Quantifying uncertainties in the measurement of tephra fall thickness
© Engwell et al.; licensee Springer. 2013
Received: 6 March 2013
Accepted: 28 June 2013
Published: 10 September 2013
The uncertainties associated with tephra thickness measurements are calculated and implications for volume estimates are presented. Statistical methods are used to analyse the large dataset of Walker and Croasdale J Geol Soc 127:17-55, 1971 of the Fogo A plinian deposit, São Miguel, Azores. Dirichlet tessellation demonstrates that Walker and Croasdale’s measurements are highly clustered spatially and the area represented by a single measurement ranges between 0.5 and 10 km2. K-means cluster analysis shows that lower thickness uncertainties are associated with closely spaced measurements. Re-examination and analysis of Fogo A fall deposits show thickness uncertainties are about 9% for measured thickness while uncertainty associated with natural variance ranges, between 10 and 40%, with an average error of 30%. Correlations between measurement uncertainties and natural variance are complex and depend on a unit’s thickness, position within a succession and distance from source. Normative error increases as tephra thickness decreases. The degree to which thickness measurement error impacts on volume uncertainty depends on the number of measurements within a given dataset and their associated uncertainty. The uncertainty in volume associated with thickness uncertainty calculated herein for Fogo A is 1.3%, equivalent to a volume of 0.02 km3. However uncertainties associated with smaller datasets can be much larger; for example typically exceeding 10% for less than 20 data points.
KeywordsTephra thickness Observational uncertainty Natural variance
In volcanology, pre-historic eruption magnitude estimates rely on measured tephra thickness data, often presented as an isopach contour map (Thorarinsson and Sigvaldon 1972; Pyle 19891995; Fierstein and Nathenson 1992; Bonadonna and Houghton 2005; Sulpizio 2005; Bonadonna and Costa 2012). Assessment of tephra volume without the need for drawing isopach maps is a new development (Burden et al. 2013). These approaches are typical of a class of multiscale modelling problems in physical sciences, where traditionally, a macroscopic model, based on simplifying symmetry or linearization considerations, is used to characterize to the first order a particular physical phenomenon. Many natural processes involve influential smaller scale effects, neglect of which can sometimes cause such macroscopic models to fail to capture important details adequately. For instance, in meteorology, complex interactions between weather systems at different spatial and time scales are expressed in sophisticated multiscale models. However, such models become intractable in the limit by grid size and time-step constraints, and by limitations in numerical modeling of uncertainties. Little work has been conducted on quantifying, even at first order, the uncertainties associated with the tephra thickness measurements on which volume estimates are based, let alone give consideration to confounding or competing multiscale effects at increasing levels of granularity or gridding. Here we aim to determine how well thickness measurements characterize deposit thickness in the context of tephra volume assessment.
The topology of a tephra fall deposit depends principally on the duration and dynamics of the eruption plume, its interaction with wind, the original total grain size distribution of the ejecta (e.g. Walker 1973; Sparks et al. 1997; Bonadonna and Houghton 2005; Rust and Cashman 2011) and aggregation processes (e.g. Brazier et al. 1983; Brown et al. 2011). Tephra thickness is also affected by factors unrelated to eruption dynamics, for example: variation in topography, re-mobilisation, bioturbation, compaction and soil formation. The common lack of consolidation of volcaniclastic deposits means that, post-deposition, deposits are often thinned or removed by grainsize-dependent erosional processes, with anomalously thick deposits forming in topographic lows. In addition, compaction, especially of fine-grained (< 2 mm) ash deposits leads to increased deposit density and a thickness decrease of up to 50% over timescales of less than two years (Guichard et al. 1993; Blong and Enright. 2011). For this reason, Bonadonna and Houghton (2005) advocate measurements of mass per unit area as opposed to thickness, removing the need to correct for compaction or density variations. This said, erosion effects can remain an issue if immediate measurements are not made and it is only usually practical to determine mass per unit area in newly erupted material recovered from collecting trays, whereas mass measurements are usually impractical with most pre-historic deposits exposed in outcrops. As a result, thickness is likely to remain the measure of choice in palaeovolcanological studies.
The controlling assumption when analyzing and modelling tephra deposits is that tephra thickness is uniform over local areas, with deposits mantling topography and occurring everywhere except on very steep slopes (Duffield et al. 1979; Wright et al. 1980; Cas and Wright 1995). Tephra volume estimates derived from isopach maps rely on thickness trends being controlled by eruptive and atmospheric rather than depositional processes. Tephra fall models (e.g. TEPHRA2, FALL3D, HAZMAP) also adopt this assumption for computational reasons. The physics behind modeled tephra thickness is limited to processes in the simulated plume and, therefore, tephra is represented as depositing on a ‘flat’ surface. While digital elevation models are required to process the spatial distribution of modeled deposits and for presenting results, actual topography has no effect on calculated deposit thickness in such models. Inverse tephra modelling techniques determine optimised parameters by using a mathematical algorithm to assist in determination of a best-fit parameter set (Connor and Connor 2006). The optimality of input parameters is determined by calculating goodness of fit of model to measurements, with the best set of parameters resulting in a goodness-of-fit value of zero. However, this value is never reached for a number of reasons, one of which being that random variation in tephra deposits are not accounted for in the model (Connor and Connor 2006). Providing estimates of uncertainty for thickness measurements would enable better understanding of controls on model fit to data and therefore of uncertainties in model results.
The Fogo A Deposit
The Fogo member A deposit was produced during a trachytic Plinian eruption approximately 5000 years ago (Moore 1990). Walker and Croasdale (1971) and Bursik et al. (1992) separated the deposit into two volumetrically dominant fallout deposits, a lower syenite-poor Plinian fall and an overlying, syenite-rich Plinian fall deposit, separated by pyroclastic density current (PDC) deposits. Bursik et al. (1992) inferred the lower syenite-poor Plinian deposit formed while there was a southerly wind. The coarser grained syenite-rich deposit has a near-axisymmetric distribution and was formed during a more intense eruptive phase during which the plume reached an estimated height of 21 km (Bursik et al. 1992).
The thickness measurements from Walker and Croasdale (1971) relate to whole deposit thickness incorporating both syenite-poor and syenite-rich pumice fall deposits and interlayered PDC deposits. The measured PDC deposits are thin (typically less than 20–30 cm) and occur only in proximal (<10 km) localities. The Fogo A deposit has a maximum total thickness exceeding 20 m on the rim of the caldera decreasing to 5 m at the coast 6 km south west of the vent (Figure 1, Walker and Croasdale 1971). Erosion, weathering and soil formation prevent deposits of less than 0.25 m being easily recognized. Information to the north and south is limited because marine sediments that might contain the Fogo A tephra layer have not been sampled.
The Fogo member A dataset of Walker and Croasdale (1971) contains 250 subaerial tephra thickness measurements distributed axi-symmetrically around the vent. A small number of proximal deposits contain thin fine-grained PDC deposits but these contribute a minor fraction of the total thickness and are not observed to have been erosive. This enables assessment of deposit heterogeneity and large-scale spatial variance. The corresponding area that a single measurement can represent is dependent on the natural variability of accumulation, thickness decay with distance, measurement spacing, outcrop identification and accessibility. To-date no studies have focused on the effect of natural variability on thickness interpretation and observational uncertainty, and therefore there is little empirical guidance on what is the appropriate resolution for measurements.
In the case of Fogo A, the tesselation results display a clustered distribution in medial distances, particularly along the coast (Figure 2A). With distance from source, measurement spatial distribution becomes more uniform. Most measurements represent an area of between 0.5 and 10 km2, but there is a large amount of scatter (Figure 2B). These findings are attributed to significant spatial variations in the density of good outcrops and difficulties in locating suitable places on São Miguel (e.g. via road access to outcrops) at which to measure thickness at greater distances from source.
For each cluster, thickness variation was calculated using the standard deviation of all enclosed measurements and is expressed as a percentage relative to cluster average thickness (Figure 3B). Within a single cluster, measurements should have related thickness and therefore low standard deviation. When the dataset is split into a greater number of clusters the percentage error decreases. With increasing cluster number, the spatial area delineated by each cluster decreases and hence the encompassed measurements are more similar in terms of their immediate depositional environment and consequently thickness estimates. The within-cluster percentage error (calculated as one standard deviation from cluster mean thickness) decreases with distance from source when 5 and 10 clusters are used showing that over similar scales, proximal deposits are more variable than distal deposits. When the data are split into twenty clusters, the errors do not decrease with distance from source but remain constant. This latter error trend is considered to directly reflect local variation. The relationship between the number of clusters and within cluster error (Figure 3C) indicates that this trend can be represented by a power-law function with lower errors when more clusters are used.
Thickness with distance from source: exponential versus power law
Regression best fit equations for the Walker and Croasdale (1971) Fogo A thickness data as a function of distance from source
Least squares fit
Least squares fit
To investigate the bifurcation in thickness decay trends at distances greater than five km and subsequent effect on function fit, the data were split into four quadrants using north–south and east–west dividing lines (Figure 4B) that intersect at the source. This choice reflects the inferred wind-influenced asymmetries of the deposit. The bifurcation is explained by azimuth-dependent differences in rate of thickness decrease around the vent. Thickness decreases more quickly to the west associated with changes in wind strength and direction in the latter stages of the eruption (Walker and Croasdale 1971; Bursik et al. 1992). R2 values are much greater for each individual quadrant compared with a single regression although some scatter is still noted (Table 1), but are similar for both power-law (Figure 4C) and exponential (Figure 4D) regression. Power-law regression fits significantly better when the dataset is split into quadrants compared to when the dataset is considered as a whole. In summary, the whole dataset is better represented by an exponential function than a power law, but splitting the data into quadrants allows much better estimation of thickness trends with distance from source with little difference in fit between exponential and power-law trends.
Natural variability of deposits
Dirichlet tessellation analysis shows that a single tephra thickness measurement typically represents a deposit area between 0.5 and 10 km2. In addition, K-cluster analysis shows significant variance in deposit thickness within an area of a few km2 unrelated to typical thickness trends related to distance from source. To investigate these trends outcrop scale variance was determined by measuring the thickness of the Fogo A deposit over a number of different spatial scales.
Typically in experiments, measurements are repeated several times to provide a spread of values used to assess the uncertainty associated with a single variable. This enables the effects of observational uncertainty related to measurement technique, specifically random errors, to be reduced. During tephra thickness data collection, usually a single measurement is made at a single location. As a consequence, such data are liable to include some outliers, with uncertainties associated with inaccuracy and imprecision when measuring a given deposit. For instance, errors can arise from incorrect identification of the limits of the tephra unit or from failure to measure thickness normal to base contact. There is also a tendency to ‘fit’ measurements (perhaps subconsciously) into convenient user-defined size bins (e.g. to the nearest 10 cm), unrelated to the true precision of measurement (Hincks et al. 2013) which, when a large amount of data is collected across a whole deposit, can contribute to an overstating of total volume. In addition, a presumptive mental model may be employed, for example that proximal deposits are thicker than distal deposits. On local scales this assumption can be false, perhaps leading to small-scale variations being unintentionally smoothed out, in the dataset.
To assess the contribution of these measurement biases to thickness and therefore volume estimation, an experiment was conducted to quantify measurement uncertainties. Five scientists measured units 1 to 6 (Figure 5) using a tape measure with a resolution of 1 mm at Loc. 2 (Figure 1). The measurements (5 for each unit) were pooled to produce a dataset for each unit at each location. Observational uncertainty is calculated as one standard deviation normalized to average thickness expressed as a percentage (Figure 6A). For each of the units there is a large spread in the measurement error (up to 65%). Unit 6 and the ‘total’ thickness are exceptions, with much less scatter in the data and comparatively low error (< 18% and < 8% respectively). There is a strong correlation between the observational uncertainty and average unit thickness (Figure 6B): thinner units have greater error and therefore larger relative uncertainty than thicker units.
Trends in observational uncertainty closely follow those of natural variance. Again the greatest exception is unit 6 which had low observational uncertainty (Figure 5) but high error associated with natural variance. This is due to the characteristics of this particular deposit. The bounds of unit 6 are well defined and the unit is darker and finer grained than surrounding units. The same is true for the syenite-poor Plinian fall deposit measurement, where unit 6 provides a definite upper bound and the base is defined clearly by the contact with the underlying palaeosol. These characteristics make measurements easier and less subjective, resulting in lower errors and observational uncertainty. As most tephra deposits have a poorly defined upper limit, due to erosion and soil formation, the error estimates provided here may be regarded as a lower limit.
Implications and Conclusions
Tephra thickness errors for the Fogo A deposit calculated using statistical methods and fieldwork
All (Power Law)
The average and standard deviation percentage error for each method are presented in Table 2. As described in Figure 4, average error decreases with an increase in number of clusters although the standard deviation varies little. The average error associated with natural variance on scales less than about 200 m2 (28%) is similar to those associated with the difference between the measurements of Walker and Croasdale (1971) and regression fit to data split into quadrants (19–37%), and also the error calculated using 20 clusters (28%). Natural variability in real world dispersion processes is therefore inferred to explain the within-cluster scatter of data and the disparity between data and regression fit.
Because measurement of such natural variance includes associated observational uncertainty, we propose a typical value of ~ 30% as appropriate to describe total measurement uncertainty. The uncertainty associated with observational measurement was quantified separately and has an average of about one-third of the total uncertainty (i.e. ~ 9%.) The interplay between observational uncertainty and natural deposit variability means it is difficult to simply deduct one from the other to produce an overall uncertainty estimate. However, from a first order estimate using propagation of errors associated with observational uncertainty, the residual error associated with natural variance is approximately 26%, showing the net contribution from observational uncertainty is small.
The estimated on land volume of the Fogo A fall deposit is 1.2 km3 (Walker and Croasdale 1971). An average measurement uncertainty of 30% yields a volume uncertainty of 1.3%, equivalent to a volume error 0.02 km3 for the Walker and Croasdale dataset comprised of 250 measurements. Thickness measurement is only one source of volume uncertainty, which will also be influenced by spatial distribution of data, functional forms of fitting isopachs to thickness data, and extrapolation, using these functional forms, of thickness into regions where there is no data. The extent these factors impact on eruption magnitude assessments and other estimates of eruption dynamics, will be discussed further elsewhere.
The uncertainties presented in Table 2 are whole deposit average uncertainties. We have shown that uncertainties vary greatly depending on the underlying topography, the thickness of the deposit and its stratigraphic position, and difficulties in detecting true stratigraphic boundaries. Therefore uncertainties will vary significantly between proximal and distal locations. A further study to compare results from this proximal study with distal deposits would be highly informative. Fine-grained units at the base of the Fogo A deposit provide indicative evidence that these errors are likely to be significant due to direct deposition onto uneven ground and decreased deposit thickness. Remobilization of fall deposits means that over short length scales, thickness variance is high, particularly for thin deposits. In addition, the dark colour and finer grained nature of pyroclastic density current deposits means that deposit boundaries are more easily identifiable.
The findings presented here highlight the need for a standardized method of measuring and reporting tephra thickness in which an estimate of uncertainty is provided for every measurement. Here, we have studied uncertainties in thickness measurements in one particular deposit. We therefore recommend that future studies of tephra thickness should endeavour to assess these uncertainties both to better characterize the uncertainty in volume estimates and also to build up a published database of many case studies.
This work was supported by an ERC Advanced Research Grant (VOLDIES) to Prof RSJ Sparks FRS. R. Burden and E. Johnston are thanked for their assistance measuring deposits in the field. The manuscript benefited from helpful discussion with R. Blong. Maps were produced using GMT software (Wessel and Smith 1991) and greatly benefitted from the assistance of B. Hemmings. We would like to thank two anonymous reviewers and Chris Newhall who provided detailed suggestions that significantly improved the text.
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