1 Introduction

Volcanic eruption is one of the most destructive natural phenomena. Its occurrence can affect places far from the eruption vent, when large volumes of volcanic materials are ejected, covering large areas with volcanic debris called tephra. Estimating the total tephra fall volume provides important data to understand the nature of volcanic activity better and predict future volcanic eruptions. It also provides information about the extent of destruction caused by the eruption, which is very useful in mitigating the phenomenon's effects.

Estimating tephra fall volume is based on the relationship between the thickness of volcanic debris and area. The changes of their values from the vent to the farthest distance that the volcanic materials reach are described using several mathematical models. Volcanic ash deposit field data are converted into isopach contour maps in cm or isopleth mass contour maps in kg/m2, which are then used to compute tephra fall volume. Several models have been formulated to show thickness as a function of the isopach area. The most used ones are the exponential (Pyle 1989), power law (Bonadonna and Houghton 2005), and Weibull (Bonadonna and Costa 2012). Fierstein and Nathenson (1992) studied some approaches to model the relationship, while Daggitt et al. (2014a) developed the AshCalc software to efficiently implement the tephra fall volume estimation and compare the results generated using different models. Takarada et al. (2001), Takarada et al. (2016) developed the segment integration method, which was used to estimate the discharge mass of the 2000 and 2014 eruptions of Usu and Ontake volcanoes, respectively. Volume estimation for recent volcanic eruptions used these models, like the work of Balangue-Tarriela et al. (2022), for estimating the volume of volcanic debris ejected by the 2020 Taal volcano eruption. Developing a new computing system that is highly accessible, efficient, and easy to implement for estimating tephra fall volume is very important.

Web-based spatial information systems have recently been the preferred platform for processing geospatial information. The information system is called WebGIS, which deals with geographic information, including geospatial analysis, within the online environment (Netek et al. 2023). Most WebGIS systems follow the Open Geospatial Consortium (OGC) standards (OGC 2022a). OGC is the collaboration of more than 450 commercial, governmental, nonprofit, and research organizations for the development and implementation of open standards. Its main objective is to unlock the power of geospatial data to solve society’s most pressing problems. WebGIS provides spatial information processing capabilities through OGC-based web services. These web services include the Web Map Service (WMS) and Web Processing Service (WPS). WebGIS renders maps by sending requests to WMS servers, which are set up to accept requests to query online databases and return map images. It also processes spatial information by sending requests to WPS servers. WPS is a service interface through which information processing can be done over the Internet (Tripathi et al. 2019).

WebGIS is generally organized following the Spatial Data Infrastructure (SDI) model (Stefanakis and Prastacos 2008), wherein the system is conceptualized as a three-layered system, as shown in Fig. 1. The three layers of an SDI, from top to bottom, are the users and applications, geospatial services and geospatial contents. The geospatial services layer provides a wide array of web services for data rendition, access, and processing. These web services process requests to access the geospatial contents layer, where the spatial database resides. Requests are formulated at the users and applications layer and then sent to the web services layer, where they are used to access the spatial database. The request results are then sent back to the users and applications layer, where they are used to display maps or vector data that may be used for further data processing.

Fig. 1
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Spatial data infrastructure (SDI) architecture

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Geological surveys and related organizations have recently stored geographically referenced information in a web-based geospatial databases. Formulating web services to access and process these web-based geospatial contents is starting to be the rule rather than the exception. Major projects for the sharing of geospatial data using the SDI model have been ongoing for years. These include the OneGeology project, which is participated in by the world’s major geological surveys to share geological maps and related geoscience data. The CCOP Geoinformation Sharing Infrastructure for East and Southeast Asia (GSi) is one of the most significant geoinformation sharing projects in Asia. The project is implemented by the Coordinating Committee for Geoscience Programmes in East and Southeast Asia (CCOP). This paper presents a WebGIS-based tephra fall volume estimation system called WebTephraCalc, which is developed following OGC standards and SDI architecture. It is a tephra fall volume calculation system with exponential, power law, Weibull, and the segment integration method using data provided by web services. Following the SDI architecture, it is interoperable with other geospatial information systems and provides an interface for accessing remote web services and geospatial contents for tephra fall volume estimation, overlaying maps of interest, and other geospatial analysis.

2 Database, web services and clients

WebTephraCalc has three main components: the WebGIS portal on the client side and web services and geospatial database on the server side, as shown in Fig. 2. It uses PostgreSQL/PostGIS (PostgreSQL 2022) database software for the storage of tephra fall data. It also provides an interface for uploading geographically referenced data like shapefiles and an option for entering tephra fall data in comma-separated values (CSV). Web services are formulated using Mapserver (MapServer 2022) and PyWPS (PyWPS 2022) software for WMS and WPS, respectively. The services are set up to accept requests to query the database and return the results to the WebTephraCalc main portal.

Fig. 2
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Main components of WebTephraCalc

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Web services clients are software used to formulate and send web services requests and accept responses dynamically. All GIS software such as ArcView and Quantum GIS can serve as web services clients. Open-source software like LeafLet (Leaflet 2022) and OpenLayers (OpenLayers 2022) are web services clients that can be easily integrated into web-based applications. WebTephraCalc uses the Leaflet WMS client to send WMS requests to display tephra fall contour maps. On the other hand, OpenLayers WPS client is used to formulate and send WPS requests to retrieve thickness data and compute isopach areas for the estimation of tephra fall volume.

2.1 Web map service (WMS)

The procedure for the formulation of WMS is described in the OneGeology cookbook (OneGeology 2022). Every WMS has its corresponding service Uniform Resource Locator (URL). WMS request is a set of instructions in the form of a Uniform Resource Identifier (URI), which consists of the service URL and other request parameters, such as the layer name (name of the data), map projection, extent, and image size. The response to the request is one or more geographically referenced images that can be displayed on a web page (OGC 2022b). Figure 3 shows the WMS GetMap request URI to display the tephra fall contour map of the Usu Volcano 2001 eruption. Zooming and panning maps involve multiple GetMap requests executed by the WMS client.

Fig. 3
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WMS GetMap requests URI to display the tephra fall contour map of the 2001 eruption of the Usu volcano

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2.2 Web processing service (WPS)

The procedure for the formulation of WPS using Python programming language is described in the PyWPS documentation (PyWPS 2022). Every WPS has its corresponding service URL. WPS requests are in the form of URI consisting of the URL and other request parameters, such as the identifier of the computation function and the name of the map layer (OGC 2022c). In this paper, WPS requests are formulated to retrieve the isopach contour line thickness or mass per unit area and compute the isopach areas. Figure 4 shows the Phyton function GetTephraFallData executing the PostgreSQL/PostGIS Structured Query Language (SQL) to get the isopach thickness and area. Figure 5 shows the WPS Execute request URI, using the identifier “GetTephraFallData”, to execute the Phyton function. The WPS returns the request result in JavaScript Object Notation (JSON) format containing the data used to quantify the ejected volcanic materials. The computation of the tephra fall volume for the selected model or method is done on the WebTephraCalc portal.

Fig. 4
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Phyton function executed by the WPS Execute request to get isopach thickness and area data for tephra fall volume estimation

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Fig. 5
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WPS Execute request URI to query the database for the needed data to compute tephra fall volume

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2.3 Accessing web services provided by other information systems

The URLs of the web services provided by an organization's information systems can be known through OGC Catalogue Services or publications on its website. WMS provides the GetCapabilities request option to get information about the service like the following URI for the Usu volcano WMS layer:

https://geohazards-info.gsj.jp/ows/tf_wms/2023/M_tephra_fall_usu_2000_3_31_table_20231208143759_I58_wms?SERVICE=WMS&VERSION=1.1.1&REQUEST=GetCapabilities&.

The request will generate a response in Extensible Markup Language (XML) format containing information which includes layer name, map projection, and map extent needed to display the map using GetMap request.

WPS provides two request options: GetCapabilities and DescribeProcess. GetCapabilities provides information about the available processes or functions. DescribeProcess provides information about a specific process and data format that will be returned. The following are examples of WebTephraCalc’s WPS GetCapabilities and DescribeProcess URIs:

https://geohazards-info.gsj.jp/webtephracalc_wps?request=GetCapabilities&service=WPS.

https://geohazards-info.gsj.jp/webtephracalc_wps?request=DescribeProcess&service=WPS&version=1.0.0&Identifier=GetTephraFallData.

Web services clients can automatically parse these requests results and provide the needed information for the GetMap and Execute requests of WMS and WPS, respectively. These make geospatial data shared through web service with published URL accessible using web applications like WebTephaCalc.

WebTephraCalc accesses the web services provided by the GSi project at https://ccop-gsi.org/main (Fig. 6). WebTephraCalc automatically accesses the web services of GSi’s tephra fall map catalog (map folder) containing tephra fall contour maps. These maps are shown on the WebTephraCalc portal with the GSi icon. Other data such as volcano geological maps, volcano crater maps, satellite images, and geological hazard maps that are publicly shared by data owners in GSi can be searched and displayed on the WebTephraCalc main portal. WebTephraCalc is also a universal WMS client and displays any map shared as WMS using the service URL.

Fig. 6
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CCOP Geoinformation Sharing Infrastructure for East and Southeast Asia (GSi) SDI with web services accessed by WebTephraCalc

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3 Tephra fall volume estimation models

Most models used to estimate the volume of tephra fall use the thickness–area relationships. Volumes of the tephra fall, V, can be derived from the following equation:

$$V = \int_{0}^{\infty } {T\left( x \right)dA} = 2\int_{0}^{\infty } {xT\left( x \right)dx}$$
(1)

where T and x are the thickness and the square root of the area A, respectively (Bonadonna and Costa 2012). The exponential, power law, and Weibull models are formulated to express thickness as a function of the isopach area, T(x). Applying Eq. 1, these models are used to estimate the volume of tephra fall, as shown in the following equations.

3.1 Exponential model

The exponential model (Pyle 1989) expresses the relationship between the deposit thickness, T, as a function of the square root isopach area, x, as follows:

$$T\left( x \right)\, = \,ce^{ - mx} ,$$
(2)

where m is the rate of decrease in thickness and c is the maximum thickness at the vent. The values of m and c are computed through least squares regression, after taking the log of the thickness versus square root area plot to linearize the equation (Daggitt et al. 2014a). The model allows the data to be divided into several segments when thinning rates significantly vary with distance from the vent, as shown by many tephra fall deposits (Pyle 1989, 1995; Fierstein and Nathenson 1992; Bonadonna and Houghton 2005; Watt et al. 2009; Daggitt et al. 2014a). Using Eq. 1, the total volume of the debris fall will be computed using the model as follows:

$$V = 2\int_{0}^{\infty } {xce^{ - mx} } dx.$$
(3)

3.2 Power law model

The power law model (Bonadonna et al. 1998; Bonadonna and Houghton 2005) shows the decrease of the tephra fall thickness, T, as a function of square root of isopach area, x, as shown by the equation:

$$T\left( x \right)\, = \,sx^{ - p}$$
(4)

where s and p are the linear scaling factor and rate of decrease in thickness, respectively. Like the exponential model, parameters s and p are found through least squares regression, after taking the log of both sides of the thickness versus square root area plot. Power law equation could not be integrated between 0 and \(\infty\) hence the proximal (× 1) and distal (× 2) limits of integration should be indicated (e.g., Bonadonna and Costa 2012). Applying Eq. 1, the volume of tephra fall will be computed using the model as follows:

$$V = 2\int_{x1}^{x2} {sx^{ - p + 1} } dx.$$
(5)

3.3 Weibull

Weibull model was the latest model used to estimate tephra fall volume. The model combines the advantages of exponential and power law, which are the integration between 0 and \(\infty\) and the accounting of the variable rate of decrease in deposit thickness, respectively (Bonadonna & Costa 2012). The model states that the relationship between thickness, T, and square root isopach area, x, is expressed as follows:

$$T\left(x\right)=\theta {\left(\frac{x}{\lambda }\right)}^{\kappa -2}{e}^{{-\left(\frac{x}{\lambda }\right)}^{\kappa },}$$
(6)

where \(\lambda\) is the decay length scale of deposit, \(\theta\) is the thickness scale and \(\kappa\) is the shape parameter. Since the variation in thinning rate is captured by the additional parameter, \(\lambda\), multiple segments computation is not necessary (Daggitt et al. 2014a). Applying Eq. 1, the tephra fall volume will be computed using the model as follows:

$$V = \int_{0}^{\infty } {T\left( x \right)dA} = 2\int_{0}^{\infty } {xT\left( x \right)dx} = \frac{{2\theta \lambda^{2} }}{\kappa }.$$
(7)

3.4 Segment integration method

The segment integration method (Takarada et. al. 2001, 2016) is a tephra fall mass estimation strategy that uses the changes in the weight per unit area of the ejected volcanic debris from the vent to the farthest distance that the materials reach. The method expresses the decay of tephra fall mass in kg/m2 as a function of area A in m2. The model is described as follows:

$$dM\left( A \right)\, = \,cA^{ - w}$$
(8)

where w is the rate of decrease of mass and c is the maximum weight per unit area at the vent. Taking the log of mass linearizes the equation and parameters w and c are estimated through least squares regression. The total tephra fall weight M is calculated as follows:

$$M = \int_{A1}^{A2} {cA^{ - w} dA} ,$$
(9)

where A1 and A2 are the first and last observed area of each segment, respectively.

3.5 Volume estimation and result outputs

The summary of volume estimation flow used by WebTephraCalc after receiving tephra fall data from WPS execution is shown in Fig. 7. WebTephraCalc computes tephra fall volume using JavaScript Math object (JavaScript 2022). The system also uses the Google Chart API Trendlines (Google Charts 2022) to display the data plots and the least squares regression lines. WPS will be executed when the “Calculate” button on the chosen data is clicked and displays the interface for volume estimation, as shown in Fig. 8. The system displays the plot between thickness against \(\sqrt{area}\) for all models, with the exception for the segment integration method, which displays the plot of mass in kg/m2 against area in m2. A function for automated segmentation is available. In using this option, segments are automatically selected using the developed algorithm that iterates through the points along the plot to determine the successive points approximating a straight line. The interface also provides manual segmentation options by directly selecting the isopachs/points along the plot line that approximates a straight line. Selecting all isopachs results to a single segment computation. The volume estimation output for each segment includes the volume in km3, weight in tons, parameters of the T(x) equation, and the mean relative squared error (MRSE) in the model's fit to the data. Total weight is computed using the set density values for all models, with the exception for the segment integration method, which directly computes total weight using kg/m2 contour map. The specific details for volume estimation using each model are explained below:

Fig. 7
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Summary of volume estimation flow followed by WebTephraCalc after receiving tephra fall data from WPS execution

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Fig. 8
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WebTephraCalc volume estimation interface showing computation result for Aso volcano eruption using the exponential model

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3.6 Exponential model

WebTephraCalc computes the least squares regression to determine the model parameters. It uses the intersection between segments as the new starting and ending values for the two adjacent segments to ensure continuity between segments in a multi-segment computation. The computation result displays the segments’ least squares regression of log thickness (m) against square root of area (km). Figure 8 shows the computation results and the two-segment plot using data from the Aso volcano eruption.

3.7 Power law model

The least squares regression is computed by the system to determine the model’s parameters. Volume estimation using the model requires the proximal and distal limits of integration. The output from this model’s computation displays the least squares line of log thickness (m) against the log of square root of area (km).

3.8 Weibull

Volume estimation using the Weibull model uses the values of λ, κ and θ. These parameters can just be determined from observations (Bonadonna and Costa 2012). Estimating their optimum values can be done by minimizing the relative squared error (RSE) function (Bonadonna and Costa 2013). RSE is computed as follows:

$$RSE\left(\lambda ,\kappa ,\theta \right)=\sum_{x \epsilon data}{\left(\frac{{T}_{observed}(x)-T(x)}{{T}_{observed}(x)}\right)}^{2}$$
(10)

where T(x) is the Weibull function, as shown in Eq. 6.

WebTephraCalc implemented the optimization method proposed by Daggitt et al. (2014a) by eliminating θ as a free parameter from the model. The optimization method expresses θ as a function of λ and κ as follows:

$$\theta \left( {\lambda ,\kappa } \right) = \lambda^{\kappa - 2} \sum\nolimits_{x \in data} {\frac{{x^{\kappa - 2} }}{{T_{observed} \left( x \right)}}e^{{ - \left( {\frac{x}{\lambda }} \right)^{\kappa } }} } \left( {\sum\nolimits_{x \in data} {\left( {\frac{{x^{\kappa - 2} }}{{T_{observed} \left( x \right)}}e^{{ - \left( {\frac{x}{\lambda }} \right)^{\kappa } }} } \right)^{2} } } \right)^{ - 1}$$
(11)

The optimization makes the model a two-parameter model only requiring the estimation of \(\lambda\) and \(\kappa\). WebTephraCalc estimates the Weibull parameters through the iterative process of minimizing the RSE function. It implements the algorithm to compute RSE using randomly selected values of λ and κ within a set range of values. This involves several computations runs on the same data, each run consisting of several iterations, constantly identifying the values with lower RSE. The error is reduced after each run, and the search range for λ and κ are adjusted to narrower ones. After several runs, the optimum values of λ and κ with the lowest RSE is determined and used in the final computation of tephra fall volume. The system sets the default number of runs and iterations per run at 5 and 2000, respectively. These values provide stable outputs after several trial runs using actual data. The optimization method also allows the display of error surface as a function of λ against κ like the one shown in Fig. 9. The figure shows the narrowed search range, shown as colored region in the plot, for the two parameters.

Fig. 9
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Iterative computation to minimize RSE function using randomly chosen values for κ and λ within a set search range

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The Weibull model inputs for the volume computation with WebTephraCalc include the number of computations runs, number of iterations per run and the initial lower and upper bounds for λ and κ. The model’s specific outputs include the estimated λ, κ, θ, RSE, MRSE and the error surface plot indicating the range of λ and κ with the lowest RSE. It also shows the final search range for the values of λ and κ.

3.9 Segment integration method

The system estimates the parameters through least squares regression on the plot of mass log in kg/m2 as a function of area in log m2. It determines the intersection between segments as segment boundary to ensure continuity between segments in a multi-segment computation. The computation result displays the segments least squares regression line of log mass (kg/m2) against log area (m2). The computation method is used to estimate debris fall mass of some volcanic eruptions in Japan.

4 Volume estimation uncertainties and WebTephraCalc limitations

Tephra volume estimation using isopach data is subjected to a series of uncertainties starting from the environmental factors such as slope and vegetation affecting tephra fall deposit thickness (e.g.Engwell et al. 2013; Blong et al. 2017; Cutler et al. 2018) to the field thickness measurement (Engwell et al. 2013; Kawabata et al. 2013; Green et al. 2016) and construction of isopachs using the field data (Engwell et al.2013; Klawonn et al. 2014). Another major source of uncertainties are the model and the extrapolation of thickness beyond the available data. Model uncertainty is inevitable, because no fitted curve perfectly describes the thinning pattern, especially when fitting is done on log-transformed square root of isopach area. The uncertainty caused by extrapolation beyond the available data is impossible to validate (Yang and Jenkins 2023). Other sources of uncertainties that can further influence the selection of model to use for volume estimation are related to distance travelled by tephra deposits from the vent. Engwell et al. (2013) observed that distance within 5 km range follow the exponential trend. However, with distance from the vent, some decay tends to fit better to either Power law (Bonadonna and Houghton 2005) or Weibull (Bonadonna and Costa 2012). Exponential model tends to be the better model to use when tephra fall data is sparse (e.g., Pyle 1995; Legros 2000; Sulpizio 2005). Power law can better reproduce the natural thinning of tephra deposits but is strongly sensitive to the proximal or distal extreme of integration. On the other hand, using Weibull for the estimation of volume using large data sets is advantageous, because the model’s best fitting depends on three parameters, and can well reproduce the gradual thinning of tephra deposits, without depending on the selection of segments and arbitrary extremes of integration (Bonadonna and Costa 2012).

Other source of uncertainty in the estimation of tephra volume is caused by the computation of isopachs areas. WPS used for computing isopachs areas use PostGIS software, where open isopachs are first closed before the computation is done. Closing open isopachs with end points that are far from each other contributes uncertainty in the volume estimation because of erroneous isopach area calculation. However, WPS for area calculation minimizes this source of error by setting a threshold for end points distance. Open isopachs with end points distance beyond the set threshold are not included in the volume calculation. WebTephraCalc cannot compute tephra fall volume when the number of closed isopachs in a map layer is less than 2. Furthermore, the system sets the maximum number of segments for volume estimation to 11. These limitations and sources of uncertainties should be considered when using WebTephraCalc and the model to choose for the computation of tephra fall volume.

5 Results and discussion

WebTephraCalc is developed at the Geological Survey of Japan (GSJ). It is created to take advantage of the ongoing global trend of saving geospatial contents on web-based databases and the formulation of web services to access and process geospatial data. WebTephraCalc accesses its own web services to obtain tephra fall data for volume computation. Following OGC standards, WebTephraCalc attains interoperability and able to access other information system’s web services and geospatial contents. WebTephraCalc accesses East Asia’s GSi information system’s web services and tephra fall map layers in its database and used for volume estimation.

During WebTephraCalc’s development, it was continuously tested using GSJ’s tephra fall data from several volcanic eruptions. Parallel computations were also done using ArcGIS and other software to check the system’s consistency and reliability. To know whether WebTephraCalc provides similar results provided by other systems, its calculation results are compared with published values. Three volcanic eruptions are chosen for this because of the availability of the eruption data and published computation results. Eruption volume estimation results from the Cerro Negro 1992 eruption (Bonadonna and Costa 2012; Daggitt et al. 2014b) and Taal 2020 (Balangue-Tarriela et al. 2022) are used to compare WebTephraCalc’s estimation for the exponential, power law and Weibull models. On the other hand, Ontake 2014 eruption results (Takarada et al. 2016), which use kg/m2 to estimate total weight, are used to compare with WebTephraCalc’s output using the segment integration method.

Table 1 shows the WebTephraCalc and published eruption results for the exponential and power law models, and the segment integration method. For the Cerro Netro 1992 eruption, WebTephraCalc generated the same value, within the first three significant figures provided by the published value, using the exponential model by Daggitt et al. (2014b). The result for the power law shows WebTephraCalc volume within 0.3% of the published value. WebTephraCalc also provides exponential and power law volume estimate within 0.2% and 1.7% of the values provided by Balangue-Tarriela et al. (2022), respectively. For the segment integration method, WebTephraCalc provides weight estimate for the Ontake 2014 eruption within 1% of the published result.

Table 1 Comparison of tephra fall volume values produced between WebTephraCalc and published results
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Comparison is also made between WebTephraCalc and the published Weibull model results for the Cerro Negro 1992 eruption (Bonadonna & Costa 2012; Daggitt et al. 2014a), as shown in Table 2. The table shows the volume, estimated λ, κ, θ values, and the MRSE provided by WebTephraCalc and the published results. WebTephraCalc Weibull volume value is the median value after five tries. For the volume estimation, WebTephraCalc shows the same results within the number of published significant figures. On the other hand, the model parameter values are within 0.7% of the published results. Furthermore, the MRSE of the fit provided by the WebTephraCalc is slightly smaller than the MRSE of the fit of the published result. This indicates the reliability of WebTephraCalc in estimating tephra fall volume using the Weibull model.

Table 2 Comparison between the Weibull results generated by WebTephraCalc and the published ones for the 1992 Cerro Negro eruption (Bonadonna & Costa 2012; Daggitt et al. 2014a)
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Table 3 shows the comparison between WebTephraCalc’s volume estimates and the results provided by Balangue-Tarriela et al. (2022) using the Weibull model for the Taal 2020 eruption. Using the λ search range to estimate Weibull parameters provided in the paper, WebTephraCalc produced a volume estimate within 1% of the published result with one run (2,000 iterations) after several tries. However, WebTephraCalc provides significantly higher results when the default setting of 5 runs, 2000 iterations per run, is used. WebTephraCalc provided a consistent volume estimate of around 0.160 km3 using the settings. This is significantly higher compared with the volume of 0.090 km3 provided by the published result. Comparing the MRSE of the fit for the two results shows that the one with consistently higher volume estimates of WebTephraCalc provides a smaller value. This indicates that a higher number of iterative computations provides a better fit for the data.

Table 3 Comparison between the Weibull results generated by WebTephraCalc and the published outputs for the 2020 Taal volcano eruption (Balangue-Tarriela et al. 2022)
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WebTephraCalc can easily be used to simulate missing data scenarios by providing options to remove selected isopachs. Several computation runs are done to determine how the system’s calculation results change when incomplete data are inputted. Figure 10 shows WebTephraCalc’s volume estimation results using the Cerro Negro 1992 eruptions data with three missing data scenarios: the last isopach is missing, the first isopach is missing, and two middle isopachs are missing. The estimated volumes are compared with the values generated using complete data.

Fig. 10
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WebTephraCalc’s volume estimation results using complete and incomplete data from the Cerro Negro 1992 eruption (Bonadonna & Costa 2012) with the exponential, Power law, and Weibull models

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The computation results using the exponential model show that the estimated volume barely changed when the last isopach is removed. The volume is 0.7% within the computed value using complete data. Engwell et al. (2013) observed that tephra fall thickness decay follows the exponential trend within the 5 km range. This makes the estimated volume using the exponential model not significantly affected when the farthest isopach is missing. On the other hand, volume value increased by around 6% using the power law model when the last isopach is missing. Bonadonna and Houghton (2005) observed that some decay fits better with power law with distance from the vent. This makes volume estimation using power law more sensitive than the exponential model when the farthest isopach is removed. The middle isopachs are very important, especially when the exponential model is used. Removing the data significantly alters the thinning pattern resulting to significant volume changes. The result shows a 15% volume change when two middle isopachs are missing using the exponential model. The power law and Weibull models are less sensitive with 8% and 5% volume change, respectively. Figure 10 shows that the volume estimated using the Weibull model when the last isopach is missing is 1% within the volume obtained using complete data. Weibull model captures the gradual thinning of tephra fall using large data sets (Bonadonna and Costa 2012). The farthest isopach have minimal effect on the overall estimation of the volume using the model. The computation results also show that the estimated volume is significantly affected when the nearest isopach from the vent is removed using all models. The exponential and Weibull models show a significant decline in volume values by 9% and 6%, respectively. For the exponential model, the c coefficient corresponding to the maximum thickness at the vent will be lower when the first isopach is missing resulting to significant decrease in volume estimate. On the other hand, the power law model shows a volume increase of 5.6%. Unlike the exponential and Weibull models, Power law requires the proximal and distal limits of integration to estimate tephra fall volume. The volume increase indicates a reduced thinning decay rate from the proximal to distal limits when the first isopach is removed. The volume estimation changes when some data are missing depend on the type of eruption. However, the results provide additional information how the system estimates tephra fall volume for a particular data characteristic. WebTephraCalc’s computation results using incomplete data generally conform with the expected changes in volume values when some data are missing using different models.

6 Conclusions

WebTephraCalc is an easy-to-use, accessible, and reliable tool for volume estimation of volcanic debris using data provided by web services. Following the SDI model, its usefulness will be further enhanced when most geological surveys store their isopach data in web-based geospatial databases and share them in the form of web services. Users only need a device (PC or mobile) with a browser and internet connection to use the system. Volcanologists can easily use WebTephraCalc to compute and compare volume estimation results using the exponential, power law, Weibull models, and the segment integration method. Computation can be done using data from the system’s own database and other distributed sources. It is shown that the calculation results using the system are in line with the ones shown in published works. WebTephrCalcs volume estimation results also conform with the expected volume value changes when some data are missing. WebTephraCalc is a functioning WebGIS system capable of displaying and overlaying a wide array of geospatial data from different sources for visual analysis. Future versions of WebTephraCalc can be used for more complex spatial analysis through the formulation of WPS for these operations.