# Dispersion Curve Interpolation Based on Kriging Method
## 1 Introduction  
Volcanic eruptions significantly impact human life and, as such, continuous monitoring of active volcanoes is particularly important. The existing method is to place fixed instruments in the study area for long-term continuous recording, such as fixed seismometers. However, real-time seismic signal imaging techniques still has many limitations in the seismology field; for example, high-precision tomography cannot be fully automated. Furthermore, such continuous monitoring cannot predict the future state of volcanoes [1]. The sampling points in the spatial-temporal domain are often sparse and, if we want to predict the state of a volcano at a moment in the past or future, we must construct a continuous temporal-spatial data surface through interpolation. There are many ways to interpolate spatial-temporal data (see, e.g., [2; 3]). Temporal-spatial interpolation techniques have been applied widely on temperature (see, e.g., [4; 5]) and atmospheric data (see, e.g., [6]). However, spatial-temporal interpolation has hardly been applied in seismology. There are two assumptions for the temporal-spatial interpolation problem. One is called the time slices approach (reduction method), and the other is the extension approach (extension method) [2; 7]. [PERSON] and [PERSON] [2] pointed out the problems of these two methods. Time aggregation has a serious negative effect on the accuracy of the reduction method. For the extension method, it is difficult to compare one temporal unit with one spatial unit. If spatial information can be expressed in time, 2D or 3D interpolation can be applied directly. In ambient noise tomography, the dispersion curve matches this condition. Ambient noise tomography is an imaging method using the information of noise in the environment [8], based on the dispersion of surface waves. Traditional seismic imaging methods utilize information from earthquakes, and the noise received by seismometers is suppressed by the filter. However, noise also contains a lot of information about the underground medium. Recently, many scientists have used tomographic methods ondifferent scales across the globe [9; 10; 11]. Due to safety issues and environmental protection, active source experiments are not advocated, particularly in areas with dense population or natural attractions. Ambient noise tomography has also been shown to work well in volcanic areas [12; 13]. Therefore, ambient noise tomography was used to generate the original shear-wave velocity (V\({}_{\text{S}}\)) image in this study. Ambient noise tomography methods can mainly be divided into those focused on the extraction of the dispersion curve, or inversion. The methods used to extract dispersion curves include frequency time analysis (FTAN) [14; 15] and the frequency-Bessel transform method [16; 17]. As regards inversion, relevant methods include the two-step method [18], direct inversion [19], and quasi-Newton method [17]. The accuracy of these inversion methods depends on the density of rays on each grid. If there are more rays passing through a grid point, the results will be more accurate. However, constrained by geographic factors, seismometers sometimes cannot be placed evenly throughout the whole study area, leading to an insufficient ray density to obtain an accurate structure. Similarly, an uneven distribution of noise sources may result in a small number of rays. Therefore, the supplementation of data is essential. In addition to investing more manpower and instruments in field work, mathematics-based interpolation methods can also address this problem, while avoiding wasted resources. Previous data interpolation studies in seismology mostly focused on the interpolation (or resampling) of seismic records (see, e.g., [20; 21; 22]). These interpolation methods aim at active source seismic methods. Therefore, it is important to develop an interpolation method for noise source imaging.