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Kauahikaua, J.P. USGS Open-File Report 98-794


Probability of Lava Inundation

We wish to estimate the probability of lava inundation for three project sites ? the existing Kulani Prison site and proposed sites B and C (fig. 1). Our analysis and the data are consistent with production of a random number of lava flows per unit time for at least the last 10,000 years. The lava flows are reasonably independent of one another; while the presence of a previous lava flow can influence the flow path of a later lava flow, the terrain is entirely composed of lava flows all of which influence later flows. The random influence of all previous flows on a later flow path has the same effect as making the later flow path relatively independent of any one previous flow.

Therefore a Poisson equation is appropriate to compute probabilities (Kauahikaua and others, 1995b). The probability of occurrence of at least one lava flow within the time interval, t, is 1 ? e-t/T , where T is the recurrence interval. Table 2 lists the recurrence intervals estimated by the three methods and the corresponding probabilities for t=50 and t=100 years for the three project sites (fig. 1). The italicized table entries for the recency adjustment are provided here only for comparison and are not intended as valid recurrence interval or probability estimates.

 

Table 2. Probability estimates for 50 and 100 years

 

 

Probability

 

 

T, years

t=50 yrs

t=100 yrs

Existing site, downslope adjustment

435

0.11

0.21

Existing site, burial adjustment

400

0.12

0.22

Existing site, recency adjustment

250

0.18

0.33

Site B, downslope adjustment

2170

0.022

0.045

Site B, burial adjustment

1670

0.030

0.058

Site B, recency adjustment

1000

0.049

0.095

Site C, downslope adjustment

4350

0.011

0.022

Site C, burial adjustment

2220

0.022

0.044

Site C, recency adjustment

1000

0.049

0.095

Probability estimates over arbitrary periods of 50 and 100 years for the three project sites adjusted in three ways for the effects of under-represented (buried) flows. The italicized, recency-adjusted estimates are less reliable than the others and are not used in this paper.

For a period of 50 years, the estimated probability of lava inundation is 11-12% for the existing prison site, 2-3% for proposed site B, and 1-2% for proposed site C. The larger differences between probabilities estimated with different adjustment methods for Site B and Site C probably reflect the larger errors expected for estimates made in significantly smaller areas. Estimates for the smaller project sites are based on fewer data, so we expect larger errors in estimating the recurrence intervals. The significantly higher probability for the existing project site is due primarily to its larger area (see figure 3).

 

Estimated lava flow dynamics

Forecasts of any eruption dynamic at Mauna Loa, such as duration or flow length, can be based only on the record of Mauna Loa eruptions since recorded observations began in 1832. Basic physical parameters for the 39 observed eruptions of Mauna Loa are tabulated in Lockwood and Lipman (1987), Barnard (1995), and Decker and others (1995). The project sites will be affected only by flank eruptions on the northeast rift zone. Table 3 lists the pertinent facts for the 17 flank eruptions since 1832, sorted by duration. Figure 6 is a plot of the fraction of flank eruptions whose duration is less than, or equal to, a specified period. Figure 7 is a similar plot of the fraction of flank eruptions whose area is less than, or equal to, a specified size.

Table 3

Table 3.              
  duration duration         Volume,
year summit, d flank, d   elev, m repose, mo Area, km2 106 m3
1877 1 1 W flank -55   1 8
1868 1 4 SW rift 1010 23 24 123
1916 0 12 SW rift 2260 16 17 31
1887 1 12 SW rift 1740 65 29 128
1942 2 13 NE rift 2800 20 34 176
1926 1 14 SW rift 2320 77 35 121
1907 1 15 SW rift 1890 37 28 121
1899 4 20 NE rift 3260 38 23 81
1852 1 20 NE rift 2560 6 33 182
1984 1 22 NE rift 2860 108 48 220
1950 1 23 SW rift 2440 12 112 376
1919 1 38 SW rift 2350 40 28 183
1935 6 40 NE rift 3690 23 33 87
1843 5 90 N flank 2990 126 45 202
1880 0 280 NE rift 3170 6 51 130
1859 1 300 N flank 2800 26 91 383

Parameters for the 17 flank eruptions of Mauna Loa between 1832 and present sorted by duration.

plot of Mauna Loa flank eruptions

Plot showing the fraction of Mauna Loa flank eruptions of equal or shorter duration than a specified period.

If we assume that future flank eruptions will be similar to the 17 between 1832 and 1984, then, using figure 6, we can forecast a probability of 0.5 (50%) that the next eruption will last 21 days or less, and 0.8 (80%) that it will last 50 days or less. We forecast a probability of 0.8 (80%) that the lava flow from the next eruption will have an area of 50 km2 or less from figure 7. Some flow area information could be obtained from pre-1832 flows that have not been covered, but this has not been done for our estimate.

plot of Mauna Loa flank eruptions

Plot showing the fraction of Mauna Loa flank eruptions of equal or smaller area than a specified period.

Terrain analysis ? computation of lavasheds

The project sites are located on Mauna Loa?s northeast rift zone, where 31% of eruptions since 1843 have been located (Trusdell, 1995). We can narrow down further the number of eruptive events that would affect a project site by delineating the "lavashed" of that site. A lavashed is the region that is uphill from each project site. The lavashed estimates included in this study were computed using the WATRSHED function in IDRISI for Windows (Eastman, 1997) on 30 m Digital Elevation Models (DEM) (USGS, 1987). The only input parameters for the operation of the WATRSHED function are the aspect of the DEM (azimuth of the slope vector) and the "angular region to either side of the uphill vector over which flow is accepted into an adjacent cell."

Lavasheds are computed from the project site uphill (rather than from the uppermost point downhill as with other geographic operators), and the angle parameter allows broadening of the acceptable uphill direction to include directions to either side of directly uphill (plus or minus half the angular parameter). If the DEM were a perfect representation of the terrain, then we could characterize a lavashed reliably with a small angular parameter. However, angular parameters of up to 90-degrees are required to smooth out errors and uncertainties in the DEM.

These errors and uncertainties are also evident by the presence of "holes" in the computed lavasheds. Lavasheds should have holes only where pits in the terrain occur. We ignore these holes, but we represent the lavashed with holes as computed in figures in this report.

To illustrate the utility of the concept, example lavasheds were calculated for a hypothetical project site consisting of the terminal lobe of flow 1 and 1A produced by the 1984 Mauna Loa eruption (Lockwood and others, 1985). The flow has already been emplaced, but, for the purposes of this example, we can pretend that it hasn?t and that we are charged with evaluating the vulnerability of this small area that was covered by the flow. Three different lavasheds were computed for the same hypothetical project site using three different values of the angular region parameter. Figure 8 shows the nested lavasheds for values of 60, 75, and 90 degrees. The 60- and 75-degree lavasheds indicate that there are two main topographic paths to the project site, but DEM uncertainties preclude tracking them fully uphill. Only the 90-degree lavashed is sufficiently general to permit outlining both paths all the way to Mauna Loa?s summit. The 90-degree lavashed is the most general and probably includes more area than is useful for our purpose. Our best choice for the angular parameter is between 75- and 90-degrees, but we will continue to use 90-degree lavasheds to err on the conservative side.

map showing lavasheds

Map showing the nested 60-, 75-, and 90-degree lavasheds computed for the terminal lobes of the Mauna Loa 1984-1 and 1984-1A flows. For comparison, the 1852, 1855, 1880, 1881, 1942, and 1984 lava flows are also shown. Flow lobes 1984-1 and 1984-1A are referenced in the text. The blank areas within the lavasheds are the "holes" mentioned in the text.

How have recent lava flows traveled compared to the computed example lavasheds? The outlines of the most recent lava flows in the area (1984, 1942, 1880, 1881, 1855, and 1852) are also plotted in figure 8. The 1855 and the 1880 flows begin within the 90-degree lavashed and flow out of it without ever entering the 75-degree lavashed. Neither enters the hypothetical project site. The 1881 and 1942 flows enter the 75-degree lavashed, and the 1881 flow also enters the 60- degree lavashed before departing all lavasheds; neither of these flows enters the hypothetical project site. The 1852 and the 1984-1 flows begin within the 75-degree lavashed, and a portion of each flow remains within it to finally enter the hypothetical project site. The 1984-1A flow is essentially a separate flow, starting at the channel breach outside the 75-degree lavashed and entering the 75-degree and 60-degree lavashed and finally the hypothetical project site. The reason that the 1984-1A flow must be considered a separate flow is that it is responding to the additional topography of the previously emplaced 1984-1 flow which changed the way a lavashed would be computed for the next flow. The DEM we used includes the topographic expression of all flows except the 1984 flow, because the DEM is based on air photos taken in the 1970s. In other words, flows earlier than 1984 did not respond to the same topography that was used to compute these lavasheds; they would have responded to the preflow topography, which we can no longer characterize.

Experience with lavasheds computed in the above way allows the following generalizations: Adjacent project sites can have overlapping lavasheds, i.e., lavasheds are not mutually exclusive. All lava flows that inundate the project site enter from the lavashed. Not all lava flows that begin within the lavashed will enter the project site. Lava flows that begin within the lavashed, then exit, do not seem to reenter; the one exception would be breakouts or channel breaches at the edge of lavasheds like the Mauna Loa 1984-1A flow. Such breakouts should be treated as separate flows that originate at the breach point. Hanley (1998) formulated similar generalizations in an application of ARC/Info FLOWDIRECTION and COSTPATH functions for lava flows produced by Pu`u `O`o. A lavashed is a reasonable means of using terrain to define the area from which lava flows can enter a project site.

Finally, we can use the nearly identical flow advance rates for the 1942 Finch (1942) and 1984 (Lockwood and others, 1985) flows to estimate probable advance rates for flows in the lavasheds of the three project sites. Figures 9 and 10 show the estimated 9-, 24-, 48-, and 72-hour warning lines based on these advance rates for each of the project sites. Lavashed maps with these warning lines can be used to estimate how much lead time is available to respond to a flank eruption that begins within the lavashed. For example, using figure 9 we can estimate that a flow produced by a vent that opened on the 9-hour warning line could enter the existing prison project site 9 hours later. Maximum warning times can also be determined from figures 9 and 10. The existing prison project site would have no more than 24 hours warning for a flank eruption that started within its lavashed. Sites B and C could have as much as 72 hours warning for a flank eruption that started within their respective lavasheds.

map of the lavashed for the exiting prison site

Map of the lavashed for the existing prison site showing the 9- and 24-hour warning lines. Note that any eruption beginning within this lavashed would produce a lava flow that could enter the existing prison project site in less than 24 hours. The 1984 lava flows are shown for comparison. The short southern lobes slowed within 15 hours.

map of the lavashed for the future prison Site B and Site C

Map of the lavashed for future prison Site B and Site C showing the 9-, 24-, 48-, and 72-hour warning lines. The 1984 lava flows are shown for comparison.

 

Conclusions and recommendations

The flow frequencies estimated in this study are proportional to the size of the area considered. The probabilities for a 50-year period are 11-12% for the existing prison site (29.4 km2), 2-3% for proposed site B (2.5 km2), and 1-2% for proposed site C (0.6 km2). Thus the higher probability of lava flow inundation at the existing site reflects primarily the relatively large size of the project site. Defining a large project site that includes a buffer zone beyond the immediate boundaries of the facility is a more realistic approach however, considering the difficulties of evacuating a 2,300-inmate prison. A lava flow that passes well outside the project areas defined for sites B and C may still necessitate evacuation if the flow threatens roads and utility lines leading to the site.

Not included in the probability calculation is the significantly shorter maximum warning time of 24 hours for the existing prison site reflecting its proximity to potential vent areas. The decision to evacuate the prison will have to be made well before it is possible to determine whether the site is directly in the path of an oncoming lava flow or not. The warning lines depicted in figure 9 show that if another eruption occurs near the 1984 vents, lava flows could reach the existing site within 9 to 24 hours. Locating the prison farther downslope at site B or C increases the possible warning time by about 48 hours (figure 10). In either case, the probability of having to evacuate the prison is significantly higher than the probability of lava actually covering the site.

The estimates and forecasts reported here are based on the best information currently available. Some of the estimated quantities, possibly the probabilities, will change as new information becomes available. Updated mapping, and more radiometric dates, which result in identification of new flows or consolidation of one or more flows into a single eruptive unit, will have a significant effect, because it will change the total number of exposed flows and therefore the recurrence interval estimates. Better models of variations in Mauna Loa?s eruptive rate with time will also change estimates of the currently appropriate recurrence interval; we assume a random eruptive rate with no time dependence in this report. More accurate DEMs may change the computed lavasheds. We recommend further mapping and dating in order to improve hazard estimates in the future.

Acknowledgements

Don Swanson, scientist-in-charge at the Hawaiian Volcano Observatory, suggested that this project as soon as the State of Hawai`i announced its intention to construct a new prison on the slopes of Mauna Loa. The authors thank Asta Miklius and Paul Okubo for very useful reviews.

References

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