Statistical distribution of tumuli on pahoehoe flow surfaces: Analysis of examples in Hawaii and Iceland and potential applications to lava flows on Mars



[1] Spatial distributions of tumuli on lava flow surfaces can be quantitatively linked to subsurface inflation processes. Three distinct styles of flow emplacement are studied: (1) lava flows undergo inflation that changes as a function of space and time resulting in random spatial distributions of tumuli, (2) lava flows with preferred pathways result in systematic clustering of inflation features, and (3) established tubes or narrow pahoehoe flows produce chains of tumuli. Statistical analyses are required to distinguish between styles one and two. Comparison of the spatial distribution of tumuli on a portion of a lava flow at Mauna Ulu with the Poisson distribution indicates that tumuli are randomly distributed on the flow surface, typical of style one emplacement where small-scale topographic variability and low slope influence development of preferred pathways that evolve over time. At Thrainsskjoldur, direct statistical comparisons to the Poisson spatial distribution are inconclusive but nearest-neighbor analysis indicates significant clustering. An initially random spatial distribution is inferred that developed preferred pathways and systematic clustering of inflation features, as with style two. At Elysium Planitia on Mars, the flow margin is significantly deficient in inflation features, while away from the margin, the flow exhibits a random spatial distribution of tumuli typical of style one emplacement. The flow is interpreted as multiple units or a single flow that migrated in space during emplacement. At the 1843 flow on Mauna Loa, tumuli occur on a linear trend, consistent with style three emplacement.

1. Introduction

[2] Pahoehoe lava flow fields are generally associated with low effusion, long-duration basaltic eruptions. Their emplacement is an extremely complex process, dominated by random influences such as variations in space and time of local flow rate and underlying topography. It is not feasible to model the details of emplacement of complex pahoehoe flow fields deterministically due to the predominance of random influences; however, the final surface morphology of a pahoehoe lava flow field provides important clues about processes that occur during emplacement [Walker, 1991; Self et al., 1998]. Certain types of features, such as tumuli, lava rises and lava rise pits, are indicators of flow inflation or endogenous growth of a lava flow [e.g., Walker, 1991]. Tumuli (Figure 1), defined as raised elliptical to circular mounds with axial cracks found on lava flows [Walker, 1991], in particular, have been identified as possible indicators of lava tube location [e.g., Guest et al., 1984; Calvari and Pinkerton, 1998; Duraiswami et al., 2001; Duncan et al., 2004] or simply a response to irregular underlying topography [Self et al., 1998].

Figure 1.

Typical tumulus. Photo taken facing north, along western margin of study area at Mauna Ulu, Hawaii (Figure 2b).

[3] Our objective in this work is to complement field studies of pahoehoe lava flows by adding new insight into the internal structure of complicated flow fields through a statistical analysis of the final surface morphology. With the exception of work by Rossi [1999], the spatial distribution of tumuli on lava flows has not been examined previously in a statistical manner. Here, we use statistical analyses to demonstrate that the final surface morphology observed in the field retains distinct spatial distributions that can be used to readily distinguish between modes of lava transport beneath the surface of the crust. This work has implications for lava flow emplacement, and consequently for defining volcanic hazards. Further, the statistical technique presented here can be used to remotely analyze lava flows on Earth as well as other planetary surfaces.

[4] Because tumuli form by the injection of lava beneath a crust, the distribution of tumuli on a flow surface may represent the distribution of a network of lava transport pathways, or preferred pathways, beneath the surface of the crust that were present during emplacement. In some instances, the network of lava transport pathways may be influenced by the curvature of the underlying surface [Rossi, 1999]. The distribution of preferred pathways may, in turn, be a function of the temporal evolution of a pahoehoe lava flow. As a longer-lived flow field evolves, initially broad preferred pathways may evolve to narrower, more well-defined tube-like systems [Self et al., 1998; Anderson et al., 1999]. Thus the final flow morphology is complicated by the preservation of features that have developed with the flow over time and inflation features observed in the field indicate pathways that were almost certainly not active contemporaneously.

[5] A typical scenario for pahoehoe flow field emplacement begins with a surface breakout from a primary lava transport system (either the main vent or an established tube). These surface breakouts are emplaced as lobes that may coalesce into a single inflating sheet flow if lava supply continues. Field observations [Hon et al., 1994] indicate that typical sheet flows have a single coherent inflating core. However, small-scale topographic variations soon dominate flow emplacement [Thordarson, 2000; Self et al., 1998]. As a result, one can envision preferential cooling around preexisting topographic highs and the development of thermally preferred pathways through the low areas. At this point, inflation becomes more restricted along these pathways and we begin to see small tumuli develop over a broad network of preferred pathways. Instead of the entire sheet flow inflating uniformly, there are small pockets of inflation scattered around on the surface. In general, these early tumuli are numerous but relatively small (<2 m high). As flow field evolution continues, some preferred pathways may become more mature at the expense of others, leading to a more tube-like distribution of lava beneath the flow surface. While the initial tumuli are frozen in time, continued inflation within the more mature tubes can result in the development of tumulus chains or ridges that can grow to substantial sizes (∼10–20 m high) if lava supply persists.

[6] Scenarios other than that described above can be envisioned, where the various stages of emplacement occur in different orders, or perhaps not at all. Regardless of the emplacement order, we propose that we can quantitatively identify three distinct styles of flow field development from the final spatial distribution of tumuli on the surface. With the first emplacement style, there is no established tube system directly beneath the flow field feeding tumulus growth, nor is there a lava feeding network that is stationary in either space or time. The lava spreads over a broad area relative to tumulus size. Flow rate and slope are both low enough that localized inflation occurs in response to variations in topography as proposed by Self et al. [1998]. However, with this style, preferred pathways are not fully developed, and may be transient. Thus the parts of the flow undergoing inflation change as functions of both space and time. The resulting spatial distribution of tumuli is truly random. Variations in tumulus size may be due to the length of time that inflation persisted in a certain location. However, the lava supply at inflation locations is not sufficient to disturb the random distribution.

[7] The second emplacement style differs from the first in that a distribution network of preferred pathways develops beneath the surface of the broad flow. These do not necessarily evolve into full fledged “tubes”, but they persist long enough to result in systematic clustering of inflation features beyond the clustering expected from a random distribution. If this situation persists long enough, one should see a pattern in the spatial clustering, trying to form a linear reflection of the distribution network within the flow. While in some cases this clustering may be obvious, it is likely that statistical analysis is required to distinguish between styles one and two.

[8] With the third emplacement style, tumuli form over relatively narrow, well defined areas that are more or less stationary in time and space. Examples include well defined, fully evolved lava tubes, or simply narrow pahoehoe flows that for one reason or another (e.g., higher flow rate, confining topography, or slope) do not form over a broad area. This formation mechanism results in a chain of tumuli on the surface. The linear (but not necessarily straight) nature is clearly discernable in plan view and there is little or no need to perform statistical analysis in this case.

[9] This study is intended as a proof of concept to demonstrate that statistical analyses can be used to distinguish between the proposed styles of flow field evolution and to guide the future collection of data. Previous studies of inflation features have provided somewhat arbitrary definitions of what exactly constitutes a tumulus and a lava rise. Here, we have defined specific criteria for each and implemented these criteria during data collection at each site. Examples of candidates for emplacement styles one and two have been taken from Mauna Ulu, Hawaii (Figure 2); Thrainsskjoldur, Iceland (Figure 3); and Elysium Planitia, Mars (Figure 4). The chain of extremely large tumuli along the eastern margin of the 1843 flow in the saddle between Mauna Loa and Mauna Kea, Hawaii (Figure 2), provides an example of style three. We have carefully collected and analyzed a discrete portion of a lava flow with numerous lava rises and tumuli in the 1969–1974 Mauna Ulu flow field at Kilauea (Figure 2b). The study area is located in the distal portion of the flow field below Holei Pali, which is characterized by hummocky pahoehoe flows fed by tubes [e.g., Swanson, 1973]. This study area was chosen because its discrete nature allows complete mapping of surface morphologies and identification of well-defined boundaries. In addition, emplacement parameters are well constrained, and flow thicknesses can be measured [e.g., Swanson, 1973; Holcomb, 1976; Byrnes and Crown, 2001]. The temporal aspect of this unit is relatively simple, with no subsequent flows overlying the portion examined.

Figure 2.

(a) Map indicating the locations of Figures 2b and 2c on the Big Island of Hawaii. (b) Map showing location of study area at Mauna Ulu, Hawaii. Heavy solid line running from Kilauea caldera to south coast of island is Chain of Craters Road. Light gray area represents deposits from the 1969–1974 Mauna Ulu eruption. (c) Locations of 14 tumuli (white) along the eastern margin of the Mauna Loa 1843 flow (light gray). Dashed lines around tumuli indicate ambiguous margins. Dashed arrow indicates the overall direction of flow. Solid bold line in the northeastern corner of the figure is Saddle Road. The upper left corner of Figure 2c is located at approximately (155°31′W, 19°42′N), and the lower right corner is located at approximately (155°30′W, 19°41′N).

Figure 3.

Map of Reykjanes peninsula in southwestern Iceland. Triangle indicates the city of Reykjavic. Major roads are shown as thin black lines. Star indicates the location of Thrainsskjoldur volcanic shield at 63.88°N, 22.2°W. Location of lava flow with tumuli (Figure 9) on Thrainsskjoldur volcano is unknown.

Figure 4.

Location map for Figure 5. (top) Shaded relief derived from Mars Orbiter Laser Altimeter data, indicating location of Figure 4 (bottom) in relation to major volcanic centers at Elysium, Olympus, Pavonis, and Arsia Montes. (bottom) Viking image of southeastern Elysium Planitia on Mars. Arrow points to a small box indicating location of MOC image in Figure 5.

[10] Limited examples of appropriate spatial information exist for other flow fields containing tumuli [e.g., Rossi and Gudmundsson, 1996]. This paucity of data is exacerbated by the lack of information on how the existing data were collected. Despite these limitations, we have examined the spatial distribution of tumuli on the distal portion of the Thrainsskjoldur flow field in Iceland [Rossi and Gudmundsson, 1996]. Availability of high-resolution Mars Orbiter Camera (MOC) and Thermal Emission Imaging Spectrometer (THEMIS) images motivated us to look for possible tumulus-dominated lava flows on the surface of Mars. Several THEMIS and MOC images of lava flows on Mars contain features that may be tumuli. We have analyzed the MOC image shown in Figure 5 of a lava flow southeast of Elysium Mons. The flow is characterized by morphologic features that fit the terrestrial definition of tumuli and lava rises (i.e., raised elliptical to circular mounds, some with axial cracks) and are similar in size to the tumuli observed on Earth [e.g., Walker, 1991; Duncan et al., 2004].

Figure 5.

(right) MOC image 20-01192 of a lava flow (darker gray) southeast of Elysium Mons, Mars (location of this flow on Mars is shown in Figure 4). Illumination is from the northwest. The direction of flow is unknown, but the arrows represent plausible choices. (left) Enlarged portion of Figure 5 (right) showing several positive relief features interpreted as tumuli.

2. Background

[11] Tumuli, such as that shown in Figure 1, have been documented in a variety of basaltic flow environments. Several interpretations for how and where tumuli occur have been postulated. Walker [1991] examined tumuli on Hawaiian pahoehoe flows and defined them as positive topographic features with an axial cleft that formed by uplift. He found that tumuli tend to form on shallow slopes. Walker [1991] defined three tumulus morphologies: shallow slope tumuli (<4°), moderate slope tumuli (slopes > 4°) and flow lobe tumuli. Shallow slope tumuli tend to be larger than moderate slope tumuli and form in clusters or trains, analogous to style three emplacement described above. Walker [1991] did not find any association between trains of tumuli and master tubes, but suggested that they may form over lesser, more transient tubes. Alternatively, Self et al. [1998] suggest that tumuli form over preexisting depressions on older pahoehoe flows, implying that the spatial distribution of tumuli should reflect the preexisting topography, consistent with our definition of style one. The resulting hummocky flows have many tumuli and are generally associated with relatively slow, discontinuous emplacement [Self et al., 1998].

[12] Tumuli were found over tubes on the 1614–1624 flows at Mount Etna [Guest et al., 1984] and the 1983 Etna flow [Duncan et al., 2004]. Duncan et al. [2004] related the location of tumuli to a series of pathways on the 1983 flow at Mount Etna and found that larger, more complex tumuli lay directly over major feeder (also known as master) tubes, while satellite and distributary tumuli lay over more transient, lesser pathways. Duraiswami et al. [2001] describe tumuli as being common on both hummocky flow units and thicker, overlying sheet lobes in the Deccan volcanic province. They interpret the alignment of tumuli to indicate development along anastomosing tube systems.

[13] Anderson et al. [1999] have proposed that a network of thermally preferred pathways under the crust of a flow could give rise to inflation features such as tumuli or lava rises on hummocky lava flows. In this context, thermally preferred pathways form in a broad sheet flow as the flow attempts to establish an efficient distribution system. Multiple thermally preferred pathways can exist contemporaneously and may migrate (spatially) over time. Thus tumuli could reflect both transient pathways as well as longer-lived tube systems. It is possible that transient pathways are influenced by underlying topography, as suggested by Self et al. [1998].

[14] In addition to tumuli, other pahoehoe surface features, such as lava rises, may also be associated with lava flow inflation [Walker, 1991]. Lava rises are generally characterized by a broad flat surface with circumferential cracks. Here we are concerned primarily with the surface distribution of tumuli. For completeness, in some instances we have considered the surface distribution of all measurable inflation features (tumuli and lava rises).

3. Method

[15] A statistical study can be employed to address the issue of whether tumulus formation occurs systematically or whether changes in flow transport in space and time significantly randomize the inflation process. In general, to perform the statistical analysis, all that is needed is a planform map of a flow area, with the locations of lava flow margins and inflation features identified. At Mauna Loa, we have used air photos and field reconnaissance to determine the location of tumuli. Data for the locations of tumuli at Thrainsskjoldur are taken from Rossi and Gudmundsson [1996]. At Elysium, locations of inflation features are determined from analysis of high-resolution MOC imagery.

[16] In order to statistically analyze the spatial distribution of inflated features on the flow unit at Mauna Ulu, we documented the height, planform shape, location, and major fractures using a Trimble ProXR real-time differential GPS, logging positions every 1 second. A precise definition of exactly what features to include when sampling a population is critical to any sound statistical study. For this study, we mapped the perimeters of every inflated feature within the boundaries of our sample area that had well-defined margins and 1 m or more of relief. This definition does not include lobes of pahoehoe constructed from many smaller individual toes. We distinguished between tumuli and lava rises, with tumuli defined as domed features usually with an axial cleft, and lava rises defined as plateau-shaped typically with marginal fractures. In addition, we measured the highest point on each tumulus or lava rise and the lowest point along the perimeter by acquiring at least 15 s of data (positions logged every second) at each point. The acquisition of real-time data from the nearest working and available base station (Upolu Point, Hawaii, 20°14′48N, 155°53′W, transmitting at 286 kHz) was inconsistent owing to distance and topography between the base station and field site. Therefore the GPS field data were differentially corrected using Upolu Point data obtained over the Internet using the Trimble Pathfinder Office 2.8 postprocessing utility. After postprocessing, the average horizontal precision for each tumulus/lava rise ranged from 0.29 to 0.44 m, and average vertical precision ranged from 0.44 to 1.07 m.

4. Results

[17] Our primary objective is to distinguish between styles one and two for subsurface lava transport resulting in inflation features (including tumuli and lava rises), by assessing whether or not there is any systematic behavior that can be inferred by their spatial distribution on the flow surface. We first assess the distribution of inflation features at Mauna Loa, then statistically assess the surface spatial distributions of inflation features at Mauna Ulu, Thrainsskjoldur, and Elysium, in order to ascertain whether they are randomly or systematically distributed.

4.1. Mauna Loa

[18] Figure 2c is a sketch of tumulus and lava rise locations near the eastern margin of the 1843 Mauna Loa flow. The heights of some of these tumuli/lava rises are approximately 10 m or more. Flow direction is from south to north, and field observations (S. Rowland, personal communication, 2003) suggest that flow field emplacement developed from the eastern margin and migrated westward. Another approximately linear distribution of tumuli/lava rises is found 0.5–1 km to the west of the eastern margin. The nonrandom, linear distribution along the eastern margin clearly represents a systematic process (e.g., a tube or a relatively narrow preferred pathway or ponding). This example represents the formation of a spatial distribution analogous to style three with little evidence for the other two styles. Consequently there is no need for a detailed statistical analysis of the nature of the randomness.

4.2. Mauna Ulu

[19] Figure 6 shows the locations of 76 tumuli and 12 lava rises on a portion of a hummocky lava flow erupted from Mauna Ulu, Hawaii, between 1969 and 1974. The lava flow overlies much older material not associated with the Mauna Ulu eruption. The field area in Figure 6 is 9.5–10 km down flow from the Mauna Ulu vent (see Figure 2b), where slopes are very small. The easternmost margin of the field area is the lateral margin of a branch of the lava flow. The western margin is distinct and is defined as the lateral margin of a later lava flow that has covered the western extent of the flow unit studied here. Byrnes and Crown [2001] tried to relate the surface morphology of flows at the Mauna Ulu flow field to the tube system mapped by Holcomb [1976]. Upflow of the area discussed in this paper, they identified inflation features, yet concluded that the units they mapped did not correlate to major tubes, but instead were related to a smaller-scale distributary network.

Figure 6.

(a) Locations of the 76 tumuli measured at Mauna Ulu (location of study area shown in Figure 2b). A trapezoid approximating the shape of the field site is shown. Symbols indicate tumuli with heights 0–0.67 m (squares), 0.67–1.17 m (pluses), 1.17–1.67 m (solid diamonds), 1.67–2.17 m (open triangles), 2.17–2.67 m (solid triangles), 2.67–3.18 m (minuses), 3.18–3.68 m (asterisks), 3.68–4.18 (open diamonds). Figures 6b–6c illustrate the irregular lateral margins of the study area, the tumuli (solid diamonds) from Figure 6a, and 12 lava rises (crosses), with different grids overlain: (b) 77 cells of equal area, (c) 148 cells of equal area, and (d) 37 cells of equal area. Gray grid cells are those that have not been used in comparisons with the Poisson distribution. For each grid, the 74 Mauna Ulu tumuli contained within the large trapezoid in Figure 6a have each been assigned to one grid cell.

[20] We begin by considering only the tumuli within the study area (lava rises are excluded). On the basis of visual inspection in the field, it appears that tumuli may tend to cluster near the eastern lateral margin of the flow unit. However, truly random spatial distributions should exhibit clustering (note that uniformly spaced features would not be random). In order to test whether this apparent clustering of tumuli is significant, we compared tumulus locations to the Poisson probability distribution. The Poisson distribution is the limiting form of the binomial used to describe random events in time or space that are relatively rare [Snedecor and Cochran, 1967]. Requirements for using the Poisson distribution are that (1) the probability of at least one occurrence of the event in a given spatial interval is proportional to the area of the interval, (2) the probability of two or more occurrences of the event in a very small area is negligible, and (3) the occurrence of an event within one spatial interval has no effect on the occurrence or nonoccurrence in another nonoverlapping interval of the same size [Larson, 1974].

[21] If the spatial distribution of tumuli within the study area is significantly different from the Poisson distribution, we can conclude that there is some systematic behavior controlling their occurrence (i.e., the clustering or spatial ordering we perceive is real). Alternatively, if the spatial distribution is indistinguishable from the Poisson distribution, we must conclude that the tumuli occur randomly. To compare the spatial distribution of tumuli to the Poisson distribution, it is necessary to impose an arbitrary grid containing cells of equal area. The study area can be grossly described by a large trapezoid (shown in Figure 6a) that includes 74 of the 76 measured tumuli. This trapezoid includes a minimal amount of area that is not part of the lava flow of interest.

[22] Figure 6b illustrates an example of an equal area trapezoidal grid. To create the equal area grid, we allowed the number and dimensions of the grid cells to vary by row. We attempted to find a combination of parameters that resulted in row heights that are as uniform as possible. The grid shown in Figure 6b contains 77 cells, each with an area of 1,383 m2. In comparing the spatial distribution of tumuli to the Poisson, we assumed that each tumulus is a single point. In fact, tumuli have a range of areal sizes. Thus we have carefully chosen our grid cell sizes such that they are sufficiently large (i.e., the area of a single grid cell is greater than the average tumulus area), while still maintaining the rare nature of occurrence (≈1 tumulus/grid cell on average). The mean value of the tumulus areas is ≈107 m2, significantly smaller than the grid cell area used in the analysis. Note that we have reported the geometric mean value of tumulus area here, as opposed to the arithmetic mean, as it is most appropriate for sample size distributions that are strongly skewed.

[23] For our comparison, we did not use the three cells at the eastern end of the bottom row of the grid in Figure 6b, as this area is almost entirely beyond the margin of the flow. This results in 74 tumuli divided among 74 equal area grid cells. As can be seen in Figure 6b, some of these cells contain no tumuli, some contain 1 tumulus, some 2 and some more.

[24] The Poisson probability distribution has only one parameter, λ, that is both the mean and standard deviation for the distribution. The probability of k discrete events (e.g., tumuli) occurring within some spatial area a is given by

equation image

For the grid established in Figure 6b, λ = 1.0 tumulus/grid cell and a = 1 grid cell. The probabilities for finding 0, 1, 2 or ≥3 tumuli in any grid are shown for p(k) in Table 1. Note that the sum of all the probabilities is equal to 1 indicating that all possible choices are represented. The p(k) × 74 entries in Table 1 show how many grid cells we would expect to find with k tumuli if they are indeed spatially random as described by the Poisson distribution. The actual number of cells in Figure 6b that contain k tumuli are given in Table 1.

Table 1. Mauna Ulu Tumulia
kp(k)p(k) × 74Actual Number of Cells
  • a

    The parameter value is λ = 1.0 tumuli/grid cell. U = 1.08.


[25] Although the actual spatial distribution of tumuli appears similar to that predicted by the Poisson distribution, there are some differences. We can quantitatively evaluate how well the Poisson distribution describes the spatial distribution of tumuli by performing a χ2 hypothesis test. The χ2 test uses the differences between the predicted and actual occurrences to estimate a test statistic, U. This test statistic is a measure of goodness of fit of the Poisson distribution and is calculated by summing the squares of the differences between the predicted and actual frequencies in each bin, normalized by the predicted frequencies. The null hypothesis to be tested is that the spatial distribution of tumuli is random. If U is less than some critical value, we must infer that this hypothesis cannot be precluded. The critical value is found from the χ2 distribution. For 2 degrees of freedom (appropriate for the 4 bins, k = 0, 1, 2, ≥3, identified in Table 1) and a significance level of 5% (i.e., there is a 5% probability of rejecting the null hypothesis, even if it is true), the critical value of U is 5.99. For the case given in Table 1, U = 1.08. We cannot therefore preclude the possibility that the tumuli occur randomly in space within the flow unit. In other words, the perceived clustering is consistent with random tumulus formation.

[26] It is not clear, however, that the grid cell size resulting in λ = 1.0 is the most appropriate choice. It may be that different choices show more clustering of tumuli (near the margins for example). Thus we have investigated both smaller (λ = 0.52, Figure 6c) and larger (λ = 2.06, Figure 6d) grid cells to test the sensitivity of our inference. Tables 2 and 3 show the results of these comparisons. Note that whereas the test statistics, U, are greater than for the λ = 1.0 case, they are still significantly less than the critical value (=5.99). Thus we still cannot preclude a random spatial distribution.

Table 2. Mauna Ulu Tumuli With Smaller Grid Cellsa
kp(k)p(k) × 143 cellsActual Number of Cells
  • a

    The parameter value is λ = 0.52 tumuli/grid cell; U = 1.84.

Table 3. Mauna Ulu Tumuli With Larger Grid Cellsa
kp(k)p(k) × 36Actual Number of Cells
  • a

    The parameter value is λ = 2.06 tumuli/grid cell; U = 2.05.


[27] The analyses thus far have only considered tumuli. However, lava rises are also indicative of inflation. Therefore, for completeness, we have repeated the analysis for all inflation features within the investigation area, comprising all 74 tumuli and the 12 lava rises. Table 4 shows the results when we use the 74 equal area grids (same grid as shown in Figure 6b). In this case, we have 86 inflation features and λ = 1.16 features/grid cell. The result is a test statistic U = 0.364. Not only is this U significantly less than the critical value, it is also less than that calculated based only on tumuli. This indicates that the Poisson is an even better fit when all inflation features are included.

Table 4. Mauna Ulu Inflation Features (Tumuli and Lava Rises)a
kp(k)p(k) × 74Actual Number of Cells
  • a

    The parameter value is λ = 1.16 features/grid cell; U = 0.364.


[28] Another way to look at this issue of spatial variability is to use the nearest-neighbors technique [Clark and Evans, 1954]. This technique is a weaker test than comparison to the Poisson distribution, but has the benefit that it eliminates the arbitrary nature of the grid sizes and simply uses the distance to the nearest neighbor for each data point. The average of the measured nearest-neighbor distances is then compared to the expected average nearest-neighbor distance for spatially random points with the same density (number of features/unit area).

[29] For the nearest-neighbor approach we use the 74 tumuli contained in the large trapezoid in Figure 6a (with total area of 106,492 m2). Thus the density of tumuli is ρ = 6.9 × 10−4 tumuli/m2. The mean value of the nearest-neighbor distances is 1.04 times that expected for a truly random distribution with the same density. This difference is not significant at the 5% level (the test statistic, c = 0.712, is less than the critical value of 1.96). Thus we again infer that the tumuli are randomly distributed in space.

[30] On the basis of the analyses above, we must infer that the inflation features, when taken as a whole, are randomly distributed in space. However, it is clear that tumuli vary dramatically in size. Excluding lava rises, Figure 7 shows histograms of tumulus heights and effective radii (=equation image). The distributions are clearly asymmetric and the lognormal probability distribution is plotted for comparison. The χ2 test for goodness of fit indicates that, indeed, the lognormal cannot be precluded in either case.

Figure 7.

Histograms of (a) heights and (b) effective radii for tumuli at Mauna Ulu field site.

[31] A question one might ask is if, perhaps there is some clustering among the tumuli within different size ranges, consistent with emplacement style two. As described in the introduction, as more stable distribution pathways develop, we may expect to see larger tumuli. We have looked at the spatial distribution of tumuli (lava rises excluded) within each of the 8 height bins in Figures 6a and 7a. Remarkably, the results for each individual height range (Table 5) are distributed almost identically to that expected from the Poisson distribution. An alternative approach to estimating tumulus size is to use the effective radius of each tumulus as a measure of the basal area. If we use effective tumulus radius as our indicator of tumulus size, we begin to see evidence for systematic behavior. Figure 8 indicates the locations of only those tumuli with effective radii >12 m. Although there is qualitative evidence of a linear chain developing, unfortunately, there are too few data points to statistically quantify this trend.

Figure 8.

Locations of tumuli measured at Mauna Ulu with effective radii greater than 12 m. Using the histogram bins in Figure 7b as a guide, those tumuli with effective radii between 12–15 m (triangles), 15–18 m (minuses), 18–21 m (asterisks), and 21–24 m (open diamonds) are shown.

Table 5. Mauna Ulu Tumuli by Height Rangea
kBin 1 λ = 0.108Bin 2 λ = 0.568Bin 3 λ = 0.649Bin 4 λ = 0.351Bin 5 λ = 0.162Bins 6, 7, and 8 λ = 0.162
  • a

    Same height ranges as bins in Figure 7a. Pred represents predicted number of grids and Act represents actual number.

  • b

    The predicted number of grids does not sum to 37 because pf rounding to the nearest whole number for each k.


[32] Visual inspection of Figure 6 suggests that there might be some clustering along the northeastern margin, and that perhaps the lack of tumuli in the flow interior is somehow compensating for this in the Poisson comparisons of Tables 13 (note that in the context of this discussion, “interior” is used in the planar sense to mean that portion of the flow surface away from the margin, not the molten material beneath the crust). To investigate this possibility, we have divided the trapezoidal grid in Figure 6b (74 cells) into two regions: margin (two cells from each row along the northeastern margin), and interior (all other cells). Excluding lava rises, when divided in this way the “margin” contains 29 tumuli in 16 equal area cells (λ = 1.81 tumuli/grid cell), and the “interior” contains 45 tumuli in 58 equal area grids (λ = 0.776 tumuli/grid cell). The concentrations of tumuli, reflected in the values for λ, are clearly different. However, because most statistical tests assume a normal probability distribution, there is no simple way to test the significance of the difference between two mean values of the Poisson distribution. What we can do is a coarse test by estimating the standard error on each mean value. Recalling that the standard deviation, s, for the Poisson distribution is equal to the mean, the standard error on a mean value is sλ = s/equation image, where n is the number of items. Thus the individual mean values, with uncertainties of ±2sλ are then, λ = 1.81 ± 0.66 and λ = 0.776 ± 0.24. Because these two uncertainty ranges do not overlap, we infer that the two mean values are significantly different. However, the higher concentration along the margin is not enough to constitute significant clustering. Whether one considers the margin and interior independently (Table 6), or as a whole, the tumuli are all still randomly distributed in space. In other words, there is no statistically significant clustering of tumuli along the eastern margin.

Table 6. Mauna Ulu Tumuli Margin and Interior
kMargin, U = 0.532Interior, U = 1.41

4.3. Thrainsskjoldur

[33] Spatial data for tumuli and lava rises were also collected by Rossi and Gudmundsson [1996] for the Thrainsskjoldur flow in Iceland (Figure 3). Rossi and Gudmundsson [1996] identify both tumuli and lava rises in their data, using similar criteria to those used here to define these features. However, it is difficult to evaluate similarities and differences between their approach and ours, except we know that (1) only flow lobe tumuli, as defined by Walker [1991], were identified (other tumuli, e.g., tumulus ridges, were excluded); (2) features were mapped primarily from air photos; (3) the lava flow is moss covered; (4) Rossi and Gudmundsson [1996] describe the lava flow as hummocky; and (5) although the exact location of the tumulus study area is not indicated, Rossi and Gudmundsson do state that the inflation features were mapped in the distal portion of the flow field (9.5 km from the vent) where slopes are very low, similar to our study site at Mauna Ulu. Figure 9 shows the locations of tumuli and lava rises (as identified by Rossi and Gudmundsson [1996]) in the Thrainsskjoldur flow field, based on Figure 10 of Rossi and Gudmundsson [1996]. Again, an arbitrary grid with equal area cells is superposed.

Figure 9.

Locations of tumuli (diamonds) and lava rises (crosses) within a lava flow at Thrainsskjoldur volcanic shield, Iceland, as determined by Rossi and Gudmundsson [1996]. A grid with equal-area cells has been overlain.

[34] The grid shown in Figure 9 comprises 408 cells, with 390 tumuli and lava rises (λ = 0.956). The predicted and actual numbers of grids with k inflation features are shown in Table 7 under trial 1. In this case, the test statistic, U = 6.21, is somewhat greater than the critical value of 5.99. However, because the test statistic was so close to the critical value, the analysis was repeated by shifting the grid slightly such that none of the cells in the last column contained any features. With this scenario, the 390 inflation features are contained within a grid of 384 cells (λ = 1.02), and U = 3.64 (trial 2 in Table 7). Additional trials indicate that the hypothesis test is extremely sensitive not only to the grid but also to small differences in how the features are binned. Thus we infer that the test is inconclusive.

Table 7. Thrainsskjoldur Tumuli and Lava Rises Trials 1 and 2
kTrial 1, U = 6.21Trial 2, U = 3.64
  • a

    The predicted number of grids does not sum to 408 because of rounding to the nearest whole number for each k.


[35] We have also conducted a nearest-neighbor analysis on the Thrainsskjoldur tumulus and lava rise locations. In this case, the average nearest-neighbor spacing is 1.10 times that expected for randomly distributed features. The test statistic c = 3.8 is greater than the critical value, thus we infer that the deviation from randomness is significant at the 5% level. This test leads to an inference of systematic behavior for the Icelandic features (i.e., randomness is precluded). Although the planform shapes of some tumuli are provided, unfortunately, we do not have sufficient size data for all the individual tumuli on this flow. Therefore we cannot determine if the systematic clustering is coming from tumuli of a certain size range.

4.4. Elysium

[36] Figure 5 is a MOC image (MOC image 20-01192, centered at 1.86°N, 186.11°W) of a portion of a lava flow on the plains southeast of Elysium Mons. The high spatial resolution of MOC images (5.81 m/pixel) makes it extremely difficult to place the image in Figure 5 in context. Older, lower spatial resolution images from the Viking mission (Figure 4) indicate that the lava flow in Figure 5 extends for over 60 km. However, the low spatial resolution of the Viking imagery (925 m/pixel) is not sufficient to determine the source of the flow, or the flow direction. From the MOC image, we have mapped a 3 × 4.5 km section of the flow. The flow itself is relatively dark with an irregular surface, little apparent mantling by dust, and few impact craters. The regional slope in this area is almost nonexistent (0.02°). The flow appears to overlie a bright unit to the southeast. The surface of the flow has many positive relief features that we interpret as tumuli because they are the same shape (circular to elliptical) and approximately the same size (<10–50 m in diameter) as terrestrial inflation features (tumuli, lava rises). In addition, many of the larger positive relief features appear to have central depressions or clefts (indicated by shadows at their summits), diagnostic of tumuli.

[37] It is possible that these positive relief features on the Mars flow are not tumuli and/or lava rises, but some other type of surface feature. We cannot rule out that they have been produced by differential erosion of the flow surface. However, we do not favor this interpretation because the features fit the terrestrial definition of tumuli and lava rises, the morphology of this flow is different than another, more clearly eroded flow in the region, there are few craters on the flow surface and they are not highly eroded, and the flow has a well-defined boundary.

[38] Every positive relief feature on the flow surface that is at least 3 pixels (∼18 m) in diameter was mapped. Relief was estimated geometrically by using shadow lengths, the solar incidence angle (36.86°), latitude and longitude. The outline of each feature was used to find its center. Figure 10 shows the center locations, with an arbitrary grid superimposed. There are 801 topographic features and 735 grid cells inside the thick black line; thus λ = 1.09. The entire area entries in Table 8 indicate the predicted and actual number of cells containing k features. The test statistic for this scenario, U = 22.01, is significantly greater than the critical value of 5.99. Thus we must infer that the features within the Figure 10 grid are not randomly distributed in space.

Figure 10.

Locations of tumuli within the section of lava flow that can be seen in the MOC image in Figure 5. A grid with equal-area cells has been overlain. Dashed line 2/3 of the way down indicates arbitrary cutoff for sample of flow interior.

Table 8. Elysium Inflation Features
kEntire Area (λ = 1.09), U = 22.01Interior (λ = 1.42), U = 1.142

[39] However, the MOC image is a random slice through the lava flow that happens to capture one distinct margin and an unknown number of internal flow units. The interior values in Table 8 illustrate the results when we look only at the features within an arbitrary sample of the interior of the flow (i.e., from the top thick black line down to the dashed line). In this case, there are 754 features within 532 grid cells (λ = 1.42), and U = 1.14. Thus we infer that the features within the interior of the flow are distributed randomly, consistent with the Poisson distribution. An important caveat to this inference is that we are necessarily limited by the resolution of the MOC image. This means that any features smaller than 3 pixels (∼18 m) would not have been identified.

5. Discussion

[40] The four regions analyzed here each provide insight into their emplacement. The tumuli and lava rises at Mauna Loa are systematically clustered along a linear trend. This spatial constraint, combined with field evidence, points to tumulus/lava rise formation consistent with what we have defined as style three in the introduction. In this case, it is likely that the flow began as a narrow branch of pahoehoe lava that may have tended to pond in topographic lows in the low slope region between Mauna Loa and Mauna Kea. However, there is no evidence that a long-lived tube beneath these tumuli/lava rises was feeding some other part of the flow. Following the inflation that resulted in the large tumuli/lava rises, the flow migrated to the west, abandoning the eastern margin of the flow.

[41] In contrast, the other three locations exhibit a large number of inflation features over broad areas. Each of these examples show some clustering typical of Poisson spatial distributions. It is not possible, simply by looking at the data, to determine whether or not the clustering is significant. A quantitative statistical analysis is required to make this assessment. On the basis of comparison with the Poisson distribution, the study area at Mauna Ulu displays a decidedly random spatial distribution of tumuli. One can argue that the effect of time evolution of a pahoehoe lava flow can mask systematic tendencies. For example, if preferred pathways develop, and then later migrate or move, tumuli formed by both stages of emplacement will be frozen in time. If enough migration of preferred pathways occurs, the resulting spatial distribution of tumuli on the surface will be random, even if each distribution network was not. The best way to address this issue is with field observations. The Mauna Ulu field site was chosen specifically for its simplicity. Certainly the lava in the study area was erupted over some finite time interval, however, the flow deposit in this area does not display the complexity of neighboring units where multiple flow events are superposed and indistinguishable.

[42] The Mauna Ulu example is the most consistent with spatial randomness. Although statistical analysis supports our impression that there is a higher concentration of tumuli near the northeastern flow margin, there is no significant clustering. It appears that preferred pathways may have started to develop, as reflected in the tumuli with effective radii > 12 m. However, the lava supply to this flow was probably cut off before the preferred pathways could reach the maturity of style two emplacement. This interpretation is consistent with our field observations that this branch of the lava flow was not active for very long, as evidenced by the lack of subsequent overlying flow branches. Adjacent flows to the south and west are significantly more complicated, comprising multiple events. For the Mauna Ulu study area, it appears that tumuli are randomly distributed on the final flow surface. Holcomb [1976] does not show tubes in the sample area chosen for this study, so we cannot compare our distribution of tumuli to any known tube system. Two probable explanations for the random spatial distribution are that small-scale preferred pathways were transient in time, or the spatial distribution of tumuli could simply reflect random preexisting variations in subsurface topography. In either case, we cannot use the tumuli to uniquely map out subsurface lava transport networks. It seems most likely that the small-scale variability of the underlying topography, combined with the very low slope, may be influencing the development of local preferred pathways that evolve and migrate within the flow over the emplacement period. This type of tumulus formation mechanism is consistent with what we have called style one.

[43] Statistical tests for the other two examples, Iceland and Mars, where we would expect our inability to constrain the sample to possibly introduce a randomizing effect, are both inconclusive with regard to the randomness of the spatial distribution of inflation features. We infer in the Iceland case that the tumuli are intimately related to preferred pathways. The inconclusive results when comparing the spatial distribution of tumuli and lava rises to the Poisson distribution, along with the clustering inferred from nearest-neighbor analysis, is consistent with our definition of style two. In this case, because the spatial distribution is so nearly Poisson, the distribution of tumuli may have started out random, however, over time, more or less persistent preferred pathways developed that led to a systematic clustering of inflation features along these pathways. It is possible that at least some of the preferred pathways are mature tubes. However, additional field observations and measurements of tumulus sizes are necessary to corroborate this inference. Another possible explanation is that the tumuli are correlated with preferred pathways that were active at different times during the eruption. As one pathway became inactive and another became active, older tumuli were frozen in time. This has the effect of randomizing the spatial distribution. However, again, without additional field evidence for multiple flow events, it is not possible to distinguish this possibility from the other.

[44] In the Mars example, the margin of the flow is deficient in positive relief features, assumed here to be tumuli, whereas the flow interior contains numerous tumuli. Comparison of the distribution of tumuli in the flow interior with the Poisson distribution points to a spatially random distribution, consistent with style one emplacement and similar to results at Mauna Ulu. However, if the flow margin is also included in the analysis, the spatial distribution is not random owing to the significant lack of tumuli in the margin. This is different from the Mauna Ulu case where we see different concentrations of tumuli along the margin, but the clustering is not significant enough to affect the overall random distribution. A plausible explanation for the difference in tumulus population between the interior and margin of the flow is that what we see in the MOC image is in fact multiple flow units or a single flow that migrated in space during emplacement. The systematic behavior we have observed could, then, be attributed to different flow characteristics between the various units (e.g., perhaps an earlier flow unit contains numerous tumuli, while some later unit contains relatively few, or vice versa).

[45] It would be particularly interesting to see if the three emplacement styles defined here preferentially occur under particular combinations of slope and effusion rate conditions. This proof of concept study has shown that collection of additional data at other flows containing tumuli is warranted. Difficulties we have encountered trying to interpret spatial data suggest that the statistical rigor of future analyses will be enhanced by ensuring that (1) the boundaries of the study area are clearly defined and associated with logical geologic boundaries of the flow unit in question; (2) definitions of the features to be measured are established before data collection begins; and (3) data collection includes the identification and measurement of every feature within the study area identified in 1 that is consistent with the definitions prescribed in 2.

6. Conclusions

[46] The study presented here has shown that tumuli are sometimes randomly and sometimes systematically distributed spatially. We have proposed that the emplacement mechanisms can directly influence whether or not the resulting spatial distribution of tumuli is random or not. We have analyzed the spatial distribution of inflation features, including tumuli and lava rises, on parts of the 1843 flow at Mauna Loa, Hawaii; the 1969–1974 Mauna Ulu flow of Kilauea, Hawaii; a lava flow on Thrainsskjoldur, Iceland; and a lava flow on the plains southeast of Elysium, Mars. The data collected at the Mauna Ulu study area were carefully constrained by the authors, whereas data for inflation features at Thrainsskjoldur were taken from the literature and data for Mauna Loa were based on aerial photographs and a brief field survey. The Martian flow is well constrained, but necessarily limited owing to the nature of remote sensing images.

[47] In the 1843 Mauna Loa example, measurable tumuli and lava rises occur as a chain along the flow margin. This spatially linear trend, combined with field evidence, is consistent with the tumulus/lava rise formation mechanism we have defined as emplacement style three. In this case, it is likely that the flow began as a narrow pahoehoe flow that may have tended to pond in topographic lows in the low slope region. It does not appear that there was a long-lived tube beneath these tumuli/lava rises that was feeding some other part of the flow.

[48] Statistical analyses are required to distinguish between styles one and two at the other three locations. At Mauna Ulu, our results are unambiguous. We have compared the spatial distribution of inflation features (tumuli alone, and tumuli plus lava rises) with the Poisson distribution. We have varied the grid size used, and have split the features into bins according to location (margin versus interior) and by size (height and effective radius). Regardless of how the data are split, all analyses point to a spatial distribution that is consistent with the Poisson. A nearest-neighbor technique also supports the inference that the final spatial distribution of inflation features at Mauna Ulu is random. We do see significantly more tumuli near the northeastern margin of the flow, as compared to the interior, and there may be a chain of larger tumuli trying to develop in the center of the study area. However, in this case, it seems most likely that the small-scale variability of the underlying topography, combined with the very low slope, may be dominating the development of local preferred pathways that evolve and migrate within the flow over the emplacement period. Lava supply was likely cut off before more mature preferred pathways could develop. This type of tumulus formation mechanism is what we have termed style one.

[49] For Thrainsskjoldur, comparison of the spatial distribution with the Poisson distribution is very sensitive to the chosen grid. Nearest-neighbor analysis indicates that a random spatial distribution is precluded, consistent with style two. In this case, the spatial distribution of tumuli may have started out random, however, over time, more or less persistent preferred pathways may have developed that led to a systematic clustering of inflation features along these pathways.

[50] At Elysium, results for the interior of the flow are consistent with the Poisson distribution, however when including the flow margin, we must reject the Poisson. A plausible explanation for the difference in tumulus population between the interior and margin of the flow is that what we see in the MOC image is in fact multiple flow units or a single flow that migrated in space during emplacement. For the Mars flow, the margin appears to be deficient in tumuli, whereas the flow interior is one or several broad lava flows with spatially random tumuli, consistent with style one.

[51] Surface distributions of inflation features are likely related to preferred pathways that feed the flow, which could be few in number and persistent in time and space, or a more distributed network that is highly transient. These lava distribution scenarios are both likely to be influenced by preexisting topographic variability, and result in surface distributions of tumuli and lava rises that are nonrandom and random, respectively. In addition, nonrandom preexisting topography, such as underlying streambeds or lava channels, may control the final surface morphology.

[52] Important factors that may influence how and where tumuli occur include effusion rate, preexisting topography and its variability, and slope. These factors vary from volcano to volcano, and sometimes within a single flow. Thus, in order to constrain the factors listed above, we need to collect data on inflation feature distribution on different slopes, on different volcanoes, and in both simple and more complex flow fields. This larger data set will provide a basis upon which we can utilize the surface morphology of flows, such as those on Mars, to constrain emplacement processes.


[53] The authors would like to thank Ryan Dunn for field assistance in Hawaii and Kelly Shockey and Antony Brian for input on spatial distribution data. Thorvaldur Thordarson, Scott Rowland, and an anonymous reviewer provided very helpful comments on this manuscript. L. Glaze and S. Baloga would also like to thank Scott Rowland for field observations, interpretation, and discussion regarding the Mauna Loa 1843 flow and Thorvaldur Thordarson for field observations and discussions regarding the Mauna Ulu study area. E. Stofan and S. Anderson would like to thank John Guest and Angus Duncan for helpful discussions.