Notable Annual Thermoelastic Strain in Vault-housed Extensometers: A Typical Case from the Kuancheng Geodynamic Observatory, North China

Notable Annual Thermoelastic Strain in Vault-housed Extensometers: A Typical Case from the Kuancheng Geodynamic Observatory, North China | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Notable Annual Thermoelastic Strain in Vault-housed Extensometers: A Typical Case from the Kuancheng Geodynamic Observatory, North China Xiaolin Yang This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-6717057/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 07 Nov, 2025 Read the published version in Pure and Applied Geophysics → Version 1 posted 9 You are reading this latest preprint version Abstract Clarifying the physical mechanisms underlying near-surface annual deformation remains a formidable challenge in crustal dynamics research. In the past few decades, quite a few extensometric observatories exhibit notable and regular annual variations in strain signals in mainland China. However, the geodynamic investigations on these prevalent but intriguing strain signals remain rare till now. Since outdoor air temperature is a key factor contributing to annual deformation of the Earth’s crust, this work therefore quantitatively elucidates the mechanism via which annual atmospheric temperature variation influences the NS and EW components of strain by analyzing data observed at the Kuancheng Geodynamic Observatory, North China, with an elastic half-space model covered by elastically thin unconsolidated layer. According to the modeled results, annual variation of atmospheric temperature with an amplitude of 16.84 ℃ is sufficient to produce a thermoelastic strain of 10 − 7 magnitude inside the mountain at a depth of 30 m. Besides, both the modeled amplitudes and phases align closely with the observed strain signals, suggesting that the annual variations of the strain observed at the Kuancheng Geodynamic Observatory mainly originate from annual variations of atmospheric temperature. The method of geodynamic diagnostics used here and obtained findings not only contribute to the quantitative interpretation of the mechanism underlying annual variations in vault-housed extensometers in mainland China but also advance our understanding of the temperature-induced deformation processes within the near-surface crustal layer. Annual variation thermoelastic strain thin unconsolidated layer atmospheric temperature vault-housed extensometer Kuancheng Geodynamic Observatory North China Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Figure 10 1. Introduction Extensometer has some unique advantages, including good continuity, high resolution (≤ 10 − 10 ), and ultrawide bandwidth (DC to tens of Hertz), and its measurements can be used to study Earth tides (Benioff, 1959), earthquake source physics (Asai et al., 2005 ), tectonic movements (Zadro & Braitenberg, 1999 ; Mentes, 2008 ; Brimich et al., 2016 ), volcanic activities (Yamazaki et al., 2013 ), Earth’s free oscillation (Park et al., 2008 ) and tsunami loading (Yanagisawa & Wakasugi, 1984 ). Considering its great application potential in geodynamics, many countries such as the United States (Bilham, 1973 ; Agnew, 2007 ), Japan (Shichi & Okada, 1979 ; Harada et al., 2003 ; Yamazaki, 2013 ), China (Wu et al., 2022 ), Germany (Kroner et al., 2005 ; Jahr et al., 2006 ; Gebauer et al., 2009 , 2010 ), Italy (Amoruso et al., 2012 ) and Hungary (Mentes, 2008 ) have widely deployed vault-houses extensometers in tectonically active regions since 1950s. Among them, mainland China initially established the extensometric observatories under the national project for earthquake prediction in the early 1970s, and to date, it has incorporated more than 110 observatories in the vault-housed extensometer network (Wu et al., 2022 ). Over the past several decades, these observatories have captured a wealth of crustal deformation phenomena, which have provided vital opportunities and "big data" security for researches on earthquake precursors, geodynamics and nontectonic deformations. Despite the complex periodic components of strain time series observed by vault-housed extensometers, the annual variation is a prevalent and more prominent rhythmic signal at the annual scale (Ben-Zion & Leary, 1986 ; Watanabe, 1991 ; Mentes, 2008 ; Teraishi, 2009). According to many existing geodynamic investigations (Ben-Zion & Leary, 1986 ; Hvoždara & Brimich, 1988 ; Watanabe, 1991 ), the physical origin of such signals can be attributed primarily to thermoelastic strain generated by annual fluctuations in surface temperature owing to the thin mountain cover at the extensometric measurement sites (usually approximately tens of meters thick). Additionally, undesired factors such as lithology (Gebauer et al., 2009 ), mountain slope (Hvoždara & Brimich, 1988 ), topographic wavelength (Ben-Zion & Leary, 1986 ; Kroner et al., 2005 ; Gebauer et al., 2010 ), cavity (Harrison, 1976 ; Takemoto, 1981 ), weathered layer thickness, and regional thermoelastic structures (Ben-Zion & Leary, 1986 ) may complicatedly modify the thermoelastic strain response of local regions. Therefore, accurate evaluation of the contribution of annual variation in surface temperature to annual variation in vault-housed extensometer is very challenging. At present, many geophysicists have adopted digital signal processing, theoretical and numerical modeling to analyze the contribution of atmospheric temperature-induced thermoelastic strain qualitatively or quantitatively. For instance, Terashi et al. (2009) quantified the correlation between strain and annual temperature variation for nine observatories in Japan using the seasonal adjustment method and showed that the correlation coefficient between the two can reach up to 0.93. Furthermore, some studies have theoretically quantified the thermoelastic strain resulting from annual temperature variations using 2D theoretical or finite element-based coupled thermal-mechanical models incorporating local topography (Hvoždara & Brimich, 1988 ). However, the above simplified models does not take into account the complex thermoelastic structures of actual strata, so that there would be a certain phase difference between the predicted and observed annual variation curves. Consequently, Ben-Zion et al. (1986) constructed a half-space thermoelastic strain model covered by a thin unconsolidated layer (BZL model hereafter), and the thermoelastic strain derived by forward modelling could reasonably explain the annual variation amplitudes and phases of annual variations observed by vault-housed extensometers in southern California (USA), highlighting the effectiveness and broad applicability of the two-layered composite model. Although quite a few extensometric observatories exhibit notable and regular annual variations in strain signals in mainland China, to date, there is a lack of targeted research elucidating the real dynamic mechanism (Yang et al., 2020 ). In view of this drawback or gap, this work takes the Kuancheng Geodynamic Observatory (KGO) as a typical case to decoding the overlooked mechanism. KGO is located in the eastern Yanshan Orogenic Belt, North China. After the installation of extensometers in 2001, notable and regular annual variations in strain have been identified in the North-South (NS) and East-West (EW) components recorded (Wang et al., 2020 ). Meanwhile, the outdoor air temperature at KGO also exhibits a similar rhythm of annual variation. Consequently, whether there is a causal relationship between the two should be explored further. Moreover, in actual earthquake prediction operations, some Chinese forecasters focusing on earthquake precursors tend to regard the amplitudes and phases of such annually varying signals as an important predictors. In view of this, the author investigated the physical correlation between annual variation in vault-housed extensometers and surface temperature at KGO according to the local topographic, lithologic features using the composite model proposed by Ben-Zion and Leary ( 1986 ). The findings could not only improve our understanding of the physical origin of the annual variation process of strain at KGO but also provide useful reference for related researches as well as rational application of predictors of annual variations observed by vault-housed extensometers in mainland China. 2. Observatory and Instrument KGO (40.59° N, 118.47° E) is located beside the Chagou Road in Kuancheng Town, Kuancheng Manzu Autonomous County, Chengde City, North China, at an elevation of approximately 338 m (Fig. 1 a). This vault-housed extensometer observatory was put into operation in 1975, mainly under earthquake precursor aspects. The folds of the geological unit where the KGO is located are relatively developed, so that the surrounding mountains are topographically complex and undulating (Fig. 1 b), and the basement of the observatory consists primarily of siliceous limestone. The in-situ area falls under a warm-temperate semi-arid and semi-humid continental monsoon climate, with mean annual sunshine hours of approximately 2825 h, mean annual evaporation of approximately 1589 mm, and maximum permafrost thickness of 2 m. This area is characterized by four distinct seasons, and abundant sunshine. The annual mean rainfall in Kuangcheng County is approximately 663 mm, with approximately 80% of rainfall concentrated in June to August (Liu 2020 ; Wang 2023 ). The mountain cover at KGO is approximately 20 to 40 m thick, and the soil horizon of the hillside is approximately 15 to 30 cm thick (Fig. 1 b) (Tan et al., 2021 ). The observatory is an artificial gallery driven into limestone. Its ground plane is shown in Fig. 1 c. The length of the main gallery is approximately 75 m. The yearly mean value of the air temperature is 12.3 ℃ in the gallery, and the yearly and daily temperature variations are less than 0.4 ºC and 0.1 ºC, respectively. The relative humidity in the gallery is approximately 72% and it is nearly constant. An SS-Y extensometer (Invar-rod type) was deployed in the gallery, with a resolution better than 5×10 − 10 and a theoretical drift rate below 10 − 6 /y (Wu & Nie, 2018 ). The thermal expansion coefficient of Invar-rod baseline is approximately 2×10 − 7 /°C, and the sampling interval is 1 min. Particularly, the baseline lengths of the NS and EW components are 28.50 m and 23.83 m, respectively, and the azimuths of the two are 0° and 90°, respectively (Fig. 1 c). Additionally, a WYY-1 outdoor thermometer was also operated (manufactured by National Institute of Natural Hazards, Ministry of Emergency Management of China), with the same sampling rate as the extensometer. 3. Data and Pre-processing Since the extensometers were re-installed at KGO in December 2015 due to the aging of previous strain sensors (Wang et al., 2020 ), this work mainly used the daily mean observations of strain and outdoor air temperature from 2016 to 2021. Figure 2 clearly shows that the raw strain contains at least three types of signals: type 1 (a variation signal with a nearly linear upward trend), type 2 (an annual variation signal, whose troughs/peaks occur in spring/fall), and type 3 (a small, high-frequency oscillatory signal in summer and fall each year). As far as the generation mechanism is concerned, the type 1 signal is attributed mainly to the linear drift of the extensometer, while the type 3 signal is attributed to rainfall interference (Wang et al., 2020 ; Zhang et al., 2025 ). To better analyze the correlation between strain and annual variation of outdoor air temperature, this work eliminates the linear trend and mean value of each observation curve, and the processed results are shown in Fig. 3 . Apparently, the annual variation of strain has an annual rhythm, and the relative change in outdoor temperature is also obviously dominated by annual fluctuations. So, is annual variation in outdoor air temperature a physical source of annual variation in strain? This hypothesis will be systemically tested in the following sections. 4. Characterization of Annual Variation in strain and Outdoor Air Temperature To accurately determine annual variation cycle and amplitude for outdoor air temperature and those in the NS and EW strain components, this work calculates the time series curves in Fig. 3 using the fast Fourier transform (FFT) method. The results show (Fig. 4 ) that all three reach the main peak value at the frequency point of 0.002739 cpd (365 d, cpd refers to cycles per day); notably, the amplitudes of the NS and EW components are 502.68 nstrain and 658.22 nstrain, respectively, and the outdoor air temperature was 16.84℃. 5. Model and Results 5.1 BZL Model and Thermal Boundary Conditions Generally speaking, the real thermoelastic strain field may be complicatedly modified by the lateral media heterogeneities, the topographic, and the cavity effects according to a number of previous studies (Ben-Zion & Leary, 1986 ; Hvoždara & Brimich, 1988 ). However, the BZL model can reasonably and effectively resolve this tough problem (Ben-Zion & Leary, 1986 ). According to this excellent model, the complex effect of topography and lateral media heterogeneities is to create the local spatial changes in the surface temperature field and in the response of the rock formation, thus a stationary thermal wave can be introduced as the thermoelastic strain source, whose wavelength is related to topography and lateral media heterogeneities in a given area (Ben-Zion & Leary, 1986 ). As for the cavity effect, many studies demonstrate that the along-vault strain component and relative long extensometer may be less affected (Harrison, 1976 ; Takemoto, 1981 ; Gebauer et al., 2009 ; Eper-Pápai et al., 2014 ). Given the two facts that the lengths of extensometers are relatively long (greater than 23 m), and the directions of extensometers parallelly align with the long axis of the vaults at KGO; therefore, the author will ignore the cavity effect in the BZL model preferred in this work. The basement of KGO is composed mainly of thin weathered or unconsolidated layer and thick bedrock, and the thermoelastic strain response of the two-layer geologic structure to surface temperature can be well resolved by the BZL model (Ben-Zion & Leary, 1986 ). In this composite model (Fig. 5 ), the thin weathered layer with a thickness of y b can cause surface temperature field attenuation and thermal conduction delay. Assuming the surface temperature field (at a depth of y = 0 m) changes locally as a standing wave (Ben-Zion & Leary, 1986 ): where, x , y , and t represent the horizontal position, formation depth, and time, respectively; k is the horizontal wavenumber, T 0 is the amplitude corresponding to the annual variation surface temperature field with wavenumber k , and ω and φ are angular frequency and initial phase, respectively. Here, k = 2π/ λ , , λ is the wavelength of the surface temperature field, and κ is the thermal diffusivity of the thin unconsolidated layer ( y b ) and thick bedrock. Particularly, it should be pointed out here that since there is no air (ground) temperature station array within a radius of tens of kilometers from KGO, this work is not able to perform frequency-wave number analysis. Additionally, considering that annual surface temperature variation in Kuancheng for many years has been similar to that of the air temperature (Liao et al., 2019 ), we approximate the amplitude of annual variation of outdoor air temperature of a single station as T 0 (Hainzl et al., 2013 ). Based on the actual angular frequency, amplitude, and initial phase of the annual temperature variation at KGO, ω , T 0 , and φ are taken as 2.0×10 − 7 rad/s, 16.84 ℃, and -2.94 rad, respectively. So, how does the temperature at the bottom boundary of the y b layer change? This requires prioritizing resolving the thicknesses of the y b layer in the azimuth of the NS and EW components of the strain, which are calculated as follows: In Eq. ( 2 ), Δ t represents the phase delay between the annual components of the calculated half-space thermoelastic strain without y b layer and observed strain; κ and τ indicate the thermal diffusivity of the y b layer, and the cycle of the annual variation, respectively. Correspondingly, the horizontal thermoelastic strain ε xx at different depths in the elastic half-space caused by the annual variation of surface temperature can be expressed as follows: where ν and β are the Poisson's ratio and thermal expansion coefficient of the bedrock, respectively. The thermoelastic strain solution includes mainly two parts: one resulting from an equivalent thermal surface traction, which decays slowly with an increase in depth in the form of ye − ky ; the other from equivalent thermal body forces, which decreases with an increase in depth in the form of e − γy . Here, a positive value of ε xx denotes tensive strain, whereas a negative value represents compressive strain. Based on this, I will preliminarily solve for the amplitudes and phases of the theoretical thermoelastic strain at KGO, aiming to determine the phase differences △ t with the observed values of NS and EW strain components, respectively. In the trial calculations, the thermodynamic parameters of limestone taken in this paper are as follows: ν = 0.25, β = 5×10 − 6 /°C, κ = 10.70×10 − 2 m 2 /d (Lei et al., 2018 ), while the surface temperature field parameters are T 0 = 16.84 ℃, λ = 3 km, φ= -2.94 rad, and ω = 2.0×10 − 7 rad/s; because the depth of the overlay of the gallery is not uniform, this work sets y = 30 m based on a trade-off strategy. By substituting the above variable values into Eq. ( 3 ), the annual variation process of theoretical thermoelastic strain can be obtained, with detailed results shown in Fig. 6 . The fluctuation pattern of the theoretical values is similar to that of the measured values. An interesting phenomenon, however, is that the phases of the latter are all lagging behind those of the former, with corresponding ∆t being 61 d and 46 d, respectively. To address the issue of the thermal diffusivity κ value for the y b layer, this work refers to experimental values for the thermal diffusivity of soil and set κ to 2.16×10 − 2 m 2 /d and τ to 365 d (Ben-Zion & Leary, 1986 ; Shi et al., 2015 ). Using Eq. ( 2 ), this work calculates the y b values in the azimuth of the NS and EW components of strain, resulting in thicknesses of 1.66 m and 1.25 m, respectively. Notably, although the thickness of the soil horizon at KGO ranges from 15 to 30 cm, the distribution of the thickness of this type of media layer is actually not uniform in the surrounding area (see Fig. 1 b); therefore, the theoretical y b value can be regarded as in the range of the influence of the standing wave of the surface temperature field. Subsequently, by substituting the values into Eq. ( 1 ), this work can obtain the temperature variations at the bottom boundary of the y b layer in the azimuth of the NS and EW strain components. The corresponding amplitudes of annual variations are 5.91℃ and 7.65℃, respectively, with initial phases of -1.89 rad and །2.12 rad, respectively (Fig. 7 ). After determining the annual variation process of the temperature at the bottom boundary of the y b layer, Eq. ( 3 ) can be used to resolve the theoretical thermoelastic strain amplitudes and phases of the NS and SW components at KGO. In the next subsection, the author will systematically explore this issue. 5.2 BZL Model Sensitivity Test to Thermodynamic Parameters and Optimal Thermal Strain Solution For the actual calculation, this work sets ω = 2.0×10 − 7 rad/s and the T 0 values in the azimuth of NS and EW components of strain to 5.91 ℃ and 7.65 ℃, respectively. After deducting the y b layer thickness, the y values in the azimuth of the NS and SW components are 28.34 m and 28.75 m, respectively. Additionally, based on the quantitative form of Eq. ( 3 ), the wavelength of the temperature field and the thermodynamic parameter of limestone ( κ and ν ) have nonlinear impacts on theoretical thermoelastic strain. However, no experimental studies have yet measured the specific values of κ, ν , and β for limestone at KGO and its surrounding regions. Furthermore, λ is influenced by regional topography and lateral media heterogeneities (Ben-Zion & Leary, 1986 ). Thus, this work empirically sets β = 7×10 − 6 /℃ and optimize the values of the other parameters within reasonable bounds using the trade-off algorithm. To visualize the combined effect of ν and κ on the amplitude of the annual variation of theoretical thermoelastic strain, I initially set λ = 3 km, and then test the combined effects of λ and κ on the theoretical values at ν = 0.30. Generally, the annual variation amplitude of theoretical thermoelastic strain tends to increase with increase in ν and κ , whereas it decreases as λ increases. Additionally, the amplitude values in the azimuth of each strain component range between 70.64 and 1604.51 nstrain (Figs. 8 , 9 ). Based on the model characteristics for parameter sensitivity tests and the conditions of KGO, this work finally determines the optimal ν and κ values of the limestone to be 0.30 and 7.74×10 − 2 m 2 /d, respectively, and the wavelengths λ of the temperature field in the azimuth of the NS and EW strain components to be 2.52 and 2.50 km, respectively. The amplitude and the initial phase φ of the annual variation of the temperature at the bottom boundary of the y b layer corresponding to each component can be substituted into Eq. ( 3 ) one by one to obtain the corresponding theoretical thermal strain response curves, and the final results are illustrated in Fig. 10 . From a macroscopic viewpoint, the amplitudes and phases of annual variations of the modeled thermoelastic strains of the NS and EW components are highly compatible with the observed values, with correlation coefficients of 0.9795 and 0.9919, respectively. This demonstrates the validity of the model and thermodynamic parameters, while rationalizing that the annual variation in surface air temperature is the main physical source of the annual variation in strain at KGO. Another phenomenon that needs to be elucidated here is that the measured amplitude of annual variation of the EW component is approximately 1.31 times that of the NS component; and the y b thickness results obtained in this work indicate that the main reason for this difference is that the equivalent thickness of the weathered layer is thinner in the azimuth of EW component than in the azimuth of NS component. When looking more closely at the local or detailed characteristics of the modeled and observed values, the two do not match exactly. For instance, in 2020, the theoretical thermal strain peaks and troughs of the NS component deviate significantly from the observed values; the deviation may be attributed to the amplitude of the annual temperature variation used in this paper being a six-year mean value, whereas, in reality, the amplitude of temperature fluctuations in each year is not a single value. In summary, the annual fluctuations in strain observed at KGO mainly result from the annual variation in outdoor air temperature. 6. Conclusions and Perspectives To quantitatively decode the physical correlation between the annual variation of outdoor air temperature and the annual variation of strain recorded at KGO, this work performs a dynamics diagnosis of the problem using a half-space model incorporating the thin weathered layer. The modeled results “roughly” but effectively show that the annual variation in outdoor air temperature of 16.84 ℃ is sufficient to generate a thermoelastic strain of 658.22 nstrain inside a mountain at a depth of 30 m, and the modeled values of amplitudes and phases are highly consistent with the observed values, which strongly suggests that the annual variation in outdoor air temperature is a physical cause of the annual variation in strain at KGO. Although the thermoelastic strain response mechanism of KGO is analyzed comprehensively in this tentative work, the surface temperature field, topography, weathered layer thickness, and thermodynamic parameters of the bedrock and soil horizon at the KGO and its neighboring regions would also affect the accuracy of modeling results to a certain extent. Therefore, it is necessary to conduct a detailed field survey or array observation of the thermodynamic structure, surface temperature field, weathered layer thickness, and outdoor air temperature field of the region where the vault-housed extensometers are located. On this basis, a more detailed and realistic 3D coupled thermal–mechanical model needs to be constructed to quantitatively reveal the thermoelastic strain caused by the annual variation in surface air temperature in depth. Finally, notably, this work has only diagnosed KGO as an example, whereas there are many other extensometric observatories in mainland China that can capture the annual variation signals (Xing et al., 2014 ). Therefore, is the annual variation processes observed at these observatories also derived from the annual variation in outdoor air temperature? This is an interesting question that deserves systematic investigation, and the author will continually analyze the associated observatories in the near future. Despite the main purpose of extensometric measurements in mainland China is to capture earthquake precursor signals, in decades of observation practice, a wide variety of non-precursor “noise” has become the “principal” components of strain time series. Therefore, the best way to scientifically identify and extract the subtle precursor signals is to enhance geodynamic investigation of annual variations and other “noise” in strain data. Declarations Acknowledgments The author is very grateful to Professor Yehuda Ben-Zion from University of Southern California for his helpful guidance about the temperature-induced crustal deformation. I also thank China Earthquake Networks Center (CENC) for very generously providing me with the strain and the outdoor temperature data at very short notice. 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A diagnostic study of annual strain variations in vault-housed extensometers at the Geodynamic Observatory Qianling, Shaanxi Province. Earthquake , 40 (2), 177-187. (in Chinese). Zadro, M., & Braitenberg, C. (1999). Measurements and interpretations of tilt-strain gauges in seismically active areas. Earth-Science Reviews , 47 (3-4), 151-187. Zhang, L., Sun, H., & An, L. (2025). Investigation of the Rainfall Effect on Strain Observed by Extensometers at the Jixian Seismostation in China. Pure and Applied Geophysics , https://doi.org/10.1007/s00024-025-03703-4. Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 07 Nov, 2025 Read the published version in Pure and Applied Geophysics → Version 1 posted Editorial decision: Revision requested 03 Sep, 2025 Reviews received at journal 03 Sep, 2025 Reviewers agreed at journal 31 Aug, 2025 Reviews received at journal 28 Aug, 2025 Reviewers agreed at journal 18 Aug, 2025 Reviewers invited by journal 08 Jul, 2025 Editor assigned by journal 31 May, 2025 Submission checks completed at journal 28 May, 2025 First submitted to journal 21 May, 2025 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-6717057","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":482190188,"identity":"7d6b3770-cd25-4c30-a485-381f4dfed903","order_by":0,"name":"Xiaolin Yang","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABAUlEQVRIie3QMUsDMRTA8Rce5JZS11csnh8h5eDoIvpRUgS7XJ0zFKkIL4vg7OJncHJOKdQl0rXgci7i0KFjBSlGOsrFjg75DxnC+xFeAFKp/xsBZGIC2oyPZGbdngQDqf28aLe83vMhBBBvjIMHOlPRQfX8MvsQ3M8PELnWUg6ZQMPGPDUTf3nRB0+9+xthlW51R3x47cStf20mriqLL0PicSaYNMkRd51GwRGyWJUKFJ3uiMKh/DmjZFkVNRga7IhG/SfpLFclhF3Owy6BuHmPwydPY7u0F1WxFnx1cpfZ987ndpzn1k7rjWkmxw4k/bp1jfOhfAK4jg2kUqlUCr4B3oZVbhFAuIoAAAAASUVORK5CYII=","orcid":"","institution":"Jinggangshan University","correspondingAuthor":true,"prefix":"","firstName":"Xiaolin","middleName":"","lastName":"Yang","suffix":""}],"badges":[],"createdAt":"2025-05-21 13:38:14","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-6717057/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-6717057/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s00024-025-03852-6","type":"published","date":"2025-11-07T15:58:15+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":86423229,"identity":"b45b0afe-3790-4bf0-8e3a-00889942c2b4","added_by":"auto","created_at":"2025-07-10 13:05:09","extension":"jpeg","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":13505406,"visible":true,"origin":"","legend":"\u003cp\u003e(a) Map showing the location of the KGO. (b) The topography of surrounding area of the observatory site. (c) Plan view of the vault. Red lines represent extensometers\u003c/p\u003e","description":"","filename":"floatimage1.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-6717057/v1/c40ab354806ee1ddeb29f549.jpeg"},{"id":86422259,"identity":"e3d4f29d-37d6-488b-948b-79da8f11e6d0","added_by":"auto","created_at":"2025-07-10 12:57:09","extension":"jpeg","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":1377410,"visible":true,"origin":"","legend":"\u003cp\u003eRaw daily mean variations in (a) strains and (b) outdoor air temperature.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eThe nstrain means 10\u003csup\u003e-9\u003c/sup\u003e. Extension is positive, and compression is negative\u003c/p\u003e\n\u003cp\u003efor strain time series\u003c/p\u003e","description":"","filename":"floatimage2.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-6717057/v1/47669e1892d910252b2e8da9.jpeg"},{"id":86421098,"identity":"039b54cc-a911-4d4e-88d7-ab59711ea669","added_by":"auto","created_at":"2025-07-10 12:49:09","extension":"jpeg","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":1599073,"visible":true,"origin":"","legend":"\u003cp\u003eDaily mean variations in (a) strains and (b) outdoor air temperature with linear trends and the mean values removed\u003c/p\u003e","description":"","filename":"floatimage3.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-6717057/v1/2d18df381e44eb36457f7e4a.jpeg"},{"id":86422262,"identity":"0c1aea4d-81aa-4bac-aded-ffdb1c2a79eb","added_by":"auto","created_at":"2025-07-10 12:57:09","extension":"jpeg","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":459305,"visible":true,"origin":"","legend":"\u003cp\u003eAmplitude spectra of (a) tunnel strains and (b) outdoor air temperature\u003c/p\u003e","description":"","filename":"floatimage4.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-6717057/v1/2a672ad1599e42236b3cecfc.jpeg"},{"id":86422261,"identity":"83940591-b808-4cee-9952-8f10e2e948a6","added_by":"auto","created_at":"2025-07-10 12:57:09","extension":"jpeg","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":2916962,"visible":true,"origin":"","legend":"\u003cp\u003eA composite model for thermoelastic strain in a half-space covered by a thin unconsolidated layer (soil or gravel). Modified from Ben-Zion and Leary (1986). Not to scale\u003c/p\u003e","description":"","filename":"floatimage5.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-6717057/v1/c8e44c766a2f8798af2fd6b9.jpeg"},{"id":86421100,"identity":"de96cad0-8d67-49f3-a2d0-db7b612d4539","added_by":"auto","created_at":"2025-07-10 12:49:09","extension":"jpeg","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":959692,"visible":true,"origin":"","legend":"\u003cp\u003eModeled annual thermoelastic strain (gray line) assuming a homogeneous half-space without the \u003cem\u003ey\u003c/em\u003e\u003csub\u003eb\u003c/sub\u003e layer and the observed the NS (red line) and EW (blue line) components of the strain\u003c/p\u003e","description":"","filename":"floatimage6.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-6717057/v1/ec816dee15652bfbc2e3ea88.jpeg"},{"id":86421104,"identity":"3d916dcd-9657-44e9-8978-5117b04b7edb","added_by":"auto","created_at":"2025-07-10 12:49:09","extension":"jpeg","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":809257,"visible":true,"origin":"","legend":"\u003cp\u003eThe corresponding temperature signals migrated to \u003cem\u003ey\u003c/em\u003e\u003csub\u003eb\u003c/sub\u003e for the NS (red line) and EW (blue line) azimuth, respectively\u003c/p\u003e","description":"","filename":"floatimage7.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-6717057/v1/b5db42ace94c28374d7f16c8.jpeg"},{"id":86421099,"identity":"5e30d7de-0524-4c9f-a321-1956c4798d51","added_by":"auto","created_at":"2025-07-10 12:49:09","extension":"jpeg","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":2033706,"visible":true,"origin":"","legend":"\u003cp\u003eAnnual amplitudes of thermoelastic strains calculated with the (a) NS and (b) EW components\u003csub\u003e \u003c/sub\u003eto illustrate trade-offs between limestone parameters of \u003cem\u003eν\u003c/em\u003e and \u003cem\u003eκ\u003c/em\u003e. The wavelength of temperature field \u003cem\u003eλ\u003c/em\u003e is set to 3 km\u003c/p\u003e","description":"","filename":"floatimage8.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-6717057/v1/394fb7609f536d69222862d3.jpeg"},{"id":86422260,"identity":"237b1bbf-e414-4697-9373-99e3185159da","added_by":"auto","created_at":"2025-07-10 12:57:09","extension":"jpeg","order_by":9,"title":"Figure 9","display":"","copyAsset":false,"role":"figure","size":2357678,"visible":true,"origin":"","legend":"\u003cp\u003eAnnual amplitudes of thermoelastic strains calculated with the (a) NS and (b) EW components\u003csub\u003e \u003c/sub\u003eto illustrate trade-offs between rock formation parameters of \u003cem\u003eλ\u003c/em\u003e and \u003cem\u003eκ\u003c/em\u003e. The Poisson’s ratio \u003cem\u003eν\u003c/em\u003e is set to 0.30\u003c/p\u003e","description":"","filename":"floatimage9.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-6717057/v1/da7c38160fbbfcf6e8b669d6.jpeg"},{"id":86421111,"identity":"d1ac2ccb-5f99-434d-90c2-beadca2e0f73","added_by":"auto","created_at":"2025-07-10 12:49:09","extension":"jpeg","order_by":10,"title":"Figure 10","display":"","copyAsset":false,"role":"figure","size":1572599,"visible":true,"origin":"","legend":"\u003cp\u003eComparisons between modeled annual thermoelastic strains in the BZL model (black lines) and observed strain changes for (a) NS and (b) EW components\u003c/p\u003e","description":"","filename":"floatimage10.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-6717057/v1/81fe2f205b7274e2f342a16e.jpeg"},{"id":95564237,"identity":"bb7bd375-2222-4f9f-a9ef-a247bb4593a6","added_by":"auto","created_at":"2025-11-10 16:09:15","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":28220332,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-6717057/v1/f7882da3-f895-4661-8e37-1db72d752e21.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Notable Annual Thermoelastic Strain in Vault-housed Extensometers: A Typical Case from the Kuancheng Geodynamic Observatory, North China","fulltext":[{"header":"1. Introduction","content":"\u003cp\u003eExtensometer has some unique advantages, including good continuity, high resolution (\u0026le;\u0026thinsp;10\u003csup\u003e\u0026minus;\u0026thinsp;10\u003c/sup\u003e), and ultrawide bandwidth (DC to tens of Hertz), and its measurements can be used to study Earth tides (Benioff, 1959), earthquake source physics (Asai et al., \u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e2005\u003c/span\u003e), tectonic movements (Zadro \u0026amp; Braitenberg, \u003cspan citationid=\"CR37\" class=\"CitationRef\"\u003e1999\u003c/span\u003e; Mentes, \u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e2008\u003c/span\u003e; Brimich et al., \u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e2016\u003c/span\u003e), volcanic activities (Yamazaki et al., \u003cspan citationid=\"CR34\" class=\"CitationRef\"\u003e2013\u003c/span\u003e), Earth\u0026rsquo;s free oscillation (Park et al., \u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e2008\u003c/span\u003e) and tsunami loading (Yanagisawa \u0026amp; Wakasugi, \u003cspan citationid=\"CR35\" class=\"CitationRef\"\u003e1984\u003c/span\u003e). Considering its great application potential in geodynamics, many countries such as the United States (Bilham, \u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e1973\u003c/span\u003e; Agnew, \u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e2007\u003c/span\u003e), Japan (Shichi \u0026amp; Okada, \u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e1979\u003c/span\u003e; Harada et al., \u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e2003\u003c/span\u003e; Yamazaki, \u003cspan citationid=\"CR33\" class=\"CitationRef\"\u003e2013\u003c/span\u003e), China (Wu et al., \u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e2022\u003c/span\u003e), Germany (Kroner et al., \u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e2005\u003c/span\u003e; Jahr et al., \u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e2006\u003c/span\u003e; Gebauer et al., \u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e2009\u003c/span\u003e, \u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e2010\u003c/span\u003e), Italy (Amoruso et al., \u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2012\u003c/span\u003e) and Hungary (Mentes, \u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e2008\u003c/span\u003e) have widely deployed vault-houses extensometers in tectonically active regions since 1950s. Among them, mainland China initially established the extensometric observatories under the national project for earthquake prediction in the early 1970s, and to date, it has incorporated more than 110 observatories in the vault-housed extensometer network (Wu et al., \u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e2022\u003c/span\u003e). Over the past several decades, these observatories have captured a wealth of crustal deformation phenomena, which have provided vital opportunities and \"big data\" security for researches on earthquake precursors, geodynamics and nontectonic deformations.\u003c/p\u003e\u003cp\u003eDespite the complex periodic components of strain time series observed by vault-housed extensometers, the annual variation is a prevalent and more prominent rhythmic signal at the annual scale (Ben-Zion \u0026amp; Leary, \u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e1986\u003c/span\u003e; Watanabe, \u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e1991\u003c/span\u003e; Mentes, \u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e2008\u003c/span\u003e; Teraishi, 2009). According to many existing geodynamic investigations (Ben-Zion \u0026amp; Leary, \u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e1986\u003c/span\u003e; Hvoždara \u0026amp; Brimich, \u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e1988\u003c/span\u003e; Watanabe, \u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e1991\u003c/span\u003e), the physical origin of such signals can be attributed primarily to thermoelastic strain generated by annual fluctuations in surface temperature owing to the thin mountain cover at the extensometric measurement sites (usually approximately tens of meters thick). Additionally, undesired factors such as lithology (Gebauer et al., \u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e2009\u003c/span\u003e), mountain slope (Hvoždara \u0026amp; Brimich, \u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e1988\u003c/span\u003e), topographic wavelength (Ben-Zion \u0026amp; Leary, \u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e1986\u003c/span\u003e; Kroner et al., \u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e2005\u003c/span\u003e; Gebauer et al., \u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e2010\u003c/span\u003e), cavity (Harrison, \u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e1976\u003c/span\u003e; Takemoto, \u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e1981\u003c/span\u003e), weathered layer thickness, and regional thermoelastic structures (Ben-Zion \u0026amp; Leary, \u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e1986\u003c/span\u003e) may complicatedly modify the thermoelastic strain response of local regions. Therefore, accurate evaluation of the contribution of annual variation in surface temperature to annual variation in vault-housed extensometer is very challenging.\u003c/p\u003e\u003cp\u003eAt present, many geophysicists have adopted digital signal processing, theoretical and numerical modeling to analyze the contribution of atmospheric temperature-induced thermoelastic strain qualitatively or quantitatively. For instance, Terashi et al. (2009) quantified the correlation between strain and annual temperature variation for nine observatories in Japan using the seasonal adjustment method and showed that the correlation coefficient between the two can reach up to 0.93. Furthermore, some studies have theoretically quantified the thermoelastic strain resulting from annual temperature variations using 2D theoretical or finite element-based coupled thermal-mechanical models incorporating local topography (Hvoždara \u0026amp; Brimich, \u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e1988\u003c/span\u003e). However, the above simplified models does not take into account the complex thermoelastic structures of actual strata, so that there would be a certain phase difference between the predicted and observed annual variation curves. Consequently, Ben-Zion et al. (1986) constructed a half-space thermoelastic strain model covered by a thin unconsolidated layer (BZL model hereafter), and the thermoelastic strain derived by forward modelling could reasonably explain the annual variation amplitudes and phases of annual variations observed by vault-housed extensometers in southern California (USA), highlighting the effectiveness and broad applicability of the two-layered composite model.\u003c/p\u003e\u003cp\u003eAlthough quite a few extensometric observatories exhibit notable and regular annual variations in strain signals in mainland China, to date, there is a lack of targeted research elucidating the real dynamic mechanism (Yang et al., \u003cspan citationid=\"CR36\" class=\"CitationRef\"\u003e2020\u003c/span\u003e). In view of this drawback or gap, this work takes the Kuancheng Geodynamic Observatory (KGO) as a typical case to decoding the overlooked mechanism. KGO is located in the eastern Yanshan Orogenic Belt, North China. After the installation of extensometers in 2001, notable and regular annual variations in strain have been identified in the North-South (NS) and East-West (EW) components recorded (Wang et al., \u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e2020\u003c/span\u003e). Meanwhile, the outdoor air temperature at KGO also exhibits a similar rhythm of annual variation. Consequently, whether there is a causal relationship between the two should be explored further. Moreover, in actual earthquake prediction operations, some Chinese forecasters focusing on earthquake precursors tend to regard the amplitudes and phases of such annually varying signals as an important predictors. In view of this, the author investigated the physical correlation between annual variation in vault-housed extensometers and surface temperature at KGO according to the local topographic, lithologic features using the composite model proposed by Ben-Zion and Leary (\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e1986\u003c/span\u003e). The findings could not only improve our understanding of the physical origin of the annual variation process of strain at KGO but also provide useful reference for related researches as well as rational application of predictors of annual variations observed by vault-housed extensometers in mainland China.\u003c/p\u003e"},{"header":"2. Observatory and Instrument","content":"\u003cp\u003eKGO (40.59\u0026deg; N, 118.47\u0026deg; E) is located beside the Chagou Road in Kuancheng Town, Kuancheng Manzu Autonomous County, Chengde City, North China, at an elevation of approximately 338 m (Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003ea). This vault-housed extensometer observatory was put into operation in 1975, mainly under earthquake precursor aspects. The folds of the geological unit where the KGO is located are relatively developed, so that the surrounding mountains are topographically complex and undulating (Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003eb), and the basement of the observatory consists primarily of siliceous limestone. The in-situ area falls under a warm-temperate semi-arid and semi-humid continental monsoon climate, with mean annual sunshine hours of approximately 2825 h, mean annual evaporation of approximately 1589 mm, and maximum permafrost thickness of 2 m. This area is characterized by four distinct seasons, and abundant sunshine. The annual mean rainfall in Kuangcheng County is approximately 663 mm, with approximately 80% of rainfall concentrated in June to August (Liu \u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e2020\u003c/span\u003e; Wang \u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e2023\u003c/span\u003e).\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eThe mountain cover at KGO is approximately 20 to 40 m thick, and the soil horizon of the hillside is approximately 15 to 30 cm thick (Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003eb) (Tan et al., \u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e2021\u003c/span\u003e). The observatory is an artificial gallery driven into limestone. Its ground plane is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003ec. The length of the main gallery is approximately 75 m. The yearly mean value of the air temperature is 12.3 ℃ in the gallery, and the yearly and daily temperature variations are less than 0.4 \u0026ordm;C and 0.1 \u0026ordm;C, respectively. The relative humidity in the gallery is approximately 72% and it is nearly constant. An SS-Y extensometer (Invar-rod type) was deployed in the gallery, with a resolution better than 5\u0026times;10\u003csup\u003e\u0026minus;\u0026thinsp;10\u003c/sup\u003e and a theoretical drift rate below 10\u003csup\u003e\u0026minus;\u0026thinsp;6\u003c/sup\u003e/y (Wu \u0026amp; Nie, \u003cspan citationid=\"CR31\" class=\"CitationRef\"\u003e2018\u003c/span\u003e). The thermal expansion coefficient of Invar-rod baseline is approximately 2\u0026times;10\u003csup\u003e\u0026minus;\u0026thinsp;7\u003c/sup\u003e/\u0026deg;C, and the sampling interval is 1 min. Particularly, the baseline lengths of the NS and EW components are 28.50 m and 23.83 m, respectively, and the azimuths of the two are 0\u0026deg; and 90\u0026deg;, respectively (Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003ec). Additionally, a WYY-1 outdoor thermometer was also operated (manufactured by National Institute of Natural Hazards, Ministry of Emergency Management of China), with the same sampling rate as the extensometer.\u003c/p\u003e"},{"header":"3. Data and Pre-processing","content":"\u003cp\u003eSince the extensometers were re-installed at KGO in December 2015 due to the aging of previous strain sensors (Wang et al., \u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e2020\u003c/span\u003e), this work mainly used the daily mean observations of strain and outdoor air temperature from 2016 to 2021. Figure\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e clearly shows that the raw strain contains at least three types of signals: type 1 (a variation signal with a nearly linear upward trend), type 2 (an annual variation signal, whose troughs/peaks occur in spring/fall), and type 3 (a small, high-frequency oscillatory signal in summer and fall each year). As far as the generation mechanism is concerned, the type 1 signal is attributed mainly to the linear drift of the extensometer, while the type 3 signal is attributed to rainfall interference (Wang et al., \u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e2020\u003c/span\u003e; Zhang et al., \u003cspan citationid=\"CR38\" class=\"CitationRef\"\u003e2025\u003c/span\u003e). To better analyze the correlation between strain and annual variation of outdoor air temperature, this work eliminates the linear trend and mean value of each observation curve, and the processed results are shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e. Apparently, the annual variation of strain has an annual rhythm, and the relative change in outdoor temperature is also obviously dominated by annual fluctuations. So, is annual variation in outdoor air temperature a physical source of annual variation in strain? This hypothesis will be systemically tested in the following sections.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003e\u003c/p\u003e"},{"header":"4. Characterization of Annual Variation in strain and Outdoor Air Temperature","content":"\u003cp\u003eTo accurately determine annual variation cycle and amplitude for outdoor air temperature and those in the NS and EW strain components, this work calculates the time series curves in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e using the fast Fourier transform (FFT) method. The results show (Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e) that all three reach the main peak value at the frequency point of 0.002739 cpd (365 d, cpd refers to cycles per day); notably, the amplitudes of the NS and EW components are 502.68 nstrain and 658.22 nstrain, respectively, and the outdoor air temperature was 16.84℃.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e"},{"header":"5. Model and Results","content":"\u003cdiv id=\"Sec6\" class=\"Section2\"\u003e\u003ch2\u003e5.1 BZL Model and Thermal Boundary Conditions\u003c/h2\u003e\u003cp\u003eGenerally speaking, the real thermoelastic strain field may be complicatedly modified by the lateral media heterogeneities, the topographic, and the cavity effects according to a number of previous studies (Ben-Zion \u0026amp; Leary, \u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e1986\u003c/span\u003e; Hvoždara \u0026amp; Brimich, \u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e1988\u003c/span\u003e). However, the BZL model can reasonably and effectively resolve this tough problem (Ben-Zion \u0026amp; Leary, \u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e1986\u003c/span\u003e). According to this excellent model, the complex effect of topography and lateral media heterogeneities is to create the local spatial changes in the surface temperature field and in the response of the rock formation, thus a stationary thermal wave can be introduced as the thermoelastic strain source, whose wavelength is related to topography and lateral media heterogeneities in a given area (Ben-Zion \u0026amp; Leary, \u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e1986\u003c/span\u003e). As for the cavity effect, many studies demonstrate that the along-vault strain component and relative long extensometer may be less affected (Harrison, \u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e1976\u003c/span\u003e; Takemoto, \u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e1981\u003c/span\u003e; Gebauer et al., \u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e2009\u003c/span\u003e; Eper-P\u0026aacute;pai et al., \u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e2014\u003c/span\u003e). Given the two facts that the lengths of extensometers are relatively long (greater than 23 m), and the directions of extensometers parallelly align with the long axis of the vaults at KGO; therefore, the author will ignore the cavity effect in the BZL model preferred in this work. The basement of KGO is composed mainly of thin weathered or unconsolidated layer and thick bedrock, and the thermoelastic strain response of the two-layer geologic structure to surface temperature can be well resolved by the BZL model (Ben-Zion \u0026amp; Leary, \u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e1986\u003c/span\u003e). In this composite model (Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e), the thin weathered layer with a thickness of \u003cem\u003ey\u003c/em\u003e\u003csub\u003eb\u003c/sub\u003e can cause surface temperature field attenuation and thermal conduction delay.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eAssuming the surface temperature field (at a depth of \u003cem\u003ey\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0 m) changes locally as a standing wave (Ben-Zion \u0026amp; Leary, \u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e1986\u003c/span\u003e):\u003c/p\u003e\u003cp\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"427\" height=\"35\"\u003e\u003c/p\u003e\u003cp\u003ewhere, \u003cem\u003ex\u003c/em\u003e, \u003cem\u003ey\u003c/em\u003e, and \u003cem\u003et\u003c/em\u003e represent the horizontal position, formation depth, and time, respectively; \u003cem\u003ek\u003c/em\u003e is the horizontal wavenumber, \u003cem\u003eT\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e is the amplitude corresponding to the annual variation surface temperature field with wavenumber \u003cem\u003ek\u003c/em\u003e, and \u003cem\u003eω\u003c/em\u003e and \u003cem\u003eφ\u003c/em\u003e are angular frequency and initial phase, respectively. Here, \u003cem\u003ek\u003c/em\u003e\u0026thinsp;=\u0026thinsp;2π/\u003cem\u003eλ\u003c/em\u003e, \u003cimg src=\"data:image/png;base64,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\" width=\"116\" height=\"29\"\u003e, \u003cem\u003eλ\u003c/em\u003e is the wavelength of the surface temperature field, and \u003cem\u003eκ\u003c/em\u003e is the thermal diffusivity of the thin unconsolidated layer (\u003cem\u003ey\u003c/em\u003e\u003csub\u003eb\u003c/sub\u003e) and thick bedrock. Particularly, it should be pointed out here that since there is no air (ground) temperature station array within a radius of tens of kilometers from KGO, this work is not able to perform frequency-wave number analysis. Additionally, considering that annual surface temperature variation in Kuancheng for many years has been similar to that of the air temperature (Liao et al., \u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e2019\u003c/span\u003e), we approximate the amplitude of annual variation of outdoor air temperature of a single station as \u003cem\u003eT\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e (Hainzl et al., \u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e2013\u003c/span\u003e). Based on the actual angular frequency, amplitude, and initial phase of the annual temperature variation at KGO, \u003cem\u003eω\u003c/em\u003e, \u003cem\u003eT\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e, and \u003cem\u003eφ\u003c/em\u003e are taken as 2.0\u0026times;10\u003csup\u003e\u0026minus;\u0026thinsp;7\u003c/sup\u003e rad/s, 16.84 ℃, and -2.94 rad, respectively.\u003c/p\u003e\u003cp\u003eSo, how does the temperature at the bottom boundary of the \u003cem\u003ey\u003c/em\u003e\u003csub\u003eb\u003c/sub\u003e layer change? This requires prioritizing resolving the thicknesses of the \u003cem\u003ey\u003c/em\u003e\u003csub\u003eb\u003c/sub\u003e layer in the azimuth of the NS and EW components of the strain, which are calculated as follows:\u003c/p\u003e\u003cp\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"365\" height=\"47\"\u003e\u003c/p\u003e\u003cp\u003eIn Eq.\u0026nbsp;(\u003cspan refid=\"Equ2\" class=\"InternalRef\"\u003e2\u003c/span\u003e), Δ\u003cem\u003et\u003c/em\u003e represents the phase delay between the annual components of the calculated half-space thermoelastic strain without \u003cem\u003ey\u003c/em\u003e\u003csub\u003eb\u003c/sub\u003e layer and observed strain; \u003cem\u003eκ\u003c/em\u003e and \u003cem\u003eτ\u003c/em\u003e indicate the thermal diffusivity of the \u003cem\u003ey\u003c/em\u003e\u003csub\u003eb\u003c/sub\u003e layer, and the cycle of the annual variation, respectively.\u003c/p\u003e\u003cp\u003eCorrespondingly, the horizontal thermoelastic strain \u003cem\u003eε\u003c/em\u003e\u003csub\u003e\u003cem\u003exx\u003c/em\u003e\u003c/sub\u003e at different depths in the elastic half-space caused by the annual variation of surface temperature can be expressed as follows:\u003c/p\u003e\u003cp\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"553\" height=\"71\"\u003e\u003c/p\u003e\u003cp\u003ewhere \u003cem\u003eν\u003c/em\u003e and \u003cem\u003eβ\u003c/em\u003e are the Poisson's ratio and thermal expansion coefficient of the bedrock, respectively. The thermoelastic strain solution includes mainly two parts: one resulting from an equivalent thermal surface traction, which decays slowly with an increase in depth in the form of \u003cem\u003eye\u003c/em\u003e\u003csup\u003e\u0026minus;\u0026thinsp;\u003cem\u003eky\u003c/em\u003e\u003c/sup\u003e; the other from equivalent thermal body forces, which decreases with an increase in depth in the form of \u003cem\u003ee\u003c/em\u003e\u003csup\u003e\u0026minus;\u003cem\u003eγy\u003c/em\u003e\u003c/sup\u003e. Here, a positive value of \u003cem\u003eε\u003c/em\u003e\u003csub\u003e\u003cem\u003exx\u003c/em\u003e\u003c/sub\u003e denotes tensive strain, whereas a negative value represents compressive strain.\u003c/p\u003e\u003cp\u003eBased on this, I will preliminarily solve for the amplitudes and phases of the theoretical thermoelastic strain at KGO, aiming to determine the phase differences △\u003cem\u003et\u003c/em\u003e with the observed values of NS and EW strain components, respectively. In the trial calculations, the thermodynamic parameters of limestone taken in this paper are as follows: \u003cem\u003eν\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0.25, \u003cem\u003eβ\u003c/em\u003e\u0026thinsp;=\u0026thinsp;5\u0026times;10\u003csup\u003e\u0026minus;\u0026thinsp;6\u003c/sup\u003e/\u0026deg;C, \u003cem\u003eκ\u003c/em\u003e\u0026thinsp;=\u0026thinsp;10.70\u0026times;10\u003csup\u003e\u0026minus;\u0026thinsp;2\u003c/sup\u003e m\u003csup\u003e2\u003c/sup\u003e/d (Lei et al., \u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e2018\u003c/span\u003e), while the surface temperature field parameters are \u003cem\u003eT\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e\u0026thinsp;\u003cem\u003e=\u003c/em\u003e\u0026thinsp;16.84 ℃, \u003cem\u003eλ\u0026thinsp;=\u003c/em\u003e\u0026thinsp;3 km, \u003cem\u003eφ=\u003c/em\u003e-2.94 rad, and \u003cem\u003eω\u0026thinsp;=\u003c/em\u003e\u0026thinsp;2.0\u0026times;10\u003csup\u003e\u0026minus;\u0026thinsp;7\u003c/sup\u003e rad/s; because the depth of the overlay of the gallery is not uniform, this work sets \u003cem\u003ey\u003c/em\u003e\u0026thinsp;=\u0026thinsp;30 m based on a trade-off strategy. By substituting the above variable values into Eq.\u0026nbsp;(\u003cspan refid=\"Equ3\" class=\"InternalRef\"\u003e3\u003c/span\u003e), the annual variation process of theoretical thermoelastic strain can be obtained, with detailed results shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e. The fluctuation pattern of the theoretical values is similar to that of the measured values. An interesting phenomenon, however, is that the phases of the latter are all lagging behind those of the former, with corresponding ∆t being 61 d and 46 d, respectively.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eTo address the issue of the thermal diffusivity \u003cem\u003eκ\u003c/em\u003e value for the \u003cem\u003ey\u003c/em\u003e\u003csub\u003eb\u003c/sub\u003e layer, this work refers to experimental values for the thermal diffusivity of soil and set \u003cem\u003eκ\u003c/em\u003e to 2.16\u0026times;10\u003csup\u003e\u0026minus;\u0026thinsp;2\u003c/sup\u003e m\u003csup\u003e2\u003c/sup\u003e/d and \u003cem\u003eτ\u003c/em\u003e to 365 d (Ben-Zion \u0026amp; Leary, \u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e1986\u003c/span\u003e; Shi et al., \u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e2015\u003c/span\u003e). Using Eq.\u0026nbsp;(\u003cspan refid=\"Equ2\" class=\"InternalRef\"\u003e2\u003c/span\u003e), this work calculates the \u003cem\u003ey\u003c/em\u003e\u003csub\u003eb\u003c/sub\u003e values in the azimuth of the NS and EW components of strain, resulting in thicknesses of 1.66 m and 1.25 m, respectively. Notably, although the thickness of the soil horizon at KGO ranges from 15 to 30 cm, the distribution of the thickness of this type of media layer is actually not uniform in the surrounding area (see Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003eb); therefore, the theoretical \u003cem\u003ey\u003c/em\u003e\u003csub\u003eb\u003c/sub\u003e value can be regarded as in the range of the influence of the standing wave of the surface temperature field. Subsequently, by substituting the values into Eq.\u0026nbsp;(\u003cspan refid=\"Equ1\" class=\"InternalRef\"\u003e1\u003c/span\u003e), this work can obtain the temperature variations at the bottom boundary of the \u003cem\u003ey\u003c/em\u003e\u003csub\u003eb\u003c/sub\u003e layer in the azimuth of the NS and EW strain components. The corresponding amplitudes of annual variations are 5.91℃ and 7.65℃, respectively, with initial phases of -1.89 rad and །2.12 rad, respectively (Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e).\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eAfter determining the annual variation process of the temperature at the bottom boundary of the \u003cem\u003ey\u003c/em\u003e\u003csub\u003eb\u003c/sub\u003e layer, Eq.\u0026nbsp;(\u003cspan refid=\"Equ3\" class=\"InternalRef\"\u003e3\u003c/span\u003e) can be used to resolve the theoretical thermoelastic strain amplitudes and phases of the NS and SW components at KGO. In the next subsection, the author will systematically explore this issue.\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec7\" class=\"Section2\"\u003e\u003ch2\u003e5.2 BZL Model Sensitivity Test to Thermodynamic Parameters and Optimal Thermal Strain Solution\u003c/h2\u003e\u003cp\u003eFor the actual calculation, this work sets \u003cem\u003eω\u003c/em\u003e\u0026thinsp;=\u0026thinsp;2.0\u0026times;10\u003csup\u003e\u0026minus;\u0026thinsp;7\u003c/sup\u003e rad/s and the \u003cem\u003eT\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e values in the azimuth of NS and EW components of strain to 5.91 ℃ and 7.65 ℃, respectively. After deducting the \u003cem\u003ey\u003c/em\u003e\u003csub\u003eb\u003c/sub\u003e layer thickness, the \u003cem\u003ey\u003c/em\u003e values in the azimuth of the NS and SW components are 28.34 m and 28.75 m, respectively. Additionally, based on the quantitative form of Eq.\u0026nbsp;(\u003cspan refid=\"Equ3\" class=\"InternalRef\"\u003e3\u003c/span\u003e), the wavelength of the temperature field and the thermodynamic parameter of limestone (\u003cem\u003eκ\u003c/em\u003e and \u003cem\u003eν\u003c/em\u003e) have nonlinear impacts on theoretical thermoelastic strain. However, no experimental studies have yet measured the specific values of \u003cem\u003eκ, ν\u003c/em\u003e, and \u003cem\u003eβ\u003c/em\u003e for limestone at KGO and its surrounding regions. Furthermore, \u003cem\u003eλ\u003c/em\u003e is influenced by regional topography and lateral media heterogeneities (Ben-Zion \u0026amp; Leary, \u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e1986\u003c/span\u003e). Thus, this work empirically sets \u003cem\u003eβ\u003c/em\u003e\u0026thinsp;=\u0026thinsp;7\u0026times;10\u003csup\u003e\u0026minus;\u0026thinsp;6\u003c/sup\u003e/℃ and optimize the values of the other parameters within reasonable bounds using the trade-off algorithm.\u003c/p\u003e\u003cp\u003eTo visualize the combined effect of \u003cem\u003eν\u003c/em\u003e and \u003cem\u003eκ\u003c/em\u003e on the amplitude of the annual variation of theoretical thermoelastic strain, I initially set \u003cem\u003eλ\u003c/em\u003e\u0026thinsp;=\u0026thinsp;3 km, and then test the combined effects of \u003cem\u003eλ\u003c/em\u003e and \u003cem\u003eκ\u003c/em\u003e on the theoretical values at \u003cem\u003eν\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0.30. Generally, the annual variation amplitude of theoretical thermoelastic strain tends to increase with increase in \u003cem\u003eν\u003c/em\u003e and \u003cem\u003eκ\u003c/em\u003e, whereas it decreases as \u003cem\u003eλ\u003c/em\u003e increases. Additionally, the amplitude values in the azimuth of each strain component range between 70.64 and 1604.51 nstrain (Figs.\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e8\u003c/span\u003e, \u003cspan refid=\"Fig9\" class=\"InternalRef\"\u003e9\u003c/span\u003e). Based on the model characteristics for parameter sensitivity tests and the conditions of KGO, this work finally determines the optimal \u003cem\u003eν\u003c/em\u003e and \u003cem\u003eκ\u003c/em\u003e values of the limestone to be 0.30 and 7.74\u0026times;10\u003csup\u003e\u0026minus;\u0026thinsp;2\u003c/sup\u003e m\u003csup\u003e2\u003c/sup\u003e/d, respectively, and the wavelengths \u003cem\u003eλ\u003c/em\u003e of the temperature field in the azimuth of the NS and EW strain components to be 2.52 and 2.50 km, respectively.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eThe amplitude and the initial phase \u003cem\u003eφ\u003c/em\u003e of the annual variation of the temperature at the bottom boundary of the \u003cem\u003ey\u003c/em\u003e\u003csub\u003eb\u003c/sub\u003e layer corresponding to each component can be substituted into Eq.\u0026nbsp;(\u003cspan refid=\"Equ3\" class=\"InternalRef\"\u003e3\u003c/span\u003e) one by one to obtain the corresponding theoretical thermal strain response curves, and the final results are illustrated in Fig.\u0026nbsp;\u003cspan refid=\"Fig10\" class=\"InternalRef\"\u003e10\u003c/span\u003e. From a macroscopic viewpoint, the amplitudes and phases of annual variations of the modeled thermoelastic strains of the NS and EW components are highly compatible with the observed values, with correlation coefficients of 0.9795 and 0.9919, respectively. This demonstrates the validity of the model and thermodynamic parameters, while rationalizing that the annual variation in surface air temperature is the main physical source of the annual variation in strain at KGO. Another phenomenon that needs to be elucidated here is that the measured amplitude of annual variation of the EW component is approximately 1.31 times that of the NS component; and the \u003cem\u003ey\u003c/em\u003e\u003csub\u003eb\u003c/sub\u003e thickness results obtained in this work indicate that the main reason for this difference is that the equivalent thickness of the weathered layer is thinner in the azimuth of EW component than in the azimuth of NS component.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eWhen looking more closely at the local or detailed characteristics of the modeled and observed values, the two do not match exactly. For instance, in 2020, the theoretical thermal strain peaks and troughs of the NS component deviate significantly from the observed values; the deviation may be attributed to the amplitude of the annual temperature variation used in this paper being a six-year mean value, whereas, in reality, the amplitude of temperature fluctuations in each year is not a single value.\u003c/p\u003e\u003cp\u003eIn summary, the annual fluctuations in strain observed at KGO mainly result from the annual variation in outdoor air temperature.\u003c/p\u003e\u003c/div\u003e"},{"header":"6. Conclusions and Perspectives","content":"\u003cp\u003eTo quantitatively decode the physical correlation between the annual variation of outdoor air temperature and the annual variation of strain recorded at KGO, this work performs a dynamics diagnosis of the problem using a half-space model incorporating the thin weathered layer. The modeled results \u0026ldquo;roughly\u0026rdquo; but effectively show that the annual variation in outdoor air temperature of 16.84 ℃ is sufficient to generate a thermoelastic strain of 658.22 nstrain inside a mountain at a depth of 30 m, and the modeled values of amplitudes and phases are highly consistent with the observed values, which strongly suggests that the annual variation in outdoor air temperature is a physical cause of the annual variation in strain at KGO.\u003c/p\u003e\u003cp\u003eAlthough the thermoelastic strain response mechanism of KGO is analyzed comprehensively in this tentative work, the surface temperature field, topography, weathered layer thickness, and thermodynamic parameters of the bedrock and soil horizon at the KGO and its neighboring regions would also affect the accuracy of modeling results to a certain extent. Therefore, it is necessary to conduct a detailed field survey or array observation of the thermodynamic structure, surface temperature field, weathered layer thickness, and outdoor air temperature field of the region where the vault-housed extensometers are located. On this basis, a more detailed and realistic 3D coupled thermal\u0026ndash;mechanical model needs to be constructed to quantitatively reveal the thermoelastic strain caused by the annual variation in surface air temperature in depth.\u003c/p\u003e\u003cp\u003eFinally, notably, this work has only diagnosed KGO as an example, whereas there are many other extensometric observatories in mainland China that can capture the annual variation signals (Xing et al., \u003cspan citationid=\"CR32\" class=\"CitationRef\"\u003e2014\u003c/span\u003e). Therefore, is the annual variation processes observed at these observatories also derived from the annual variation in outdoor air temperature? This is an interesting question that deserves systematic investigation, and the author will continually analyze the associated observatories in the near future. Despite the main purpose of extensometric measurements in mainland China is to capture earthquake precursor signals, in decades of observation practice, a wide variety of non-precursor \u0026ldquo;noise\u0026rdquo; has become the \u0026ldquo;principal\u0026rdquo; components of strain time series. Therefore, the best way to scientifically identify and extract the subtle precursor signals is to enhance geodynamic investigation of annual variations and other \u0026ldquo;noise\u0026rdquo; in strain data.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003e\u003cem\u003eAcknowledgments\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe author is very grateful to Professor Yehuda Ben-Zion from University of Southern California for his helpful guidance about the temperature-induced crustal deformation. I also thank China Earthquake Networks Center (CENC) for very generously providing me with the strain and the outdoor temperature data at very short notice.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eFunding\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe author declare that no funds, grants or other support were received during the preparation of this manuscript.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eConflict of interest\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe authors have no relevant financial or non-financial interests to disclose.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAuthor Contributions\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eXiaolin Yang: Conceptualization; Data analysis; Investigation; Methodology; Model; Writing—original draft.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eAgnew, D. C. (2007). 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Investigation of the Rainfall Effect on Strain Observed by Extensometers at the Jixian Seismostation in China. \u003cem\u003ePure and Applied Geophysics\u003c/em\u003e, https://doi.org/10.1007/s00024-025-03703-4.\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"pure-and-applied-geophysics","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"paag","sideBox":"Learn more about [Pure and Applied Geophysics](https://www.springer.com/journal/24)","snPcode":"24","submissionUrl":"https://submission.nature.com/new-submission/24/3","title":"Pure and Applied Geophysics","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Annual variation, thermoelastic strain, thin unconsolidated layer, atmospheric temperature, vault-housed extensometer, Kuancheng Geodynamic Observatory, North China","lastPublishedDoi":"10.21203/rs.3.rs-6717057/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-6717057/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eClarifying the physical mechanisms underlying near-surface annual deformation remains a formidable challenge in crustal dynamics research. In the past few decades, quite a few extensometric observatories exhibit notable and regular annual variations in strain signals in mainland China. However, the geodynamic investigations on these prevalent but intriguing strain signals remain rare till now. Since outdoor air temperature is a key factor contributing to annual deformation of the Earth\u0026rsquo;s crust, this work therefore quantitatively elucidates the mechanism via which annual atmospheric temperature variation influences the NS and EW components of strain by analyzing data observed at the Kuancheng Geodynamic Observatory, North China, with an elastic half-space model covered by elastically thin unconsolidated layer. According to the modeled results, annual variation of atmospheric temperature with an amplitude of 16.84 ℃ is sufficient to produce a thermoelastic strain of 10\u003csup\u003e\u0026minus;\u0026thinsp;7\u003c/sup\u003e magnitude inside the mountain at a depth of 30 m. Besides, both the modeled amplitudes and phases align closely with the observed strain signals, suggesting that the annual variations of the strain observed at the Kuancheng Geodynamic Observatory mainly originate from annual variations of atmospheric temperature. The method of geodynamic diagnostics used here and obtained findings not only contribute to the quantitative interpretation of the mechanism underlying annual variations in vault-housed extensometers in mainland China but also advance our understanding of the temperature-induced deformation processes within the near-surface crustal layer.\u003c/p\u003e","manuscriptTitle":"Notable Annual Thermoelastic Strain in Vault-housed Extensometers: A Typical Case from the Kuancheng Geodynamic Observatory, North China","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-07-10 12:49:04","doi":"10.21203/rs.3.rs-6717057/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2025-09-03T10:43:17+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-09-03T04:02:50+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"276409107919092647811534848117180734805","date":"2025-09-01T02:22:28+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-08-28T14:51:46+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"337811950747219876076563341362261732458","date":"2025-08-18T09:40:27+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2025-07-08T07:02:12+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2025-05-31T15:06:06+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2025-05-28T12:16:33+00:00","index":"","fulltext":""},{"type":"submitted","content":"Pure and Applied Geophysics","date":"2025-05-21T13:26:39+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"pure-and-applied-geophysics","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"paag","sideBox":"Learn more about [Pure and Applied Geophysics](https://www.springer.com/journal/24)","snPcode":"24","submissionUrl":"https://submission.nature.com/new-submission/24/3","title":"Pure and Applied Geophysics","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"508075b4-0df2-4d5a-ba0b-5d7c55a18112","owner":[],"postedDate":"July 10th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"published-in-journal","subjectAreas":[],"tags":[],"updatedAt":"2025-11-10T16:06:14+00:00","versionOfRecord":{"articleIdentity":"rs-6717057","link":"https://doi.org/10.1007/s00024-025-03852-6","journal":{"identity":"pure-and-applied-geophysics","isVorOnly":false,"title":"Pure and Applied Geophysics"},"publishedOn":"2025-11-07 15:58:15","publishedOnDateReadable":"November 7th, 2025"},"versionCreatedAt":"2025-07-10 12:49:04","video":"","vorDoi":"10.1007/s00024-025-03852-6","vorDoiUrl":"https://doi.org/10.1007/s00024-025-03852-6","workflowStages":[]},"version":"v1","identity":"rs-6717057","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-6717057","identity":"rs-6717057","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}