will work for all data.
The practical value of the inflection point in a curve of logs vs. log measurement extent (e.g., Fig. 3) is as an estimate of the scale at which a phenomenon must be measured to ensure that most of the variation present in the real world is also present in one's data. This value is jeopardized if, as the extent of measurement increases beyond the variance scale, the variance continues to increase. Thus, the slope of the curve of logs vs. log measurement extent at large extents can be used as an index of the "solidity" of our estimation of the v-scale. If this slope is small, the inflection point represents a solid estimate of the variance scale. If, in contrast, the slope is substantial, this is an indication that the standard deviation continues to increase with an increase in the extent of measurement, and our estimate of the v-scale is weak. In general, the slopes calculated for our temporal data are small, whereas the slopes for our spatial data are substantial (Table B1). The estimates of the spatial v-scale should therefore be taken with a grain of salt.
Comparisons Among Scales
When these definitions are applied to our data, the different methods often yield different estimates of scale. These differences are due to the nature of the data (in particular, the shape of the spectrum), and are not intrinsic to the definitions themselves. The sole exception to this statement is the derivative scale, which yields a value that is approximately 5% high, as noted above. This slight bias is negligible compared to the order-of-magnitude differences commonly found when these definitions are applied to real-world data. The precise manner in which a data series interacts with our definitions of scale is complex. However, we have conducted a preliminary exploration. First, we calculated the scale for a simulated monochromatic, sinusoidal signal that varies with a 10-m wavelength, sampled as it would be on our transects. As expected, each definition of scale yields a reliable 5-m scale for each of the transect lengths for each of the p-, f-, w-, v-, and i-scale calculations.
We next applied our definitions
to simulated 1/f-noise data following a method suggested by Hastings
and Sugihara (1993). To construct each data series, we first created an
appropriate Fourier series by choosing amplitude coefficients for a harmonic
series of frequencies, fi. The coefficient for each harmonic
is the absolute value of a sample chosen from a normal distribution with a mean
of zero and a variance equal to 1/fi.
Each harmonic in the Fourier series was then assigned a random phase, uniformly
distributed between 0 and 2
.
The inverse Fourier transform of this series yielded one realization of a 1/f-noise
data set with spectral exponent
.
Thirty such data sets were created for each value of
for each method of calculating scale, and the average and standard deviation
of calculated scales were determined.
values ranging from 0 to 2 were tested, and the results are given in Fig. 7.
In all cases, the slope of log
vs. log measurement extent was substantial for large values of a, suggesting
that the estimates of the v-scale shown here are "weak," as discussed
above.
Note that all of our definitions for the scale of variability can be related through the spectrum. This interrelationship is simply a mathematical acknowledgment of the fact that any definition of the scale of variability is tied to a description of pattern. A spectrum can be calculated for any natural pattern (periodic or otherwise) (Priestley 1981, Bendat and Piersol 1986), thus the existence of a spectrum does not imply that a pattern is periodic.
LITERATURE CITED
Bendat, J. S., and A. G. Piersol. 1986. Random data: analysis and measurement procedures (Second Edition). John Wiley and Sons, New York, New York, USA.
Hastings, H. M., and G. Sugihara. 1993. Fractals: a user's guide for the natural sciences. Oxford University Press, Oxford, UK.
Priestley, M. B. 1981. Spectral analysis and time series. Academic Press. New York, New York, USA.
Tennekes, H., and J. L. Lumley. 1972. A first course in turbulence. MIT Press, Cambridge, Massachusetts, USA.
Table B1. Parameters related to the "solidity" of the v-scale estimate.
High Slope
Low Slope
Ratio
Solar irradiance
0.012
102.00
0.000117
Intertidal body temp.
0.045
204.00
0.000220
Upwelling index
0.050
7.16
0.006980
Sea surface temp.
0.018
317.00
0.000057
Significant wave height
0.007
251.00
0.000028
Force
short
0.405
8.44
0.048000
medium
0.189
5.55
0.034100
long
0.300
4.10
0.073000
Wave force index
short
0.176
3.46
0.050800
medium
0.201
3.45
0.058300
long
0.195
6.06
0.032200
Temperature
short
0.446
6.47
0.068900
medium
0.274
5.81
0.047200
Chlorophyll
medium
0.488
6.11
0.079900
Diversity
short
0.108
6.71
0.016000
medium
0.087
5.30
0.016400
long
0.165
7.33
0.022900
Mussel density
short
0.128
3.45
0.037100
medium
0.244
6.75
0.036100
long
0.258
7.22
0.026600
Mussel disturbance
short
0.074
1.82
0.040700
medium
0.078
4.83
0.016100
long
0.140
8.05
0.017400
Mussel recruitment
medium
0.159
2.95
0.053900
Predators
short
0.095
5.47
0.017400
medium
0.695
6.32
0.011000
long
0.035
2.38
0.014700
Grazers
short
0.052
4.52
0.011500
medium
0.298
11.30
0.026200
long
0.112
3.84
0.029200
Notes: The lower the ratio of slopes, the better the estimate. High slopes > 0 indicate that variability continues to increase with measurement scale, and the calculated v-scale thus incorporates only a fraction of the total variability.
![]() |
FIG. B1. The integral scale is affected by the extent of measurement for 1/f-noise processes. To produce these curves 1/f-noise signals with a maximum extent of 5000 m were sampled at a range of measurement extents. |
![]() |
FIG. B2. A schematic representation of the factors used to calculate a derivative for discretely sampled data (see text). |
![]() |
FIG. B3. Estimation of the derivative using finite difference leads to an overestimate of the derivative scale. The ratio of the sampled, calculated scale to the actual scale is shown as a function of the relative magnitude of the measurement interval to the wavelength of the sinusoidal signal. |