## 1 Introduction

[2] The Miyake-jima volcano is a basaltic-andesitic stratovolcano located at the volcanic front of the Izu-Mariana subduction zone and is one of the most active volcanoes in Japan (Figure 1). Eruptions in the past 600 years have occurred at intervals of 21 to 69 years [e.g., *Miyazaki*, 1984; *Tsukui and Suzuki*, 1998; *Geshi et al*., 2002]. The caldera that formed at the summit of Miyake-jima in 2000 has a diameter of 1.7 km and a depth of 450 m [e.g., *Geshi et al*., 2002; *Geshi*, 2009]. Many feeder and non-feeder dikes are exposed in the walls of the caldera [*Geshi et al*., 2010]. *Geshi et al*. [2010] found systematic differences between the shapes of the feeder and the non-feeder dikes and suggest that these differences are primarily related to the free surface affecting the upper geometry of the feeder dikes. Using a finite-element model, *Geshi et al*. [2012] also show that the thickness variations of non-feeder dikes depend on abrupt changes in stiffness among layers and relate also to the volumetric flow rate of the magma.

[3] In all these considerations, it is important to know the overpressure of the magma in the dikes. Here overpressure is defined as the total magmatic pressure in the dike minus the minimum principal compressive (maximum tensile) stress *σ*_{3} acting on the dike walls. Many have estimated the magma overpressure based on dike thickness (or aperture for a magma-filled dike fracture) and the assumption that the host rock behaves as elastic [e.g., *Delaney and Pollard*, 1981; *Pollard et al*., 1983; *Pollard and Segall*, 1987]. In such studies, the following fundamental equation [e.g., *Sneddon and Lowengrub*, 1969] is used:

where *E, v*, *a*, and *p*_{0} are, in that order, Young's modulus and Poisson's ratio of the host rock, the half-length of the dike, and the magma overpressure in the dike. Also, *x* is the coordinate of the observation point used in the measurement of the opening displacement of the dike-fracture wall *u _{y}* and is measured from the center toward the tip of the dike (Figure 2). Equation ((1)) is valid for internal magma overpressure

*p*

_{0}acting on a line crack in an infinite elastic body and also for an extension fracture in an infinite elastic body subject to an external tensile stress (

*p*is then the magnitude of the tensile stress). Here equation ((1)) gives the crack shape formed by the overpressure and/or external stress. For example, the magnitude of the regional tensile stress field has been estimated from the aspect (length/aperture) ratios of tension fractures in the rift zone of Iceland [

_{0}*Gudmundsson*, 1983]. Since equation ((1)) is very simple and suggests a linear relation between the crack shape and the driving pressure or stress, it has also been applied to estimate the overpressure associated with observed dikes in the field, although more general solutions exist [e.g.,

*Gudmundsson*, 2011;

*Gudmundsson et al*., 2012;

*Kusumoto et al*., 2013].

[4] When using equation ((1)), we need to know the total length of the dike and the elastic constants of the host rock. We can measure the elastic constants of the host rock or estimate their typical values from tables in textbooks and/or data handbooks [e.g., *Bell*, 2000; *Lama and Vutukuri*, 1978]. However, in general, it is difficult to determine the center and/or the total length of a dike since only parts of most dikes can be observed and measured in the field [e.g., *Gudmundsson*, 1984; *Gudmundsson et al*., 2012]. Length, in this sense, has two meanings: the dip dimension for dikes seen in vertical sections and the strike dimension for dikes seen in lateral sections (plan view). The intended meaning is normally clear from the context, but defined where necessary.

[5] In the present paper, the origin of the coordinate system is not at the center of the dike but rather at its tip. For this purpose, we introduce a new parameter, *q*, related to the location of the origin (Figure 2). The opening fracture displacement *u _{y}*(

*x*) at a given location

*x*is thus expressed by

*u*(

_{y}*q*) and

*q*, respectively. We propose a new method for estimating the total length (strike or dip dimension) and the maximum thickness of the dike (or dike segment) using the values of

*q*and

*u*(

_{y}*q*). We apply the new method to data on non-feeder dikes from the caldera walls of Miyake-jima in Japan and estimate the dike aspect ratios, magma overpressures, and fracture toughness.