Protocols for characterizing aeolian mass-flux profiles

Aeolian Research 1 (2009) 19–26

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Aeolian Research journal homepage: www.elsevier.com/locate/aeolia

Protocols for characterizing aeolian mass-flux profiles J.T. Ellis a,*, B. Li b, E.J. Farrell b, D.J. Sherman b a b

Department of Geography, University of South Carolina, Callcott Building, Columbia, SC 29208, USA Department of Geography, Texas A&M University, College Station, TX 77843, USA

a r t i c l e

i n f o

Article history: Received 29 August 2008 Revised 10 February 2009 Accepted 18 February 2009

Keywords: Aeolian saltation Sampling methods Sediment traps Field-based measurements Non-linear regression analysis

a b s t r a c t This paper describes a protocol for characterizing and analyzing the vertical mass-flux profiles above erodible beds in natural aeolian environments. Three main areas of methodological inconsistencies are explored, using two field data sets, to demonstrate the variability of results caused by methodological choices associated with: (1) inconsistent representation of sediment trap elevations; (2) erroneous or sub-optimal regression analysis; and (3) inadequate or ambiguous bed elevation measurements. The recommended methodology is physically based, produces results that most closely represent the measured data, and will permit results from different field and laboratory studies to be compared using a standardized convention. The recommended protocol suggests the following: (1) measure vertical flux profiles with as many traps as feasible with shorter traps near the bed; (2) measure bed elevation before, during, and after data collection; (3) use the geometric mean to represent trap centers; (4) use the saltationenhanced aerodynamic roughness length when approximating the bottom elevation of traps deployed at the bed (0 mm); (5) plot vertical mass-flux data with height above the bed as the independent variable; and (6) fit vertical mass-flux curves using non-linear, exponential curve fitting. Ó 2009 Elsevier B.V. All rights reserved.

1. Introduction The details of non-linear interactions between saltating grains and near surface winds and the accurate representation of characteristic flux profiles are important to understanding the physics of grain behavior, modeling sand transport rates, and predicting abrasion potential. Aeolian flux profiles were perhaps first measured by Bagnold et al. (1936), although his data described only the non-linear character of the distribution. Subsequent to his work, many complementary laboratory (Butterfield, 1999; Dong and Qian, 2007), field (Greeley et al., 1996; Namikas, 2003), and numerical (Anderson and Haff, 1988, 1991; Zheng et al., 2004) studies have been conducted. In evaluating previous studies, it is evident that even after seven decades of research there is no standard protocol for the measurement, analysis, and representation of aeolian massflux profiles. In some cases, it is not possible to recognize exactly how results were obtained. This makes comparisons between, or interpretations of, results from different studies difficult, at best. The purpose of this paper is to describe several methodological ambiguities observed in the literature and to recommend a standard protocol so that future studies may produce commensurate mass-flux profile data. From our review of published research, we identify three main points of methodological inconsistency, error, or deficiency: (1)

* Corresponding author. Tel.: +1 803 777 1593; fax: +1 803 777 4972. E-mail address: [email protected] (J.T. Ellis). 1875-9637/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.aeolia.2009.02.001

inconsistent representation of sediment trap elevations; (2) erroneous or sub-optimal regression analysis; and (3) deficient or ambiguous bed elevation measurements. Each of these points will be discussed in turn, and two data sets will be used to demonstrate the potential variability of results stemming from different protocol choices alone. The results of these analyses are used to formulate the protocol recommended herein.

2. Background 2.1. Trap Elevations There are multiple approaches to measuring aeolian saltation in the laboratory and in the field (Rasmussen and Mikkelsen, 1998; Li and Ni, 2003; Ellis et al., 2009), many of which aim to discern the distribution of moving sand above the surface. When mass flux is measured with non-point devices, such as typical impact sensors or sand traps, one must choose a representative elevation to characterize the vertical distribution. For example, a Safire-type impact sensor (Baas, 2004) measures grain impacts over a 0.02 m vertical length. The sand traps used by Namikas (2003) had vertical openings of 10, 20, or 40 mm and were 60 mm wide. The Safires and sand traps each integrate transport, or grain impacts, over their vertical dimension, but for the purposes of representing flux distributions, the data are reported as if they were obtained at a point. Several methods have been employed to designate an appropriate elevation for such points. Namikas (2003) and Rasmussen and

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J.T. Ellis et al. / Aeolian Research 1 (2009) 19–26

Mikkelsen (1998), for example, used the geometric mean of the top and bottom elevations of each trap. Ni et al. (2002) used the arithmetic mean, and Chen et al. (1996) plotted the trap data using the elevation of the trap top. For the bottom compartments, Namikas (2003) used a grain-size derived, roughness length estimate (z0 = 2d/30, where d is grain size) and Rasmussen and Mikkelsen (1998) used a saltation-enhanced aerodynamic roughness length instead of 0 mm as the bed elevation. Each of these options produces different representations of vertical flux distributions. We suggest that where large numbers of relatively ‘short’ (vertical heights of approximately 0.02 m, for example) traps or sensors are used, the choice of a representative elevation is less important because the potential differences in results from using alternative elevation representations are reduced. 2.2. Regression Analysis Representing the continuous vertical distribution of mass flux with curve fitting is typically accomplished using regression analysis to find best-fit relationships between flux data and elevation. The application of this technique to flux data is paramount, as the regression equations are usually the primary empirical source for calibration or verification of computer and mathematical models. Elevation should be the independent variable in such analyses (Namikas, 2003; Rasmussen and Mikkelsen, 1991). As is often the case in Earth science applications, however, dependent and independent variables are sometimes reversed, perhaps because of the powerful intuition to plot elevation on the ordinate (Mark and Church, 1977; Williams, 1983; Wilkinson, 1984; Bergeron and Abrahams, 1992; Bauer et al., 1992). Unfortunately, the results of regression analysis are different when the independent and dependent variables are switched. Such a reversal seems to have been the case in the analyses of Greeley et al. (1996), for example. The mass-flux profiles depicted by Gerety and Slingerland (1983), White (1996), Ni et al. (2002), Liu et al. (2006), Zhang et al. (2007), and Dong and Qian (2007) also plot elevation above the surface on the ordinates of their graphs, but their texts are ambiguous regarding the method employed to derive the curves. Another critical regression issue is the selection of the optimal regression model. It has been traditional to use linear approaches to find least-squares relationships for mass-flux profile data, including the linearization of data by taking logarithms. It is recognized, however, that linear regression, using logarithmically transformed data, introduces a bias when the data are de-transformed (Lee et al., 2004; Jansson, 1985). Where the potential for such bias arises and where such biases might affect interpretation, it is preferable to use a non-linear least-square method to fit curves to the data. For example, non-linear regression analysis is commonly used for the development of suspended-sediment rating curves (Crawford, 1991; Asselman, 2000; Horowitz, 2003). Lyn (2000) compared linear and non-linear regression methods for Rouse concentration profiles and found that non-linear regression replicated data best on linear–linear plots, and that linear methods worked best on log–log transformations (where de-transformation was not performed). These results suggest that consideration of a non-linear regression option is warranted. 2.3. Bed Elevation Accurate and precise knowledge of trap or instrument elevations above the bed surface is critical for quantitatively characterizing the mass-flux profile. It is relatively simple to establish such elevations when devices are first installed or before a measurement period begins, and this is commonly practiced. Elevation changes during or over a measurement period, however, are almost never reported. Although it is possible that bed elevations,

thus device heights, may be constant over some finite time interval, it is unlikely that such changes never occur, especially during non-sand feed laboratory experiments (Zou et al., 2001) or during intense transport conditions in the field. Sherman et al. (1998), for example, measured centimeter-scale fluctuations over approximately 17 min periods in a coastal aeolian environment. Under conditions when there is substantial transport, millimeter to centimeter scale bed elevation changes over similar time scales are probably common.

3. Methods In an effort to recommend a standardized protocol for measuring and analyzing vertical profiles of transporting sand, we considered field-based data to investigate the influence of the aforementioned methodological ambiguities. 3.1. Field Data Data from two field studies were used to formulate and assess our recommendations. We first considered data obtained from arrays of 7–10, vertically stacked hose-style traps (Pease et al., 2002) that were deployed on a sand sheet in the Guadalupe County Park on the coast of northern Santa Barbara County, California (cf., Bauer et al., 1998, for details on field site). The vertical opening of the lowest trap was either 10 or 25 mm, depending on the particular deployment. Openings of the other traps ranged from 25 to 100 mm, with the size of the opening increasing upward. Total stack height of the traps ranged from 0.375 to 0.600 m above the bed. All traps were 100 mm wide. Sand was collected through eleven 15-min data runs during periods of active saltation over a flat, unobstructed surface with a fetch length and width exceeding 60 m. Sand samples from each trap and each run were weighed and dry-sieved at 0.25 phi intervals to obtain mass and sand size data. The average, mean grain size for all runs was 0.39 mm, and this value is used for the analysis herein. Corresponding wind data were not collected during this experiment. We used data from the field experiment described in Namikas (1997), conducted in the Oceano Dunes State Vehicle Recreation Area (about 15 km north of the Guadalupe site) to assess our recommendations. This is one of a few field-based studies which reports comprehensive vertical profiles of wind-blown sand. Vertical transport data were obtained from an array of 15 traps with openings of 10–40 mm high, with the size of opening increasing upward. Maximum trap height above the bed was 0.35 m. All traps were 60 mm wide. Sand was collected during eight runs with durations from about 6 to 34 min during periods of saltation over a flat, unobstructed sand surface (with ripples present) with a fetch length approximately 100 m. Sand samples from each trap were weighed and dry-sieved at 0.25 phi intervals. Because of the small sample sizes obtained from some traps, many of the samples were combined. Mean grain size for the first six runs was 0.25 mm and for the last two runs, 0.26 mm. 3.2. Data Analysis 3.2.1. Normalizing Vertical Profiles of Trapped Sand The absolute quantity of sand captured in each trap was converted to a percent relative to the total mass trapped during a particular run. The normalization process allows direct comparison of profiles obtained from different transport conditions and environments. In hydrological environments, normalization is typically based on a reference (maximum) sediment concentration, with applications in laboratory studies (Rouse, 1938), rivers (Grams et al., 2006), estuaries (Clarke and Elliott, 1998), and coastal

J.T. Ellis et al. / Aeolian Research 1 (2009) 19–26

environments (Lee et al., 2004), for example. There is currently no accepted view on how to establish a similar reference concentration for aeolian saltation. The convention has been to normalize the measurements using either the total mass (Dong and Qian, 2007; Chen et al., 1996) or the total mass measured closest to the bed (Butterfield, 1999). We use the former approach. Many vertical profile measurements are obtained using traps with openings of different heights; therefore it is also necessary to normalize for trap height:

Q ni ¼ ½ðQ i =QÞ=ðhti  hbi Þ100

ð1Þ

where Qni is the normalized flux percent per 10 mm of the opening height of trap i, Qi is the mass flux caught by trap i, Q is the total mass flux measured during the run, and hti and hbi are the elevations of the top (t) and bottom (b) of trap i. 3.2.2. Trap Elevation When quantifying the vertical distribution of sand transport, it is necessary to represent that transport at one elevation above the bed for each trap (or other at-a-point sensors). We used the Guadalupe data with vertically integrated hose traps with openings of 10, 25, 50, or 100 mm to compare the results obtained when using the two centering methods that are most commonly employed; the arithmetic mean (AC; Eq. (2)) and the geometric mean (GC; Eq. (3)):

AC ¼ ððhti þ hbi Þ=2Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi GC ¼ hti  hbi

ð2Þ ð3Þ

Multiple solutions to overcome the methodological challenge of applying Eq. (3) to traps deployed on the bed (i.e. for hbi = 0) have been proposed (Namikas, 2003; Rasmussen and Mikkelsen, 1998). Where appropriate we estimated the saltation-enhanced (Rasmussen and Mikkelsen, 1998) roughness length to determine the bottom trap elevations and obtain geometric means (GC). We assumed hbi = 0 when calculating arithmetic means (AC). 3.2.3. Regression Analysis Normalized mass-flux percents and trap elevations were used as the dependent (y) and independent (x) variables, respectively, to determine the regression coefficients (a and b) using exponential (Eq. (4)), logarithmic (Eq. (5)), and power (Eq. (6)) relationships for each data run and for both centering methods:

y ¼ a  expðbxÞ

ð4Þ

y ¼ a  logðbxÞ

ð5Þ

b

ð6Þ

y ¼ ax

The analysis was repeated using the trap elevations as the dependent variables and, therefore, the normalized mass fluxes as the independent variables to determine the impact of ‘‘backward” regression analysis. Finally, we assessed the differences between linear and non-linear regression analyses of the profile data. 3.2.4. Bed Elevation In order to estimate the effects of ignoring bed elevation changes during a particular sampling period, we assessed the impacts of a maximum, undocumented change of ±25 mm from the original surface. We calculated the absolute percent error (err|%|) associated with a reference transport rate (Qr) that arises from changing the bed elevation by 1.0 mm increments across the 50 mm range of potential error:

errj%j ¼ absðQ e =Q r  1Þ100

ð7Þ

where Qe is the transport rate associated with a bed elevation change of (h  hr), h is the true elevation caused by the bed eleva-

21

tion changes and hr is the original reference elevation (that was assumed to be unchanged). 3.2.5. Assessing Recommendations Following the aforementioned analysis, we re-analyzed the data of Namikas (2003) using our recommendations and compared the results with those that he obtained to demonstrate the degree to which our protocol may influence results. Namikas (2003) reported the characteristics of mass-flux profiles for nine data sets. For each of his vertical traps, Namikas (2003) normalized the transport data by total flux, plotted the results in his Fig. 7, and fitted logarithmic, power, and exponential curves to the profiles. Results were reported in his Table 2. We digitized the averaged normalized flux data (Fig. 7; solid squares) for our re-analysis following the protocol of Sherman and Farrell (2008). 4. Results and Discussion 4.1. Trap Elevation For a 25 mm high trap deployed on the bed, using the grainroughness length to determine hbi, a geometric mean trap center of 0.81 mm is estimated. This elevation, although mathematically correct, is physically improbable, as it is less than three sand grain diameters high (for Namikas, 1997). Our results indicate that about 15% to almost 40% of the total transport occurs within the range of elevation spanned by our bottom traps (Table 1). It is therefore inconceivable that such transport is occurring within two or three grain diameters of the bed surface. The saltation-enhanced roughness length (z0) caused by saltation varies with particle size and shear velocity (Sherman and Farrell, 2008; Dong et al., 2003; Raupach, 1991) and can be estimated using the wind velocity profile (Bauer et al., 1992). Wind was not measured during the Guadalupe experiment, so we estimated z0 = 1.0 mm. Our estimation corresponds with Rasmussen and Mikkelsen (1998) and is conservative when compared to previous studies (Sherman and Farrell, 2008; Baas and Sherman, 2005). We assumed for this study that it was constant at 1.0 mm. When hbi = 1.0 mm for the bottom trap and hti = 25.0 mm, a center elevation of 5.0 mm is estimated when using Eq. (3) (GC). This elevation may still be too low (physically), but we are unable to devise a defensible alternative approach. We therefore used 1.0 mm for the bottom elevation of the lowest traps for geometric centering, and 0 mm for deriving the calculated arithmetic mean. Table 2 presents the trap arrangement for Run 1 from the Guadalupe experiment, and includes the calculated arithmetic and geometric trap centers.

4.2. Regression Analysis We used linear and non-linear regression to fit curves to the 11 Guadalupe data sets using Eqs. (4)–(6), with both arithmetic (Eq. (2)) and geometric trap (Eq. (3)) centering. The results are summarized in Table 3. The r2 values for each method average 0.99 for all runs. For both trap centering approaches, the error sum of squares (SSE) associated with the curve fitting is smaller (by more than an order of magnitude) with the non-linear regressions than with linear regression. For all methods, Run 8 contributes disproportionately to the SSE (cf., Table 3). Because the distribution of vertical transport for Run 8 is so different from the other data sets, we believe that these measurements may be contaminated as a result of mishandling samples in the field. Omitting Run 8 from the analysis reduces the respective average SSE values. An alternative approach to assessing the viability of the different regression methods is to compare the relative magnitude of residuals between observed and predicted masses found for the

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J.T. Ellis et al. / Aeolian Research 1 (2009) 19–26

Table 1 Transport data for the Guadalupe data runs. The bottom of Trap 1 was always flushed with the bed (hb = 0) and Traps 2 and higher were stacked directly on top of each other. Trap height is (hti  hbi). Trap 1

Trap 2

Trap 3

Trap 4

Trap 5

Trap 6

Trap 7

Trap 8

Run 1

Trap height (mm) Total elevation (mm) Mass (g) Mass (%) Qni

25 25 189.93 28.92 11.57

25 50 127.32 19.39 7.76

25 75 90.29 13.75 5.5

25 100 64.93 9.89 3.95

50 150 88.41 13.46 2.69

50 200 48.87 7.44 1.49

100 300 39.51 6.02 0.6

100 400 7.44 1.13 0.11

Trap 9

Trap 10

Run 2

Trap height (mm) Total elevation (mm) Mass (g) Mass (%) Qni

25 25 33.14 32.57 13.03

25 50 23.07 22.67 9.07

25 75 15.67 15.4 6.16

25 100 10.53 10.35 4.14

50 150 11.73 11.53 2.31

50 200 5.17 5.08 1.02

100 300 2.45 2.41 0.24

Run 3

Trap height (mm) Total elevation (mm) Mass (g) Mass (%) Qni

25 25 97.16 29.4 11.76

25 50 76.24 23.07 9.23

25 75 50.05 15.15 6.06

25 100 34.02 10.3 4.12

50 150 37.76 11.43 2.29

50 200 20.13 6.09 1.22

100 300 11.68 3.53 0.35

100 400 3.41 1.03 0.10

Run 4

Trap height (mm) Total elevation (mm) Mass (g) Mass (%) Qni

25 25 178.08 20.09 8.04

25 50 162.85 18.37 7.35

25 75 135.4 15.27 6.11

25 100 97.57 11.01 4.40

50 150 129.96 14.66 2.93

50 200 79.34 8.95 1.79

100 300 75.49 8.52 0.85

100 400 27.77 3.13 0.31

Run 5

Trap height (mm) Total elevation (mm) Mass (g) Mass (%) Qni

25 25 438.76 24.55 9.74

25 50 357.79 20.02 7.94

25 75 234.13 13.10 5.20

25 100 172.58 9.66 3.83

50 150 238.25 13.33 2.64

50 200 148.65 8.32 1.65

100 300 142.91 8.00 0.79

100 400 54.25 3.04 0.30

100 500 15.17 0.85 0.08

Run 6

Trap height (mm) Total elevation (mm) Mass (g) Mass (%) Qni

25 25 615.4 29.87 11.81

25 50 391.45 19.00 7.51

25 75 266.09 12.92 5.11

25 100 195.22 9.48 3.75

50 150 251.96 12.23 2.42

50 200 146.46 7.11 1.41

100 300 140.59 6.82 0.67

100 400 52.95 2.57 0.25

100 500 18.1 0.88 0.09

100 600 5.68 0.28 0.03

Run 7

Trap height (mm) Total elevation (mm) Mass (g) Mass (%) Qni

25 25 368.01 20.71 8.19

25 50 396.81 22.33 8.83

25 75 258.82 14.57 5.76

25 100 189.34 10.66 4.21

50 150 236.64 13.32 2.63

50 200 142.28 8.01 1.58

100 300 137.33 7.73 0.76

100 400 47.46 2.67 0.26

100 500 16.18 0.91 0.09

100 600 4.93 0.28 0.03

Run 8

Trap height (mm) Total elevation (mm) Mass (g) Mass (%) Qni

10 10 33.34 20.11 20.11

25 35 44.25 26.69 10.67

25 60 31.02 18.71 7.48

25 85 21.31 12.85 5.14

25 110 13.02 7.85 3.14

50 160 14.54 8.77 1.75

50 210 5.58 3.37 0.67

100 310 2.76 1.66 0.17

100 410 0.01 0.01 0.00

Run 9

Trap height (mm) Total elevation (mm) Mass (g) Mass (%) Qni

10 10 17.65 15.93 15.93

25 35 27.62 24.93 9.97

25 60 18.48 16.68 6.67

25 85 12.4 11.19 4.48

25 110 7.24 6.54 2.61

50 160 8.54 7.71 1.54

50 210 3.68 3.32 0.66

100 310 15.17 13.69 0.14

Run 10

Trap height (mm) Total elevation (mm) Mass (g) Mass (%) Qm

25 25 86.63 33.68 13.47

25 50 56.54 21.98 8.79

25 75 39.17 15.23 6.09

25 100 26.51 10.31 4.12

50 150 28.82 11.21 2.24

50 200 13.11 5.10 1.02

100 300 5.4 2.10 0.21

100 400 1.01 0.39 0.04

Run 11

Trap height (mm) Total elevation (mm) Mass (g) Mass (%) Qni

25 25 76.87 38.95 15.58

25 50 46.29 23.45 9.38

25 75 27.72 14.04 5.62

50 125 29.85 15.12 3.02

50 175 12.63 6.40 1.28

100 275 3.17 1.61 0.16

100 375 0.85 0.43 0.04

Table 2 Trap elevations for Run 1 calculated using the arithmetic (AC) and geometric (GC) methods. The cumulative height of the stack of traps (measured from the trap top) is hti. Run 1

Trap 1

Trap 2

Trap 3

Trap 4

Trap 5

Trap 6

Trap 7

Trap 8

hti (mm) AC (mm) GC (mm)

25 12.50 5.00

50 37.50 35.36

75 62.50 61.24

100 87.50 86.60

150 125.00 122.47

200 175.00 173.21

300 250.00 244.95

400 350.00 346.41

bottom trap (for our data, the bottom trap averaged about 27% of the total mass transport). These results are also presented in Table

3. Similar to the SSE results, the non-linear regression outperforms the linear regression regarding the percent error in predicted

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J.T. Ellis et al. / Aeolian Research 1 (2009) 19–26

Table 3 Comparison of linear and non-linear regression analysis and arithmetic (AC) and geometric (GC) centering. r2, error sum of squares (SSE), and percentage error in predicted bottom trap mass transport are calculated for each Guadalupe run, the average of all runs, and the average of all runs except Run 8. Linear fit

Run Run Run Run Run Run Run Run Run Run Run

1 2 3 4 5 6 7 8 9 10 11

Mean Ex. Run 8

Non-linear fit

Linear fit

Non-linear fit

Linear fit

AC

GC

AC

GC

AC

GC

AC

GC

AC

r2

r2

r2

r2

SSE

SSE

SSE

SSE

Error in predicted bottom trap (BT) flux (%)

0.99 1.00 1.00 1.00 1.00 1.00 1.00 0.92 1.00 1.00 0.99

1.00 0.99 1.00 0.99 1.00 1.00 1.00 0.92 1.00 1.00 0.99

1.00 1.00 0.99 0.98 0.99 0.99 0.96 0.98 0.99 1.00 1.00

1.00 1.00 0.99 0.97 0.99 1.00 0.95 0.97 0.99 1.00 1.00

0.48 1.12 0.74 1.52 1.07 7.02 4.29 186.47 2.64 1.74 0.19

0.46 4.31 2.28 2.72 0.89 4.86 6.02 208.31 3.62 5.61 0.10

0.36 0.04 0.67 1.05 0.59 0.76 4.01 7.98 1.96 0.06 0.18

0.07 0.41 1.42 1.61 0.80 0.28 5.06 10.85 3.24 0.13 0.13

1.92 7.81 3.49 13.04 4.29 21.53 16.63 50.97 6.70 8.27 1.41

4.09 15.60 10.27 18.04 0.32 17.70 22.12 51.67 6.39 16.87 6.15

2.12 0.62 3.03 6.96 1.61 2.73 12.68 5.97 3.44 0.56 0.57

0.62 1.89 4.23 8.34 2.99 1.32 13.78 7.54 4.75 0.75 0.51

0.99 1.00

0.99 1.00

0.99 0.99

0.99 0.99

18.85 2.08

21.84 3.19

1.61 0.97

2.18 1.31

12.34 8.51

15.42 11.75

3.66 3.43

4.25 3.92

bottom trap flux. If we repeat this analysis without Run 8 data, the predicted error is more than twice as large for the linear regression than for the non-linear regression. These results lead us to conclude that non-linear regression is better suited to fit the distribution curves. In order to assess which trap centering method produced the best fit to the data, again with emphasis on predicting transport near the bed and by removing Run 8, we fit exponential, logarithmic, and power (Eqs. (4)–(6)) function curves to the Guadalupe data sets with both centering methods. Table 4 summarizes the results of this analysis, including the means and standard deviations for r2, a and b constants for the different curves, and SSE and relative error for the bottom trap. The mean profiles representing vertical saltation system, using the average a and b constants presented in Table 4, are plotted in Fig. 1. The curves fit using the exponential distribution produce the highest r2 values, the smallest error sum of squares, and the least error at the bottom trap. Curves fit with the exponential distribution that use the arithmetic trap centers produce the smallest SSE, although the absolute difference is small. It is believed that the geometric centering used for the bottom trap still produces a value that is too low to properly represent the saltation system. The differences between the two sets of error sum of squares were compared using Student’s t-test. The results suggest that the means are from the same population (n = 11, t = 0.336, and p = 0.74): the mean error differences associated with the two methods are statistically trivial. We believe that this result supports a recommendation to employ the geometric centering because it better represents an exponential distribution of the saltating grains. We assessed the magnitude of error that results from using mass transport rather than elevation as the independent variable

Non-linear fit GC

AC

GC

in least-squares curve fitting by non-linear regression analysis by re-analyzing the Guadalupe data using the exponential distribution with non-linear regression geometric mean trap centers. SSE and the error in predicted bottom trap increases from 2.18% and 4.25% to 62.21% and 20.37%, respectively. The degree of differences increases substantially with decreasing r2. It is unlikely that anyone would argue for the misapplication of regression as described, however we believe it worthwhile to note the potential for increased error in cases where this mistake may have been made and the results published. 4.3. Bed Elevation The relative change in mass flux (Q/Qr) associated with unrecognized bed elevation change (eb) is found by solving:

Q=Q r ¼ ebðhhr Þ

ð8Þ

We converted relative measurement errors to absolute % error (err|%|) using Eq. (8) and the results from Run 6 are depicted in Fig. 2, as an example. For this example, we assumed that the relative error is constant for any elevation above the surface, but the absolute error declines rapidly away from the bed. We believe that 10 mm scale bed elevation changes are probably quite common in field, and perhaps laboratory environments, and would cause 10–15% errors in reported transport if everything else was perfect. These errors increase to about 23–33% with a 20 mm unrecognized or unrecorded bed elevation change. The error depicted in Fig. 2 is asymmetric around a 0 mm bed elevation change, larger errors occur when the bed accretes. The issue of bed elevation change is complicated (and almost unavoidable) be-

Table 4 Average r2 ðr 2 Þ; regression intercept (a), slope (b), and error sum of squares (SSE); and the mean error percent of the bottom trap (mean error (%) BT) for the exponential, logarithmic, and power distributions while employing the arithmetic (AC) and geometric (GC) centers. The associated standard deviations (r) are also provided. r2

rr2

Intercept (a)

r (a)

Slope (b)

r (b)

SSE

r (SSE)

Mean error BT (%)

r Error BT (%)

Exponential

AC GC

0.99 0.99

0.01 0.02

15.06 13.77

3.85 3.52

0.02 0.01

0.004 0.004

1.61 2.18

2.41 3.27

3.66 4.25

3.66 4.17

Logarithmic

AC GC

0.96 0.94

0.03 0.05

3.86 3.19

0.90 0.84

21.75 18.33

4.24 3.94

4.87 6.49

3.05 3.96

5.97 8.17

5.30 7.30

Power

AC GC

0.88 0.81

0.06 0.09

60.72 27.00

22.50 7.35

0.61 0.44

0.08 0.06

15.48 24.17

5.73 5.65

8.38 7.91

4.98 4.25

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J.T. Ellis et al. / Aeolian Research 1 (2009) 19–26

30 Exponential; AC Logarithmic; AC Power; AC Exponential; GC Logarithmic; GC Power; GC

25

35 20

Normalizzed Flux (%)

30

25 15

20

15 10

10

0

5

10

15

20

5

0 0

50

100

150

200

250

300

350

400

450

500

Height (mm) Fig. 1. Average, normalized representations for the Guadalupe vertical saltation runs. The exponential, logarithmic, and power regression distributions are shown by the dotted, solid, and dashed lines, respectively, and assume a 1.0 mm roughness length. Trap elevations were determined using the arithmetic (AC) and geometric (GC) methods and are represented by the black and grey lines, respectively. Inset shows elevations close to the bed (0–20 mm) where transport is greatest.

cause there is no unambiguous way to treat bed elevation irregularities caused by, for example, the presence of aeolian ripples

Percent Error of Normalized Mass Transport

45

4.4. Recommended Protocols

40 35 30 25 20 15 10 5 0 -25

whose heights are commonly in the range of 10–20 mm. This analysis also assumes that all other trap spacings are perfectly known.

-20

-15

-10

-5

0

5

10

15

20

25

Bed Elevation Change (mm) Fig. 2. Percent error (err|%|) of normalized mass transport associated with changing bed elevations using Run 6 data as the example.

Based on the analyses previously described, we recommend the following protocol for experimental design and subsequent data analysis. First, in the design of field experiments where mass-flux data will be obtained with at-a-point source devices, it is important to minimize the vertical spacing of the measurements and to maximize the total number of traps or sensors within the saltation system. This approach reduces the impacts of any errors associated with trap centering methods and maximizes the data available for subsequent curve fitting. Short trap openings, especially for those deployed on or near the surface, are recommended. Traps with larger openings are reasonable for use at higher elevations where the flux profile analysis becomes less sensitive to small centering errors. It is also recommended that bed surface changes be monitored throughout saltation events so that elevation changes can be recognized and accommodated in subsequent analyses. During data analysis, we recommend that flux data be normalized by both trap geometry and total flux. The first normalization corrects for measurements made with traps of different heights

J.T. Ellis et al. / Aeolian Research 1 (2009) 19–26

and widths and may be unnecessary when employing traps with uniform dimensions nevertheless, we suggest normalizing for the purposes of comparing results from different studies. The second normalization accomplishes a similar purpose – ameliorating comparison of flux data gathered under different transport conditions. We suggest that trap centers should be estimated using the geometric mean, with the elevation of the bottom of the lowest trap represented with an estimate of saltation-enhanced roughness length. If appropriate wind data are available, we recommend following one of the methods discussed in Sherman and Farrell (2008). If the necessary wind data are not available we suggest using 1.0 mm as an estimate for saltation-enhanced aerodynamic roughness length, following Rasmussen and Mikkelsen (1998). Analysis of the Guadalupe data indicates that the vertical flux gradient is best represented by an exponential distribution. We recommend that the exponential curve be fit using the non-linear, least-squares method. 5. Re-Analysis of Namikas (2003) Data We fit an exponential curve to the Namikas (2003) normalized flux digitized data (his Fig. 7) and we were able to reproduce his value of a (our a, or intercept) within 0.1% and his value of b (our b, or slope) within 2%. The small differences are attributed to errors in digitizing. To assess the impacts of our protocol on these data, we substituted our recommended saltation-enhanced aerodynamic roughness length (1.0 mm) for the grain-roughness length used by Namikas (2003) for the elevation of the bottom trap and fit an exponential distribution using a non-linear leastsquares-curve-fitting routine. The Namikas (2003) value for a was 6.36 and our re-analysis yielded a value of 10.60. His value for b was 20.60 and ours was 35.22. For the two curve fittings, his r2 was 0.97, whereas ours was 0.99. We also calculated the error sum of squares for the two methods – the SSE from his approach was 15.63 and that from our protocol was 1.29. Finally, we compared the prediction errors obtained for mass flux at the bottom trap. Our method produced an error of 3.0%, a substantial improvement over the 36.6% error found with the approach used by Namikas (2003). These results indicate that there is a substantial enhancement in curve fitting obtained using our recommended protocol. Of particular importance is the order of magnitude reduction of error for normalized transport rates at the bottom trap.

2. 3.

4.

5.

25

the potential error associated with selecting a trap centering method because the difference between the instrument top and bottom is typically minimal. Repetitive measurements of bed elevation should be made during data collection and trap elevations adjusted accordingly. Use the geometric mean (Eq. (3)) to represent the trap center because it is physically more representative of the non-linear distribution of flux above the bed. Use an estimate (1.0 mm recommended herein) of the roughness length for the bottom trap when the bottom trap is deployed at 0 mm and corresponding wind velocity data are not available. If data are available, we suggest using an estimated saltation-enhanced roughness length (cf., Sherman and Farrell, 2008). Estimate the continuous vertical mass-flux distribution using a non-linear, least squares, exponential curve (Eq. (4)).

Baker (1992, p. 80) indicates that ‘‘controlled experiments, at least for the most interesting phenomena, are not possible in much of the Earth sciences, including geomorphology”. The pervasiveness of this attitude has, to some degree, promoted a laissez faire attitude toward some of the basic tenets associated with the development of geomorphological knowledge. One such tenet is that the results of scientific experiments should be replicable. As we have demonstrated, the lack of a standard protocol for the measurement and analysis of vertical mass profiles for aeolian saltation precludes replicability. We attest that the protocol described herein represents a meaningful step toward rationalizing the study of wind and sand systems by minimizing the influences of different methodologies on our experimental results. Acknowledgements Data from Guadalupe were collected with the financial support from the National Science Foundation, Geography and Regional Science (SBE-9511529; Co-PI’s: Bernard Bauer and D.J.S.) and with field assistance by B. Bauer, Steven Namikas, and Jianchun Yi. S. Namikas was very generous and supportive of our use of his data for our analysis. Ideas in this paper originated when analyzing data collected in Esposende, Portugal; an experiment supported by a Fulbright Senior Scholar Fellowship to D.J.S. with field assistance by Helena Maria Leite Granja. The quality of the article was greatly improved by two anonymous reviewers. All errors and/or omissions are solely the fault of the senior author.

6. Summary and Conclusions Our review of the literature indicates substantial variability in the methods used to measure and analyze vertical profiles of saltating sand. Using a data set obtained during a field experiment in Guadalupe County Park, CA, we compared several methodological approaches that are commonly cited. From these results we suggest a protocol to analyze flux profiles that is physically meaningful, statistically robust, and that match observations closely. We tested the resulting protocol by re-analyzing the field data published in Namikas (2003) and by comparing the new results to those obtained in the original analysis. We found a slight improvement in r2 values and a substantial improvement in estimations of flux near the bed. Based on these analyses, we recommend a standard protocol when considering vertical sand transport over an unconsolidated bed that comprises the following: 1. Vertical flux profiles should be measured with as many vertical traps as feasible. Smaller traps (vertical heights) are ideal, particularly close to the bed. At-a-point sensors are recommended for finer resolution studies. Use of these instruments decreases

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