We apply a mathematical model to interpret and interpolate tephra thickness data. Spline interpolation is a methodology whereby the interpolant is a piecewise polynomial, i.e., the function is defined by multiple sub-functions that each apply to a certain interval of the domain. In the 2-D case of cubic B-spline interpolation, the solution can be envisaged in terms of deformation of a thin elastic ‘plate’ under tension (Inoue 1986). The surface of this notional plate is locally pulled towards each data point by the numerical equivalent of a spring whose individual stiffness is inversely proportional to data variance (Inoue 1986). Here, we fit cubic B-splines to tephra thickness data using the FORTRAN code of Inoue (1986).
The cubic B-splines method of Inoue (1986) is based on minimization of a weighted sum of the least squares spline fit to the data points (d
p
)(p = 1 ….. n) and the first and second derivatives. The first derivative is associated with the spline tension and serves to minimize fluctuations at the spline boundaries, while the second derivative is related to the roughness of the spline. The aim is to find a continuous distribution approximation ϕ, which minimizes the approximation error. To this end, the smoothing fit to the data is determined by the least squares norm ⊓ composed of data residuals, the first and second derivatives:
$$ \sqcap =\parallel {R}^2+\parallel {J}^2={\displaystyle \sum_{p=1}^n}{W}_{\mathrm{p}}{\left({\phi}_{\mathrm{P}}-{d}_{\mathrm{p}}\right)}^2+\frac{1}{l_{\mathrm{u}}^2}{\displaystyle \underset{\Omega}{\iint }}\left[{W}_1\left({\phi}_x^2+{\phi}_y^2\right)+{W}_2\left({\phi}_{xx}^2+2{\phi}_{xy}^2 + {\phi}_{yy}^2\right)\right]\ dx\ dy $$
(1)
where R refers to the misfit of the function to the data, J represents the roughness, ϕ is the smoothing function, Ω is the domain, W represent weightings, l
u is a unit length, and subscripts x, y, etc. express differentiations. The optimum smoothing function \( \widehat{\phi} \) through the thickness measurements is defined over the area of interest (Ωx × Ωy) in the x-y plane, where x refers to the longitude and y the latitude of the area in question. The smoothing function in tensor product form for the cubic B-spline basis functions Fi(x) and Gj(y) is described as:
$$ \widehat{\phi}\left(x,y\right) = {\displaystyle \sum_{i=1}^{M_x+3}}{\displaystyle \sum_{j=1}^{M_y+3}}{c}_{\mathrm{i}\mathrm{j}}{F}_{\mathrm{i}}(x){G}_{\mathrm{j}}(y) $$
(2)
where Mx and My are the number of coefficients in the x and y domains, respectively, and c
ij are the unknown coefficients. The cubic B-spline bases have equally-spaced knots, i.e., divisions of the area in question, ξ
− 3 … ξ
0, ξ
− 1 … ξ
M, ξ
M … ξ
M + 3 and, as an example, the nonzero parts of the basis F
i
(x), can be written as:
$$ {F}_{\mathrm{i}}(x)=\left\{\begin{array}{c}\hfill \begin{array}{cc}\hfill 0\hfill & \hfill \left(x<{\xi}_{i-4}\right)\hfill \\ {}\hfill {B}_1\left[\left(x-{\xi}_{i-4}\right)/f\right]\hfill & \hfill \left({\xi}_{i-4}<x<{\xi}_{i-3}\right)\hfill \end{array}\hfill \\ {}\hfill \begin{array}{cc}\hfill {B}_2\left[\left(x-{\xi}_{i-3}\right)/f\right]\hfill & \hfill \left({\xi}_{i-3}<x<{\xi}_{i-2}\right)\hfill \\ {}\hfill {B}_3\left[\left(x-{\xi}_{i-2}\right)/f\right]\hfill & \hfill \left({\xi}_{i-2}<x<{\xi}_{i-1}\right)\hfill \end{array}\hfill \\ {}\hfill \begin{array}{cc}\hfill {B}_4\left[\left(x-{\xi}_{i-1}\right)/f\right]\hfill & \hfill \left({\xi}_{i-1}<x<{\xi}_i\right)\hfill \\ {}\hfill 0\hfill & \hfill \left({\xi}_i\le x\right)\hfill \end{array}\hfill \end{array}\ \right. $$
(3)
where f is the knot interval and B
i
(r) are the cubic polynomials:
$$ \begin{array}{c}\hfill {B}_1(r)=\frac{r^3}{6}\hfill \\ {}\hfill {B}_2(r)=\frac{-3{r}^3+3{r}^2+3r+1}{6}\hfill \\ {}\hfill {B}_3(r)=\frac{3{r}^3-6{r}^2+4}{6}\hfill \\ {}\hfill {B}_4(r)=\frac{-{r}^3+3{r}^2-3r+1}{6}\hfill \end{array} $$
(4)
By substituting Eq. 2 into Eq. 1, and assuming ∂⊓/∂c
j
= 0 for j = 1, …, M, i.e., ⊓ is minimized, M linear coefficients for the coefficients c can be written in matrix form whereby:
$$ \mathbf{K}c=\mathbf{b} $$
(5)
The coefficient matrix K is decomposed as:
$$ \boldsymbol{K}={\boldsymbol{K}}^{\boldsymbol{d}}+{\boldsymbol{K}}^{10}+{\boldsymbol{K}}^{01}+{\boldsymbol{K}}^{20}+2{\boldsymbol{K}}^{11}+{\boldsymbol{K}}^{02} $$
(6)
where
$$ {K^{\mathrm{d}}}_{{\mathrm{i}\mathrm{ji}}^{\prime }{\mathrm{j}}^{\prime }}={\displaystyle \sum_{p=1}^n}{w}_{\mathrm{p}}{F}_{\mathrm{i}}^{\mathrm{p}}{G}_{\mathrm{i}}^{\mathrm{p}}{F}_{{\mathrm{i}}^{\prime}}^{\mathrm{p}}{G}_{{\mathrm{j}}^{\prime}}^{\mathrm{p}} $$
(7)
and
$$ {K}_{{\mathrm{i}\mathrm{j}\mathrm{i}}^{\prime }{\mathrm{j}}^{\prime}}^{\mathrm{hl}}=\frac{W_{k+1}}{l_u^2}{\displaystyle \underset{\Omega_{\mathrm{x}}}{\int }}{F}_{\mathrm{i}}^{(k)}{F}_{{\mathrm{i}}^{\prime}}^{(k)}dx{\displaystyle \underset{\Omega_{\mathrm{y}}}{\int }}{G}_{\mathrm{j}}^{(l)}{G}_{{\mathrm{i}\mathrm{j}}^{\prime}}^{(l)} dy $$
(8)
and the vector b is given as:
$$ {b}_{ij}={\displaystyle \sum_{p=1}^n}{w}_p{F}_i^p{G}_i^p{d}_p $$
(9)
Here, w
p are weights that describe uncertainty in thickness measurements d
p. Further explanation and derivations are provided in Inoue (1986).
The method requires four fitting parameters; tension (τ), roughness (ρ), number of divisions of the area (i.e., spline knot spacing), and measurement weights (w
p). Applying the method, we try to satisfy two conflicting aims, namely producing a good fit to the data while avoiding excessive local variation in the resultant surface that becomes unphysical. Tension is added to the spline to minimize distortions of the surface between data points, such that the fitted surface remains physically meaningful and plausible for subsequent analysis. The tension value can vary between 0 and 1. When no tension is applied to the data (τ = 0), unphysical bumps characterized by large thickness variations can appear in areas far from constraining data points. However, applying high tension (τ = 1) can cause sharp, unrealistic angles in the surface at data points (Appendix 1). Following previous studies (e.g., Inoue 1986; Bauer et al. 1998), τ is set to 0.99 to prevent anomalous highs forming in extrapolated areas with little or no data.
The knot spacing is determined based on the deposit extent and number of measurements. For well-documented deposits, more knots can be used to show local variation in thickness trends. When data are poorly spaced across the deposit extent, fewer knots are used, so a smoother surface is produced reflecting global rather than local thinning trends (see Appendix 2 for further details). In the examples presented here, a knot spacing of 10 km is used.
In the first instance, weights of one were applied (w
p = 1), so that all data points had equal weighting. Varying the weighting of all thickness measurements by the same amount results in a volume estimate that varies by the same factor. However, an advantage of the spline method is that each measurement may be weighted individually, for example, according to measured thickness uncertainty or according to proportional error. But, in general, values of uncertainty are not presented for thickness measurements reported in the volcanological literature. However, uncertainty estimates are available for the Fogo Member A example (Engwell et al. 2013) and here, in addition to being weighted equally, the Fogo A dataset is also weighted according to these uncertainties, as described below.
Roughness (ρ) controls the balance between fit to data and overall model surface roughness. Roughness values that are too small give large residuals relative to observational errors on individual measurements. However, large roughness values give too much weight to measurements with no consideration to global trends in data (Inoue 1986). Typical values used for ρ lie between 0.1 and 1000, where 0.1 gives a very smooth fit (essentially a linear best fit plane through the data) while a roughness of 1000 produces exaggerated variations in the surface (Fig. 1). Small values of roughness are sufficient where there is a little local variation in data. Higher roughness values are required to reflect large variations in data, but too high values of roughness obscure the global trend. Therefore, choice of roughness value is guided to some extent by the visual credibility of the resulting fitted thickness trend surface. In the method, we determine the most appropriate value by using roughness values that range over six orders of magnitude (0.01, 0.1, 1, 10, 100, and 1000) and assess the resulting map based on fit to data points and visual credibility. Surfaces that are too flat and therefore do not accurately represent global changes in data, and those that are too rough with overly complicated contour patterns are discarded. Thus, there is some element of subjective judgment in parameter choice with a spline-derived pattern that has—through adjustment of the roughness parameter—a broad similarity to the decision process for visually-identified, hand-drawn isopach maps. The spline method provides results that produce the lowest objective data-fitting error, either in terms of global trends in the data or for highlighting very local variations in deposit thickness, depending on the choice of spline parameters.
Fitted results are in the form of a gridded dataset of interpolated thickness values across the specified x-y domain. This dataset is processed using GMT software (Wessel and Smith 1991), where the function grdcontour is used to contour results at user specified thickness, and grdvolume is used to determine area within each contour to enable the production of log thickness versus square root isopach area plots. The volume between each isopach contour and the interpolated surface was quantified by direct integration of the spline-derived surface. Volumes were derived from the cubic B-spline isopach maps assuming exponential thickness decay following Pyle (1989) and Fierstein and Nathenson (1992). These methods do not enable extrapolation of the thickness trends to areas where there is no deposit exposure, and therefore estimates constitute a minimum erupted volume, thus allowing direct comparison with volumes determined by integration of the spline surface. Consideration of uncertainties associated with choice of thickness decay assumption (e.g., exponential (Pyle 1989), power law (Bonadonna and Houghton 2005), or Weibull (Bonadonna and Costa 2012)) is discussed in detail in Klawonn et al. (2014a, b) and is not pursued further here.
The method was applied to a number of examples representing a range in eruptive styles and magnitude (Table 1). 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. The lower syenite-poor Plinian deposit formed while there was a southerly wind, while the coarser grained syenite-rich deposit has a near-axisymmetric distribution and an inferred Plinian plume height of 21 km (Bursik et al. 1992). Walker and Croasdale (1971) present measurements of the combined thickness of the two Plinian deposits and inferred a bulk deposit volume of 1.2 km3. The Walker and Croasdale (1971) Fogo member A dataset is large, with 250 measurements, distributed axisymmetrically around the vent, and as such the dataset represents one of the best in the world with regards to spatial density of measurements. However, the distribution of measurements is limited to Sao Miguel Island, and therefore only the very proximal to medial deposits are represented, with the furthermost measurement ~20 km from source. A number of measurements of ‘zero’ thickness were reported by Walker and Croasdale (1971), indicating eastern and western extents of the deposit. The deposit is poorly constrained to the north and south, where deposition occurred over the sea. This said, because dispersion is inferred to be near uniaxial, thickness trends are well represented by measurements on land. The study of Engwell et al. (2013), quantified uncertainties for the Fogo member A deposit, and therefore it was possible to weight measurements directly to uncertainties measured in the field. Engwell et al. (2013) showed that for the Fogo A deposit, the uncertainty, y (%), in thickness, x (cm), follows a power-law:
$$ y=29{x}^{-0.3} $$
(10)
Table 1 Fogo member A isopach map of Walker and Croasdale (1971): Comparison of differences between measured thicknesses and spline interpolation of published isopachs; mean and standard deviation of percentage differences for distributions shown in Fig. 2 (first row)
Therefore, interpolation was conducted by weighting data points according to this relation, with thinner distal deposits given less weight than thicker proximal deposits.
The Askja 1875 Layer D deposit is the main product of a 6.5 h-long rhyolitic Plinian eruption (Sparks et al. 1981). The deposit is coarse grained and well sorted, and proximally can be separated into five subunits (Carey et al. 2010). The medial–distal deposit is comprised of subunits D1, D3, and D5 (Carey et al. 2010) and covers an area of 7500 km2 (Sparks et al. 1981). Published isopachs of the deposit are elongate to the east, showing dispersal was strongly wind controlled. Thickness values were digitized from Sparks et al. (1981). The 136 measurements are well distributed in proximal areas and in distal reaches are focused in bands near perpendicular to the crosswind axis.
Thickness data for the rhyolitic 1.8 ka Taupo Plinian Pumice deposit were digitized from Walker (1981), who inferred a bulk deposit volume of 9 km3. Published isopachs show a strong easterly elongation, with maximum thickness displaced 20 km downwind from inferred source. In comparison to the Askja Layer D and Fogo member A eruption, there are no very proximal thickness values available, due to source location within Lake Taupo. The dataset comprises 180 measurements that are well dispersed across the medial portion of the deposit, becoming more widely spaced in distal portions.
Thickness data for the 1995 Cerro Negro (81 measurements), El Chichon 1982 Layer B (69 measurements), and Mount St. Helens May 18th 1980 (235 measurements) eruptions were obtained from the IAVCEI Commission on tephra hazard modeling (http://www.ct.ingv.it/Progetti/Iavcei/results.htm). The basaltic Cerro Negro eruption consisted of numerous small explosions over the course of 13 days and resulted in a deposit with an estimated bulk volume of 0.003 km3 (Hill et al. 1998). Ash was deposited up to 30 km to the west, with published isopachs showing a distinct bilobate distribution.
The El Chichon Layer B deposit resulted from a major trachyandesitic Plinian eruption with an observed column height of >17 km (Carey and Sigurdsson 1986). The resultant plume was dispersed ENE–WSW, with published isopachs showing a circular distribution. Maximum deposit thickness is found 6 km downwind from source, and the inferred bulk deposit volume is 0.79 km3 (Carey and Sigurdsson 1986). Thickness measurements are well distributed across the deposit extent, with spatial distribution decreasing from proximal to distal reaches.
Finally, the Mount St. Helens eruption of May 18th 1980 began as a result of a large landslide that released pressure from a shallow magma chamber resulting in a lateral blast. Large pyroclastic density currents (PDCs) followed, covering an area of 600 km2 (Sparks et al. 1986). From these currents, a large co-PDC plume formed, dispersing fine-grained ash to the north and east of the source (Sparks et al. 1986). Approximately 30 min after this phase, Plinian activity began (Sparks et al. 1986), with an average column height of 16 km, transporting dacitic material to the east (Sarna-Wojcicki et al. 1981; Carey and Sigurdsson 1982). The eruption lasted 9 h producing a deposit that extended 1000 km east of the source. The data used here describe both the co-PDC and Plinian deposit. While in most other examples, deposit extent is poorly constrained, in the Mount St. Helens example the downwind deposit extent is well constrained by 47 zero thickness values.