Statistical estimation and generalized additive modeling of rock glacier distribution in the San Juan Mountains, Colorado, United States

Authors


Abstract

[1] Our goal is to quantify rock glacier abundance and analyze the topographic controls of rock glacier distribution patterns. For this purpose we use statistical estimation techniques and generalized additive models relating rock glacier occurrences to terrain attributes. Significant applied results include the ability to determine water equivalence and denudation rates. The statistical estimation of regional rock glacier abundance based on local interpretation of air photos is an efficient alternative to costly rock glacier inventories. These presence-absence data (N = 2933) also allow us to analyze the partly nonlinear topographic controls on rock glacier distribution with generalized additive models. We apply these techniques in the San Juan Mountains (2874 km2 above 3400 m), Colorado, where we obtained a total rock glacier surface area of 70 km2 corresponding to a water equivalence in the order of 0.50–0.76 km3. Estimated rock glacier debris volumes imply postglacial denudation rates on the order of 0.5–1.1 mm yr−1 within the talus sheds of rock glaciers. The distribution model shows the nonlinear factors of local slope, slope of the contributing area, local curvature, and size of the contributing area controlling the probability of rock glacier occurrence. The model yields an area under the receiver-operating characteristics curve of 0.91, which indicates an excellent fit. On the basis of the present results the integration of terrain attributes with remote-sensing data will be the next step toward automatic mapping of rock glaciers in vast mountain areas.

1. Introduction

[2] Rock glaciers are mesoscalar geomorphological and geocryological features of high mountains (Figure 1). Barsch [1996] characterized them as the geomorphological expression of creeping mountain permafrost. Rock glaciers are particularly frequent and well-developed in dry, continental mountain areas, where they also constitute an important, but often unheralded store of water [Gorbunov, 1983; Schrott, 1996; Trombotto et al., 1999; Trombotto, 2000; Brenning, 2005a, 2005b]. Their occurrence is related to local and regional climatic and topographic controls including thermal conditions, talus production and transport, and past glaciations [White, 1973, 1979a, 1979b; Olyphant, 1983; Parson, 1987; Frauenfelder et al., 2003; Brenning, 2005a]. Quantitative information on their distribution can be used to infer denudation rates [Barsch, 1977a, 1977c; Schrott, 1996; Barsch and Jakob, 1998; Brenning, 2005a] and to provide a first-order approximation of the distribution of mountain permafrost [e.g., Barsch, 1978; Schrott, 1996; Lieb, 1998; Urdea, 1998; Frauenfelder et al., 2001; Janke, 2005].

Figure 1.

Upper Camp Bird Rock Glacier in Imogene Basin, Colorado (37.94°N, 107.73°W). Elevation at the snout is ∼3460 m. The highest point to the top left is Chicago Peak, 4080 m. This rock glacier is typical of rock glaciers in the San Juan Mountains in plan, size, and location relative to treeline and the alpine and nival zones. (Photo courtesy of L. Dexter.)

[3] The objective of the present work is to quantify and analyze the spatial distribution of rock glaciers in the San Juan Mountains of Colorado, USA, as related to local topography. For this purpose we perform a statistical sample survey and fit a generalized additive model. The sample survey for the quantification of the altitudinal distribution of rock glaciers is based on air photos and digital elevation models (DEMs). This approach has previously been applied by Brenning [2005b] and Brenning et al. [2005] in the dry Andes.

[4] We use a semiparametric regression technique to identify local topographic and topoclimatic controls on rock glacier distribution and describe the topographic niche of these landforms. Specifically, we apply generalized additive models with a logistic link function to discriminate between rock glaciers and other areas based on terrain attributes. This technique is an extension of the logistic (linear) regression approach applied by Brenning and Trombotto [2006] to model rock glacier distribution.

[5] Generalized additive models (GAMs) are an extension of generalized linear models in that they conveniently include nonlinear relationships between a response and multiple explanatory variables [Hastie and Tibshirani, 1990]. The additive structure of these models and the possibility of including both linear and nonlinear effects leads to a more flexible, yet still interpretable model structure. The risk of overfitting GAMs is however low compared to highly flexible machine-learning techniques such as artificial neural networks or classification trees [see, e.g., Hand, 1997].

2. Study Area

[6] The San Juan Mountains of southwestern Colorado are an unglacierized mountain area with a maximum elevation of 4363 m at Uncompahgre Peak. They are situated at the continental divide between the Colorado and Rio Grande river basins. Our study area comprises 2874 km2 above 3400 m in the San Juan Mountains (37.47°N–38.20°N, 107.10°W–107.95°W; Figure 2). The highest and most rugged topography is concentrated in the western and northern part of the San Juan Mountains.

Figure 2.

Overview of the study area.

[7] The mountain range is mainly composed of Tertiary volcanic rocks and intrusions followed by Mesozoic and Palaeozoic sedimentary rocks and Precambrian metavolcanic and metasedimentary rocks, which predominate in the southern part of the study area [White, 1973]. The Tertiary volcanics correspond to lavas and breccias of numerous early Oligocene stratovolcanoes, middle Oligocene calderas with associated bedded flows and pyroclastics up to 2 km thick, and late Oligocene to Miocene lavas and breccias [White, 1973]. After Pliocene fluvial dissection of the mountain range, Pleistocene glaciations forged the volcanic highland into a series of cirques, arêtes and horns and widened the now U-shaped valleys. Cirques in northerly exposures are best developed. Late Wisconsin deglaciation began about 16,000 years ago, and most cirques had been vacated by glaciers between 9000 and 8000 years ago [Carrara and Andrews, 1975; White, 1979a, 1979b; Meierding and Birkeland, 1980; Carrara et al., 1984]. The late Pleistocene equilibrium line altitude (ELA) showed a depression of 1000–1250 m in the southern portion of the San Juan Mountains [Leonard et al., 2005].

[8] The timberline in the study area lies between 3400 and 3500 m; it is the boundary between subalpine forests and grasses, herbs, shrubs and krummholz of the alpine zone. Only some small ice fields and perennial snowbanks exist in the study area.

[9] The continental midlatitude climate of the study area is characterized by an annual precipitation of 1180 mm at Red Mountain Pass (RMP, 3414 m; Figure 2), and significantly lower amounts at lower elevations (based on data for 1982–2005; data provided by the United States Department of Agriculture Natural Resources Conservation Service). The modern 0°C isotherm of the mean annual air temperature (MAAT) is situated at ∼3200 m (RMP: −1.6°C).

[10] White [1973] mapped 756 rock glaciers in the San Juan Mountains. Only descriptive summary statistics of topographic and geological characteristics of the rock glaciers and cirques have been published [White, 1973, 1979a, 1979b], while the actual database of the inventory is unavailable today. According to White [1973], almost all rock glaciers in the study area are situated above 3350 m, but the lowest feature extends down to 2975 m. The upper limit of rock glacier distribution was found to be at 4109 m. White [1973] noted a rock glacier density increase westward to a maximum in the central uplands around Silverton, where elevations are highest in the entire range. Also, he found a preference for ubac slopes (vector mean of orientation: 357°, i.e., north). The mean elevation of inactive rock glaciers is 150 m lower than that of active ones [White, 1973].

[11] In a more recent study, Fitzgerald [1994] related the surface morphology of 12 rock glaciers in the Mount Sneffels Wilderness Area in the San Juan Mountains to site conditions using morphometric analyses and morphodynamic simulation studies. The results suggest that surface morphology, a geomorphological expression of rock glacier dynamics, varies in response to basin geometry.

3. Methods

[12] The methodological approach followed here is twofold: First, a stratified sample survey is used to generate land cover data sets based on air photo interpretation. The sample is then used for fitting a morphometry-based generalized additive model as an analytical tool. This model is calibrated with the results of statistical area estimation and used to predict potential occurrences of rock glaciers.

[13] All statistical analyses were performed within the data analysis environment R [Ihaka and Gentleman, 1996; R Development Core Team, 2005] and its gam package [Hastie, 2005]. Terrain parameters were computed with the free terrain analysis software SAGA GIS (provided by O. Conrad and A. Ringeler, Department of Geography, University of Göttingen, Germany, 2004, http://www.saga-gis.org/) [Olaya, 2004].

3.1. Stratified Sample Survey and Statistical Estimation

[14] The statistical land cover survey is based on point estimators for a stratified random sampling design [Thompson, 2002] using DEM-driven stratification with respect to three elevation classes (3400–3600 m, 3600–3800 m, >3800 m) and four aspect classes (N, E, S, W). In total, 2933 sampling points were evaluated.

[15] The survey is based on digital air photos (USGS digital ortho quarter quads of 1 m resolution) and a DEM of the Shuttle Radar Topography Mission (SRTM; resolution 1″, projected and resampled to a resolution of 30 m).

[16] The land cover of each sampling point was classified as “intact rock glacier”, “debris”, “exposed bedrock”, “forest”, “grass/shrubs” or, rarely, “lake” or “snowbank”, based on the air photos only. No distinction was made between active (i.e., advancing) and inactive (stationary) rock glaciers because of the limitations of air photo interpretation. In contrast to intact rock glaciers, which contain ice, relict rock glaciers have melted out and can be recognized from their collapsed appearance and irregular surface structure [Barsch, 1996; Ikeda and Matsuoka, 2002]. Standard methods of statistical point estimation are used to determine the relative and absolute surface areas occupied by the land cover types [see Thompson, 2002].

[17] We use the rock glacier area estimates to derive further indicators of the hydrological and geomorphological significance of rock glaciers in the study area. First, we use simple assumptions on average rock glacier thickness (20 m [see White, 1973; Degenhardt and Giardino, 2004]), ice content (40–60% by volume [see, e.g., Barsch, 1977c, 1996]) and ice density (0.9 g cm−3 [see, e.g., Paterson, 1994]) to obtain a first-order estimate of rock glacier water equivalence. Furthermore, we calculate denudation rates in the sense of average vertical lowering rates within the contributing areas of rock glaciers with a sediment budget approach assuming that a rock glacier's debris content equals the amount of material supplied by mass-wasting processes since deglaciation [Barsch, 1977c; Schrott, 1996; Barsch and Jakob, 1998; Brenning, 2005a]; these estimated denudation rates exclude fine sediments and dissolved matter that can be transported by surface and groundwater flow. Our estimates are based on data from an entire mountain range, which makes them independent of local effects on individual rock glaciers. We compute the denudation rates for different scenarios of average rock glacier age (12,000 or 9000 years) and volumetric debris content of rock glaciers (40–60%) and an average rock glacier thickness of 20 m. The sampled rock glaciers and their contributing areas are digitized from the air photos to determine their sizes.

3.2. Generalized Additive Modeling

[18] Generalized additive models (GAMs) with a logistic link function are a semiparametric, nonlinear extension of ordinary logistic regression models [Hastie and Tibshirani, 1990]. The latter are generalized linear models for a binary outcome variable Y representing, for example, the presence (Y = 1) versus the absence (Y = 0) of rock glaciers [Brenning and Trombotto, 2006], landslides [Ohlmacher and Davis, 2003] or other environmental phenomena [Keating and Cherry, 2004; Lewkowicz and Ednie, 2004; Luoto and Hjort, 2004; Luoto et al., 2004]. Logistic GAMs have also recently been applied in vegetation studies [McKenzie et al., 2003; Segurado and Araújo, 2004] and geomorphology [Luoto and Hjort, 2005; Hjort and Luoto, 2006]. These models can be used to statistically analyze and predict the conditional probability π(x) = P(Y = 1∣x) given a set of covariates or explanatory variables x. The logistic link transforms this probability into a logit,

equation image

for π(x)∈]0, 1[.

[19] While the logits are modeled linearly in logistic regression, GAMs are of a more general form,

equation image

where f is a (possibly nonlinear) transform of the explanatory variables. Transforms can be obtained by using, e.g., cubic smoothing splines as in the present work, or local polynomial regression [Hastie and Tibshirani, 1990], though individual covariates can still be modeled linearly. The dependence of the predicted probability π(x) on x is however nonlinear because of the logistic link transformation even if f is linear.

3.3. Explanatory Variables

[20] Only positional and morphometric attributes (and their transforms) were included as covariates in the exploratory analysis and model selection process. Table 1 lists all variables considered (for details on algorithms, see Zevenbergen and Thorne [1987], Quinn et al. [1991], and Wilson and Gallant [2000]). These variables also reflect, to some extent, regional and local climatic trends as well as inherited geomorphological characteristics of late Quaternary origin (e.g., thickness of morainic deposits); these are not explicitly included in the model and act therefore as hidden confounder variables. Table 2 summarizes selected variables for the different land cover classes.

Table 1. Variables Used for Generalized Additive Modelinga
VariableDescription
  • a

    Local morphometric parameters were computed using the method of Zevenbergen and Thorne [1987], and properties of the contributing areas were derived with the multiple flow direction algorithm of Quinn et al. [1991]. In addition, the same variables were computed from a smoothed DEM using a mean filter with radius 150 m.

Local Morphometry
Eastingin meters
Northingin meters
Elevationin meters
Local slopein degrees
“North exposedness”cos(aspect)
“West exposedness”−sin(aspect)
Overall curvaturepositive values: convex; in °/100 m
Profile curvaturepositive values: convex; in °/100 m
Plan curvaturepositive values: divergent; in °/100 m
Solar radiationpotential annual incoming solar radiation in kWh/m2
 
Characteristics of the Contributing Area
Sizesize of the contributing area or talus shed in m2
Slopeaverage slope of the contributing area in degrees
“North exposedness”cosine of the mean aspect of the contributing area
“West exposedness”negative sine of the mean aspect of the contributing area
Heightaltitudinal difference (relief) within the contributing area
Table 2. Descriptive Summary Statistics of the Stratified Random Sampling Points Used for Distribution Modelinga
 Rock GlaciersDebrisGrass/ShrubsForestExposed Bedrock
  • a

    The values represent the median for each land cover class and, in parentheses, the lower and upper quartiles. Lakes, N = 11, and snowbanks, N = 2, are not included.

Number of pointsN = 84N = 329N = 1154N = 663N = 677
Elevation, m asl3793 (3699–3865)3801 (3671–3884)3700 (3607–3810)3496 (3441–3555)3836 (3719–3917)
Local slope, deg18 (13–26)28 (21–34)17 (10–25)17 (11–25)29 (17–36)
Local curvature, °/100 m−13 (−30–7)−8 (−27–18)1 (−14–15)0 (−14–12)10 (−14–40)
Radiation, kWh/m21838 (1636–2016)1970 (1586–2256)2192 (1986–2343)2093 (1876–2284)2010 (1556–2301)
Contributing area, km20.013 (0.010–0.024)0.007 (0.004–0.013)0.008 (0.004–0.016)0.009 (0.005–0.018)0.004 (0.002–0.008)
Height of contributing area, m102 (68–130)65 (30–110)41 (17–83)46 (22–91)34 (13–69)
Slope of contributing area, °28 (24–32)29 (23–33)18 (11–24)17 (10–23)27 (17–32)

[21] Terrain attributes derived from a smoothed DEM (mean filter, radius 150 m) were also included in the analysis since these are more likely to represent general features of the topographic niche of rock glaciers than the original data. Since GAMs may include a covariable either as a linear or nonlinear term, four variants of each covariable were available, corresponding to the combinations of linear/nonlinear effects and smoothed/nonsmoothed DEMs.

[22] Model selection was performed by automatic stepwise forward variable selection based on the Akaike Information Criterion (AIC), which penalizes for the number of covariables. In the variable selection procedure, the set of admissible models was restricted to include (either none or) only one of the four mentioned variants of each terrain attribute. The resulting automatically generated model contained both the height (relief) and the size of contributing areas as covariables; since these variables are highly correlated (Pearson correlation coefficient 0.77), the variable that produced the lowest AIC was selected.

3.4. Prediction and Model Assessment

[23] A model's overall capability of discrimination was measured by the area under the receiver-operating characteristic (ROC) curve [Hand, 1997], which is abbreviated as AUROC. The AUROC can range between 0.5 (no separation) and 1.0 (complete separation).

[24] After fitting the generalized additive models, these were used for the spatial prediction of rock glacier probabilities based on terrain attribute grids. In order to obtain areas with “likely” occurrence of rock glaciers, a probability threshold was determined that complies with the condition of predicting a total potential rock glacier area equal to the area obtained by statistical estimation.

[25] Model predictions are compared to mapping results in terms of ROC curves and AUROC values, i.e., regarding the trade-off between false positives and false negatives independently of a particular decision threshold used for predicting class memberships.

4. Results

4.1. Rock Glacier Distribution

[26] The general hypsometric distribution of subalpine and alpine vegetation as described earlier is also reflected by our area estimates (Tables 3 and 4). Vegetation-free areas increase above 3600 m and dominate above 3800 m, where they characterize the periglacial or subnival zone. Intact rock glaciers remain very scarce (<1%) below 3600 m.

Table 3. Distribution of Land Cover Types in the San Juan Mountains Within Each Elevation or Aspect Classa
 Rock GlaciersDebrisGrass/ShrubsForestExposed Bedrock
  • a

    Values are in km2.

West11.274.0247.9243.2156.1
South2.652.8328.8174.699.0
East22.762.0366.7244.6131.6
North33.685.8193.0197.3133.4
>3800 m22.590.9184.11.1224.8
3600–380036.3119.7603.074.2224.8
3400–360011.364.0349.2784.270.5
Regional total70.1274.71136.3859.6520.1
Table 4. Distribution of Land Cover Types in the San Juan Mountains as Percentage of the Total Area Within Each Elevation or Aspect Classa
 Rock GlaciersDebrisGrass/ShrubsForestExposed Bedrock
  • a

    Lake and snowbank are missing to 100%.

West1.510.133.833.121.3
South0.48.049.726.415.0
East2.77.544.129.415.8
North5.213.329.930.520.6
>3800 m4.317.335.10.242.8
3600–38003.411.256.57.021.1
3400–36000.95.027.261.25.5
Regional total2.49.639.529.918.1

[27] The statistical estimation procedure yielded a total intact rock glacier area of 70 km2 (approximate 90% confidence interval: 57–83 km2), which is equivalent to 2.4% of the total area above 3400 m. Rock glaciers are most abundant in northerly exposed areas, where they cover 5.2%, and above 3800 m, where they occupy 4.3% of the surface (Figure 3). The amount of water stored within rock glaciers in the study area is on the order of 0.50–0.76 km3 of water equivalence depending upon an assumed 40–60% ice volume.

Figure 3.

Rock glacier distribution in the San Juan Mountains according to statistical estimation, expressed as percentage of the total area of each stratum.

[28] The regional average denudation rates estimated from the sediment budgets of rock glacier talus sheds range between 0.54 and 1.08 mm yr−1 for the different scenarios (Table 5).

Table 5. Denudation Rates Within Contributing Areas of Rock Glaciers in the San Juan Mountains for Different Mean Rock Glacier Ages and Rock Glacier Debris Contentsa
Debris ContentRock Glacier Age
12,000 years9000 years
  • a

    Values in parentheses indicate lower and upper quartiles of estimated denudation rates for the rock glacier sample. Values are in mm yr−1.

40%0.54 (0.26–0.76)0.72 (0.34–1.01)
60%0.81 (0.39–1.14)1.08 (0.51–1.52)

[29] We estimated the rock glacier size distribution of the entire population by weighting the individual sizes based on the probability distribution imposed by the stratified sampling design and the proportionality of rock glacier sampling probability to rock glacier size. The resulting average rock glacier size is 0.064 km2 (median: 0.051 km2). An estimated 88% of the rock glaciers in the San Juan Mountains are smaller than 0.1 km2. The 90% confidence interval of the total rock glacier area results in a population of about 900–1300 features in the entire study area, which clearly exceeds the number of rock glaciers identified by White [1973].

4.2. Generalized Additive Model

[30] The stepwise variable selection yielded a set of eight relevant covariables, namely elevation, easting, northing, north exposedness, curvature, (local) slope, mean slope of the contributing area and the size of the contributing area. Only the curvature and north exposedness variables selected by this automatic procedure are derived from the smoothed DEM, the others from the original SRTM DEM. The contributing area and both slope variables were selected by the stepwise procedure as nonparametric smoothing splines, the other five variables as linear terms. The model structure is visualized in Figure 4 in terms of exemplary one-dimensional sections through the probability prediction surface.

Figure 4.

Illustration of the predictions of the rock glacier distribution model as related to (a) elevation, (b) easting and northing, (c) north exposedness, (d) curvature, (e) local slope (solid line) and slope of the contributing area (dashed line), and (f) size of the contributing area.

[31] The model structure can be used to characterize the topographic structure of the optimal rock glacier niche in the San Juan Mountains. First, the model represents the general pattern of rock glacier distribution as related to elevation and north exposedness that has been observed earlier, and it detects an overall decreasing rock glacier frequency in eastward and southward direction. Moreover, local slope and curvature are important predictors, with gently sloping (<15°) concave skyward and convergent locations being more likely to host a rock glacier than those with a convex or divergent curvature.

[32] Conversely, the contributing areas of rock glaciers are characterized by high slope angles (>25°) that are required for an effective gravitative mass transport to the rock glacier's rooting zone. Contributing areas are required to be larger than ∼0.01 km2 to host rock glaciers, but otherwise there appears to be no great effect of catchment size.

[33] The rock glacier distribution model achieved an AUROC of 0.91, which is an “excellent” to “outstanding” value according to Hosmer and Lemeshow [2000]. It exceeds the value of 0.84 achieved by Brenning and Trombotto [2006] in the Andes. A probability threshold of 0.31 results in a total predicted rock glacier area that is equal to the estimated total area of 70 km2. This threshold results in an overall misclassification error rate of 5% (or an overall accuracy of 95%) and a sensitivity of 45% at a specificity of 98% (Table 6; the sensitivity is the portion of correctly classified positives, the specificity the portion of correctly classified negatives). All possible combinations of specificities and sensitivities that can be obtained with different decision thresholds are depicted in the ROC curve (Figure 5).

Figure 5.

ROC curve of the logistic regression model discriminating rock glacier distribution against debris surfaces and grass/shrub vegetation.

Table 6. Confusion Matrix and Accuracy of the Rock Glacier Prediction Model Estimated on the Learning Samplea
 Predicted 
01SumAccuracy
  • a

    Here 0 indicates nonrock glacier, and 1 indicates rock glacier.

Observed 014483514830.98
Observed 14638840.45
Sum1494731567 
Accuracy0.970.52 0.95

5. Discussion

5.1. Geomorphological and Hydrological Significance of Rock Glaciers

[34] Intact rock glaciers and their talus sheds occupy an important portion (about 12%) of the area above 3600 m in the San Juan Mountains, especially in northerly exposed cirques. These areas play a key role for water supply in the drier lowlands from snowmelt and groundwater [Klein and Barnett, 2003]. Within this part of the hydrological system, rock glaciers and mountain permafrost in general act as long-term and seasonal stores of water; their contribution especially to late summer runoff in dry mountain areas needs further investigation [Burger et al., 1999; Williams et al., 2006].

[35] Rock glaciers in the San Juan Mountains are less abundant and smaller in size than in higher and/or drier mountain areas such as the dry Andes or the mountains of central Asia [Gorbunov, 1983; Brenning, 2005a, 2005b]. In the Andes of central Chile and Argentina, Brenning [2005a] statistically estimated specific densities between 4 and 6.7% of the mountain area, compared to 1.7% (locally 5%) in the Zailijskiy Alatau of Kazakhstan and Kirghizia (based on the work of Gorbunov [1983]); in these areas, some rock glaciers are greater than 1–1.5 km2. Relict rock glaciers in the Southern Carpathians occupy about 1.1% of the area above their lower limit (locally up to 5% [Urdea, 1998]), while intact rock glaciers in the Swiss and Italian Alps reach specific densities of 0.3 and ∼0.8%, respectively [Barsch, 1977a; Guglielmin and Smiraglia, 1998] (note that the original area estimates of Guglielmin and Smiraglia [1998] are estimated as the maximum length multiplied by the rock glacier width; according to the rock glacier geometry present in the San Juan Mountains, these estimates are about 50% too high on average, a corrected value is reported here). The specific density of rock glaciers in the San Juan Mountains (2.4%) is at an intermediate level compared to the Swiss and Italian Alps on one side presenting a significant glacierization, and higher, but drier high mountains such as the Andes on the other side, where higher unglacierized headwalls supply debris for abundant rock glacier development.

[36] The estimated water equivalence of rock glaciers in the San Juan Mountains corresponds to 0.18–0.26 km3 of water per 1000 km2 of high mountain area above the lower limit of rock glacier distribution. This value is lower than in the semiarid Andes (0.23–0.37 km3 per 1000 km2), where glaciers are important features above 4000 m, and much higher than in the strongly glacierized Swiss Alps (∼0.03 km3 per 1000 km2 [Barsch, 1977b; Brenning, 2005b]).

[37] The estimated denudation rates in the high, vegetation-free talus sheds of rock glaciers are of the same order of magnitude as values obtained in the semiarid Chilean Andes (∼0.75 mm yr−1) with similar approaches, but appear to be higher than in the arid Andes (∼0.5 mm yr−1), where the limited availability of water probably reduces denudation [Brenning, 2005a]. On the basis of rock glaciers in the Colorado Front Range, Caine [1974] calculated weathering rates ranging from 0.71 to 1.06 mm yr−1, which is very similar to the range obtained here (0.54–1.08 mm yr−1). Barsch [1977c] obtained similar to higher values than ours for rock glaciers in the Swiss Alps, and in West Greenland, Humlum [2000] calculated headwall retreat rates of at least 2 mm yr−1 for talus-derived rock glaciers and at least 5 mm yr−1 for rock glaciers classified as glacier-derived. Humlum [2000] summarizes further estimates of headwall retreat rates derived from rock glaciers. However, different lithologic and climatic conditions in these areas as well as different methods for calculating denudation rates make it difficult to compare these estimates. André [1997] points out that retreat rates may have varied at the same place by several orders of magnitude since deglaciation, depending on the predominating weathering processes such as rapid retreat due to postglacial stress relaxation, moderate retreat due to periglacial frost shattering and very slow biogenic flaking.

5.2. Topographic Controls

[38] In the San Juan Mountains, in addition to overall trends related to the thermal regime (elevation) and regional (topographic, geological and climatic) trends, our semiparametric approach allows us to identify several process-related topographic controls on rock glacier abundance. First, we interpret the favorability of northerly exposed areas for rock glacier development as the combined result of (present-day) radiative controls on ground thermal regime and the preferential northerly orientation of Pleistocene cirques as an inherited favorable topographic effect. A strong preference of poleward facing slopes in the distribution of relict rock glaciers has also been observed by Urdea [1998] in the Southern Carpathians. Brenning and Trombotto [2006] found that radiation, but not aspect, is a statistically relevant control on rock glacier distribution. Its effect however depends on the altitude. Earlier, Morris [1981] derived an empirical relationship between the degree of development (elongation) of rock glaciers in the Sangre de Cristo Mountains (southern Colorado) as target variable and radiation shading, jointing and altitude as explanatory variables related to debris supply as well as rock glacier maintenance. He found that these variables have greatest explanatory power in a three-way interaction, indicating that the effects of these variables are not additive.

[39] The slope and size of the contributing area show interesting nonlinear effects in our model, indicating that steep contributing areas that are of at least ∼0.01 km2 in sized are favorable for rock glaciers occurrence (Figure 4). This may be interpreted based on the following general considerations related to debris supply. In general, the average slope of a contributing area influences the intensity of mass wasting and the transport velocity toward the rooting zone of a rock glacier. The larger and steeper the talus shed is, the more talus will therefore be supplied, and the more suitable will a talus shed be for rock glacier development. Pleistocene cirques are the prototype of relatively large contributing areas with steep headwalls, and they therefore present excellent conditions for rock glacier development. These favorable topographic niches are detected by the present model based on the variables representing slope and size of the contributing area.

[40] The local topographic attributes of slope and curvature describe the position of a rock glacier and a rock glacier's shape itself. Relatively gentle slopes where talus can accumulate are zones of preferential rock glacier occurrence. A concave (convergent and/or concave skyward) overall curvature also identifies areas of talus concentration as a consequence of convergent talus transport and/or downslope decreasing transport velocities. These process-related topographic conditions that influence rock glacier distribution are also included in the set of explanatory variables selected by the automatic stepwise procedure.

[41] A univariate examination of morphometric characteristics of cirques in the San Juan Mountains by White [1973, 1979a, 1979b] revealed that cirque wall height, wall roughness (based on the number of crenulations in the contour lines) and maximum cirque wall elevation had a significant effect on the presence or absence of a rock glacier in a cirque. Other variables, including the slope of the cirque floor and the size and orientation of the cirque had no significant effect in these analyses of rock glacier distribution. Though the slope of the cirque floor and the size and orientation of the cirque as used by White are not identical to our slope, aspect and contributing area variables, we attribute the significant effects of these variables in our model to two possible reasons: (1) Nonlinearities were not represented in White's analyses; (2) White's analyses are limited to the cirques of the San Juan Mountains, while the present study is performed on the landscape level, which includes a greater variety of rock glacier and nonrock glacier sites.

[42] Regarding the influence of morphometric site characteristics on rock glacier properties rather than the presence/absence, White [1973, 1979a, 1979b] did not find statistically significant controls (other than altitude and orientation) on rock glacier morphometry, form, type and activity that are of quantitative importance. The categorical variables of “form”, “type” and “activity” used by White [1973] are difficult to define accurately and in some cases, such as the status of activity or the classification as “ice-cored” or “ice-cemented” rock glacier, they can hardly be derived from (unitemporal) aerial photography. These shortcomings, along with the attempt to apply univariate statistics to complex natural phenomena, may explain the weak relationships found by White [1973, 1979a, 1979b].

[43] In our approach, by contrast, we focus on an objective response variable, the presence versus the absence of a rock glacier, as a multivariate function of terrain parameters. This allowed us to simultaneously study the relationships between rock glaciers and several of their site characteristics. In addition, computational terrain analysis now provides a great number of process-related terrain attributes that can be accurately computed from the available SRTM DEMs.

5.3. Methodological Considerations

[44] While nonlinearities were difficult to include in the logistic regression model of rock glacier distribution of Brenning and Trombotto [2006], generalized linear models provide a more convenient and flexible framework for semiparametric modeling of geomorphological distribution patterns. Increased flexibility however implies a greater potential of overfitting, i.e., of adapting the model to random structures in the data. In the context of generalized linear models, overfitting may result from highly flexible local smoothers, e.g., polynomials of high degree that tend to oscillate. This is however not the case in the present study (Figure 4). Also, in a comparable study manual nonlinear transformations of explanatory variables did not produce an overfit in the logistic regression model of Andean rock glacier distribution, according to model evaluation on independently mapped rock glacier inventories [Brenning and Trombotto, 2006].

6. Conclusions

[45] This study demonstrates that logistic generalized additive models work very well to determine the distribution of rock glaciers and their developmental controls. This is achieved by discriminating them from other land cover types and then quantifying surrounding slope characteristics and morphology. Indeed, these are powerful analytical tools that can be readily used in geomorphology and permafrost studies. Beyond the analytic purpose of our rock glacier distribution model, the next step is to integrate terrain attributes with remote-sensing data to automatically map rock glaciers in vast mountain areas (compare to the work by Paul et al. [2004] in the case of debris-covered glaciers). Furthermore, data generated can be readily applied to studies of water supply of melting mountain permafrost and to calculate regional denudation rates.

[46] Our model generated an AUROC of 0.91 indicating an excellent statistical fit in our prediction of rock glaciers. The primary controls are local slope and curvature, and the slope and size of the talus shed. We have shown that in the San Juan Mountains of Colorado rock glaciers cover near 5% of total area above treeline and in northerly exposures, and that they hold water equivalent which we estimate to be 0.50–0.76 km3, a significant volume in the arid mountains of the southwestern United States. We calculate regional erosion rates, based upon total postglacial coarse sediment buildup in rock glaciers and loose debris in the study area, to be between 0.5 and 1.1 mm yr−1. This compares well with erosion rates for high mountain permafrost areas around the world as summarized by Goudie [1995].

[47] Brenning's [2005a, 2005b] studies in the Andes have produced the same kinds of results with differing values. This calls for further investigation of rock glaciers and mountain permafrost in many more environments to develop a basic understanding of this landform and for applications in water supply and global change studies.

Acknowledgments

[48] The authors are grateful to students at the Institut für Geographie, Universität Erlangen enrolled in a course dedicated to geospatial modeling of high mountain environments who contributed greatly to the sample survey and air photo interpretation. We are also thankful for a Fulbright Senior Scholarship which allowed D. Friend to be in Germany AY 2004–2005. We acknowledge the comments made by T. Hengl and an anonymous reviewer, which helped improve the present work.

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