Map showing the existing Kulani Prison site, as well as Site B and Site C, proposed for the new prison. Also shown is the extent of the study area for which mapping and dating of individual flows is used for the probabilistic calculations. (Hilo and Mauna Loa are shown as reference points.)
Our probabilistic estimates must be computed for specific project sites and require GIS digital maps and dates of individual lava flows and volcanologic structures in the area of interest. We use the best available geologic data for the general area, which includes the project sites and surrounding areas and is collectively referred to as the study area, to evaluate lava flow hazards specific to each project site. The three project sites are labeled "Existing," "Site B," and "Site C" (Wilson, Okamoto & Associates, Inc., 1998, written communication) in figure 1 within the boundary of the study area.
Specifically, the hazard we investigated is inundation by lava, defined as the entrance of an active lava flow into the boundaries of a project site. No distinction is made as to whether the lava flow enters from outside the project site or whether it is generated from a vent within the project site. Probabilities are calculated for at least one incident of lava inundation within arbitrary periods of 50 and 100 years. Probabilities for other periods may be calculated easily with the Poisson equation given in a later section. Probabilities are computed from estimates of recurrence interval or its reciprocal, event frequency.
Site-specific Recurrence
The probability of any event can be estimated from a time series of those events (e.g., Davis, 1986). Therefore the probability of lava inundation of a project site in the future can be based on a time series of lava inundation of that project site in the past. Specifically, we wish to estimate the average recurrence interval for lava flows reaching the project site. The average recurrence interval is the reciprocal of the average flow frequency. The most direct way to estimate either quantity is to excavate the site, date all flows found, and compute the average time interval between flows. The flows beneath the site are obviously the result of any eruption that could possibly produce a lava flow capable of reaching the site across any topographic obstacles now or in the past. One may think of the excavation findings as the result of the most realistic lava flow simulation experiment imaginable for this specific project site. Excavation is, of course, not practical, so we must search for the best alternative.
a) Geologic map (J.P. Lockwood and F.A. Trusdell, unpublished mapping) of the area surrounding the three project sites shown in figure 1. The lava flows are color-coded into 1,000-year intervals and represent ages from 14 years to 14,000 years before present. b) Legend for the geologic map.
The most comparable data set can be obtained by detailed geologic mapping of the surface of the study area. Such a geologic map is shown in figure 2 (J.P. Lockwood and F.A. Trusdell, unpublished geologic mapping). The map portrays the contacts of 49 lava flow units within the study area, 19 of which have been dated by radiocarbon methods or observed. Ages for the remaining 30 units are estimated from stratigraphic relations. Based on all data regardless of quality, the average recurrence interval for lava flows found on the surface within the study area is about 260 years (3.8 flows per thousand years [table 1]). For the 19 dated flows (oldest is 10,400 years) composing 37% of the total number of mapped flows, the average recurrence interval is about 200 years (10,400 yrs * 0.37/19).
cumulative | ||||||||
fractional | ||||||||
interval, yrs | study area | existing | existing+ | Site B | Site B+ | Site C | Site C+ | cover |
0-1000 | 11 | 4 | 6 | 1 | 1 | 1 | 1 | 0.196 |
1000-2000 | 4 | 2 | 2 | 0 | 0 | 1 | 1 | 0.423 |
2000-3000 | 2 | 2 | 2 | 1 | 1 | 0 | 0 | 0.437 |
3000-4000 | 4 | 3 | 3 | 0 | 0 | 0 | 0 | 0.533 |
4000-5000 | 2 | 1 | 1 | 0 | 0 | 0 | 0 | 0.631 |
5000-6000 | 6 | 1 | 4 | 1 | 1 | 0 | 0 | 0.832 |
6000-7000 | 5 | 1 | 4 | 0 | 0 | 0 | 0 | 0.852 |
7000-8000 | 5 | 1 | 1 | 0 | 0 | 0 | 0 | 0.906 |
8000-9000 | 3 | 0 | 3 | 0 | 0 | 0 | 0 | 0.934 |
9000-10,000 | 2 | 1 | 2 | 1 | 2 | 1 | 1 | 0.988 |
10,000-11,000 | 3 | 1 | 1 | 0 | 0 | 0 | 0 | 0.995 |
11,000-12,000 | 1 | 0 | 1 | 0 | 1 | 0 | 0 | 1 |
>12,000 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
Ave. /1000 yrs | 3.8 | 1.3 | 2.3 | 0.31 | 0.46 | 0.23 | 0.23 | |
STDev /1000 yrs | 1.9 | 0.89 | 1.3 | 0.43 | 0.57 | 0.36 | 0.36 | |
TOTAL | 49 | 17 | 30 | 4 | 6 | 3 | 3 | |
Area, sq. km | 552.85 | 29.36 | 345.22 | 2.492 | 34.95 | 0.5968 | 15.6 |
Number of flows within 1,000-year intervals for each of the project sites considered. Data are derived from J.P. Lockwood and F.A. Trusdell (unpublished mapping).
Specific recurrence-interval estimates for the three project sites can be estimated in a similar manner. First, all map units within each project site were selected and tabulated. In order to compensate for flows that are completely buried beneath a project site, all map units directly downslope from that site were also selected; the result is three expanded project sites (referred to here as the downslope adjustment; Kauahikaua and others, 1995a). Average recurrence intervals and areas are tabulated in table 1 (headings followed by "+" are the expanded project sites); the average number of flows per thousand years is plotted versus area in figure 3.
Number of flows per thousand years is proportional to the area of the project sites listed in Table 1.
The square symbols in figure 3 represent the three expanded project sites and the entire study area. The diamond symbols represent the three actual project sites, with the flow numbers obtained from the corresponding expanded sites. The number of flows exposed on the surface per thousand years shows a direct proportionality to the area over which they are exposed - approximately 0.64 flows per thousand years per 100 sq. km. The proportionality must be due to the random distribution of lava flow plan dimensions (width or length) in this region.
We are, however, more interested in the flow frequency for all flows not only exposed at the surface, but also buried beneath each project site for probability computations. The flow frequency for the expanded sites (as determined by the downslope adjustment) also shows a direct proportionality to project site area - 7.07 flows per thousand years per 100 sq. km. We again have the intuitive result that lava flow frequency (including exposed and buried flows) is approximately proportional to project site area, with the implication of random distribution of lava flow plan dimensions.
In the preceding analysis, we introduced the concept of downslope adjustment as one method of compensating for the under representation at the ground surface of older map units that are progressively covered by younger units. However, there is a problem with this method. For example, the surface of Mauna Loa is being covered at an approximately exponential rate of 40% in the first thousand years (Lipman, 1980; Lockwood and Lipman, 1987; Trusdell, 1995; see Kauahikaua and others, 1995b, for a discussion of exponential coverage rates). At this rate, more than 99% of Mauna Loa?s surface older than 10,000 years is covered, and we can no longer get an accurate idea of the number of flows emplaced before that time. The picture gets increasingly foggy even after 1,000 years. We need a more general method than downslope adjustment to modify the frequency distribution of flows now at the surface for the number of flows that are completely buried by younger flows and no longer visible.
Plot showing the number of exposed flows equal to, or older than, a specified age versus the fractional coverage by younger flows. The linearity of these data suggest that we can use coverage rate to estimate the number of flows covered during each thousand year period. A histogram of adjusted flow frequencies is also plotted in figure 5.
We can compute the coverage rate specifically for these flows within the study area. Figure 4 plots the cumulative fractional coverage at the beginning of each 1,000-year period (from (figure 3) produced per unit time, then the number of older flows not covered by younger flows should be inversely proportional to the cumulative fractional coverage of the younger flows. This is similar to saying that you see fewer unpainted tiles as you paint over a tiled floor; the number of unpainted tiles is inversely proportional to the area of paint applied. A straight line (y = -43.765x + 49.348) fits the data well in figure 4, consistent with our hypothesis but not proving it. The number of flows exposed beneath younger ones is estimated by 49.348 ? 43.765 * [cumulative fractional coverage] so the total number of buried flows can be estimated in turn as 43.765 * [cumulative fractional coverage]. Using this relation to estimate the number of unseen flows within each time interval, the adjusted estimate for the entire study area is 7.4 flows per thousand years, nearly twice the unadjusted estimate of 3.8 flows per thousand years (table 1). Figure 5 shows histograms of the exposed and adjusted flow frequencies in 1,000-year groupings. The adjustment (here termed the burial adjustment) is not perfect and may be better within the first 10,000 years than for the more distant past.
Histogram of observed and burial-adjusted flow ages found within study area, 1,000-year intervals.
We derived the burial adjustment for the entire study area and have already shown that flow frequency (both exposed and buried [figure 3]) is proportional to area considered. Thus we feel confident that we can apply this general factor to subareas, such as the three project sites. That is, the estimated frequency of lava flows, including buried flows, is 1.95 times (7.4/3.8) the frequency of unburied flows now exposed at the surface in the study area. Using the burial adjustment, we get an estimate of 2.5 flows per thousand years for the existing project site, 0.6 flows per thousand years for Site B project site, and 0.45 flows per thousand years for Site C project site. Normalized by area, the adjusted estimate is 7.3 flows per thousand years per 100 sq. km (R^{2}=1.0). This compares well with the 7.07 flows per thousand years per 100 sq. km. (R^{2}=0.998) obtained by the downslope adjustment.
A third method of flow frequency adjustment for the effect of under-representation by burial is to use only data from the most recent and best known period (here referred to as the recency adjustment; Kauahikaua and others, 1995a). Older periods are acknowledged as under-represented and therefore not included in any computation. The most recent period is recognized as the best known and least likely to be under-represented. For the present study, this would result in an estimated flow frequency of 11 per thousand years for the study area (based on the 0-1000 year row in table 1). The area-normalized flow frequency is 10.8 flows per thousand years per 100 sq. km (R^{2}=0.997) using project site entries in the 0-1000 year interval row of table 1. We could also use the statistic of three flank eruptions since 1832 on the northeast rift zone near the project sites to estimate an equivalent flow frequency of 18 flows per thousand years. Kauahikaua and others (1995a) recognized that this simple method of adjustment generally yields the largest flow frequency estimate of any method, because every histogram of lava flow age distributions is shaped like the one labeled "observed" in figure 5. The number of flows exposed at the surface is generally highest in the most recent past and decreases backward in time.
This method also relies on a presumption that the most recent past is most representative of Mauna Loa?s eruptive behavior in the near future. Discomfort with this method stems from its reliance on only a small part of the available data. How can we differentiate between a volcano whose eruptive frequency is changing with time and a volcano whose eruptive frequency is random and has had a recent period of more frequent activity? Even with a complete data set, the distinction may be difficult (Ho, 1996). The problem is compounded in our case by the increasing incompleteness of the data set further into the past. The dates on a lava flow sequence drilled through near Hilo form the only data set complete enough over several tens of thousands of years to assess the constancy of Mauna Loa?s eruption rate. Beeson and others (1996) show that the rate at which lava flows overran the drill site in the last 86,000 years was fairly constant at about one flow per 4,000 years. While this is not definitive, it is the only indication available. It does not support a pattern of increasingly frequent eruptions for Mauna Loa; we believe, therefore, that the recency adjustment will underestimate recurrence intervals and should not be used if there are better options available.
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