Method for calculating freezing depth of tunnel surrounding rock in cold region based on freezing index

Abstract

The invention relates to the technical field of cold region tunnel freezing prevention, in particular to a method for calculating the freezing depth of surrounding rock of a cold region tunnel based on a freezing index, which comprises the following steps: S1, selecting at least one section of the cold region tunnel as a test tunnel, and taking the cold region as a research region; s2, setting a plurality of monitoring points in a negative temperature section in the test tunnel hole along the longitudinal direction, and calculating annual freezing indexes of all the monitoring points; s3, monitoring to obtain the maximum radial freezing depth of all the monitoring points; s4, correlation fitting analysis is carried out on the annual freezing indexes and the maximum radial freezing depths of all the monitoring points, and a correlation formula or a relation curve of the annual freezing indexes and the maximum radial freezing depths in the tunnel holes in the research area is obtained; and S5, monitoring to obtain the annual freezing index of the portal of the proposed tunnel or the existing tunnel in the research area, and obtaining the freezing depth of the surrounding rock of the proposed tunnel or the existing tunnel according to the related formula or the relation curve. According to the invention, the freezing depth of the surrounding rock in the proposed or existing cold region tunnel can be evaluated.

Description

Technical Field

[0001] The invention relates to the technical field of antifreeze of tunnels in cold regions, and in particular to a method for calculating freezing depth of surrounding rocks of tunnels in cold regions based on freezing index. Background Art

[0002] More than half of my country's land area belongs to cold regions, which mainly include permafrost areas and seasonal frozen soil areas, accounting for 20% and 55% of the land area respectively; tunnels in cold regions are called cold region tunnels due to the geological conditions of low temperature and frozen soil.

[0003] In rock underground engineering, the surrounding rock mass that changes its stress state due to the influence of excavation is called surrounding rock. The freezing depth of surrounding rock refers to the thickness of the ice layer formed by the freezing of water in the surrounding rock when the ground temperature is below 0°C. In cold-region tunnel engineering, the freezing depth of surrounding rock is an important basic parameter for the design of tunnel anti-frost heave structure, drainage and thermal insulation layer.

[0004] With the rapid development of China's infrastructure, the number of tunnels in cold regions is increasing, and the problem of tunnel frost damage is becoming more and more obvious. The freezing of surrounding rock will produce frost heave force, which will cause cracks inside the surrounding rock, thus affecting the stability of the tunnel. For tunnels in seasonally frozen areas, due to the heating effect of the surrounding rock, that is, the surrounding rock temperature is positive, in order to make the surrounding rock freeze, the lining must be frozen through first, and then the surrounding rock temperature can be gradually reduced to negative temperature, thereby causing the surrounding rock to freeze.

[0005] For tunnels in seasonally frozen areas, even if the temperature inside the tunnel is low, if the lining is not frozen through, the surrounding rock will not freeze; even if the length of the tunnel with negative temperature is long, it may be due to the cold air flowing from the outside flowing into the tunnel, thereby increasing the length of the tunnel with negative temperature, which does not mean that the lining will definitely be frozen through. Therefore, for the antifreeze parameters of cold zone tunnels, it is not reasonable to consider the antifreeze parameters only from the length of the tunnel with negative temperature inside the tunnel. The most direct and reasonable way should be to consider the freezing depth of the surrounding rock. By evaluating the freezing depth of the surrounding rock, a reference can be provided for the antifreeze design of the surrounding rock of cold zone tunnels, and it can also be used to quickly evaluate the safety of the proposed tunnel or existing tunnel structure.

[0006] The method for measuring the surrounding rock depth in related technologies is usually to conduct actual measurements of the surrounding rock, including drilling sampling, geological radar detection, etc. These methods can measure the freezing depth of the surrounding rock of existing tunnels, but cannot predict, evaluate or measure the freezing depth of the surrounding rock of a planned tunnel.

[0007] Therefore, there is an urgent need for a method to conveniently evaluate and calculate the freezing depth of surrounding rock in cold-region tunnels. Summary of the invention

[0008] The purpose of the present invention is to provide a method for calculating the freezing depth of surrounding rock of cold-region tunnels based on freezing index, aiming to improve the problem in the prior art that it is inconvenient to evaluate the freezing depth of surrounding rock of planned and existing cold-region tunnels.

[0009] To achieve the above object, the present invention provides the following technical solutions:

[0010] A method for calculating freezing depth of surrounding rock of tunnel in cold regions based on freezing index comprises the following steps:

[0011] S1. Select at least one cold region tunnel as a test tunnel, and take the cold region where the test tunnel is located as a research area;

[0012] S2, setting a plurality of monitoring points at longitudinal intervals in the negative temperature section of the test tunnel, and obtaining the annual freezing index at all the monitoring points;

[0013] S3, monitoring and obtaining the maximum radial freezing depth of all monitoring points;

[0014] S4. Perform correlation fitting analysis on the annual freezing index and the maximum radial freezing depth at all monitoring points to obtain the correlation formula or relationship curve between the annual freezing index and the maximum radial freezing depth in the tunnels in the study area;

[0015] S5. Monitor and obtain the annual freezing index of the proposed tunnel entrance or the existing tunnel in the study area, and obtain the freezing depth of the surrounding rock of the proposed or existing tunnel based on the relevant formula or relationship curve.

[0016] Furthermore, in step S2, the calculation formula of the annual freezing index at each monitoring point is: Among them, FI is the annual freezing index, t i is the negative value of the average daily temperature, and n is the total number of days with negative average daily temperature throughout the year.

[0017] Furthermore, each monitoring point is provided with a temperature sensor which can automatically store temperature data. Each temperature sensor monitors temperature data multiple times at even intervals every day, and calculates the daily average temperature value of each day, so as to obtain all negative values ​​of the daily average temperature of all monitoring points in a complete year.

[0018] Furthermore, in step S3, a plurality of the temperature monitoring holes are opened at intervals along the circumference of the tunnel at each monitoring point; the freezing depths of all the temperature monitoring holes at each monitoring point are first determined, and then the maximum radial freezing depth at each monitoring point is obtained.

[0019] Furthermore, the depth of each temperature monitoring hole is greater than the historical maximum freezing depth of the strata in the study area.

[0020] Furthermore, when determining the freezing depth of each temperature monitoring hole, at least three temperature sensors are evenly spaced in each temperature monitoring hole along the hole depth direction, and a temperature sensor is arranged near the opening of each temperature monitoring hole and at the deepest part;

[0021] Based on the temperature measurement data and depth data of the temperature sensor in each temperature monitoring hole, the depth position of 0°C inside the corresponding temperature monitoring hole is calculated and determined by linear interpolation. The depth position of 0°C inside each temperature monitoring hole is the freezing depth in the corresponding temperature monitoring hole.

[0022] Compared with the prior art, the present invention has the following beneficial effects:

[0023] 1. This scheme is based on the relevant formula or relationship curve between the freezing index and the freezing depth of the surrounding rock. By monitoring the annual freezing index at the entrance of the proposed tunnel or the inside of the existing tunnel in the study area, the freezing depth of the surrounding rock of the proposed tunnel or the existing tunnel can be quickly calculated. At the same time, based on the relevant formula and relationship curve, the critical freezing index of the surrounding rock of the cold-zone tunnel in the region can be obtained, thereby effectively guiding the cold-resistant and frost-resistant tunnels in the cold zone.

[0024] 2. Freezing inside the tunnel surrounding rock requires a certain period of continuous negative temperature, otherwise the surrounding rock frost heaving force cannot be formed; therefore, when designing the anti-freezing of the surrounding rock of cold-region tunnels, it is necessary not only to consider the temperature conditions inside the tunnel, but also to consider the continuity of the negative temperature conditions, that is, the duration. This scheme combines the two organically based on the freezing index, and provides a new solution to whether the surrounding rock is frozen, that is, whether to consider the surrounding rock frost heaving force. BRIEF DESCRIPTION OF THE DRAWINGS

[0025] The accompanying drawings are used to provide a further understanding of the present invention and constitute a part of the specification. Together with the embodiments of the present invention, they are used to explain the present invention and do not constitute a limitation of the present invention. In the accompanying drawings:

[0026] Figure 1 It is a flowchart of a method for calculating freezing depth of surrounding rock of tunnel in cold regions based on freezing index of the present invention;

[0027] Figure 2 is a relationship curve between the maximum radial freezing depth and the freezing index of the tunnel in the study area in Example 1 of the present invention; Figure 3 This is a relationship curve between the maximum radial freezing depth and the freezing index of the tunnel in the study area in Example 2 of the present invention. DETAILED DESCRIPTION

[0028] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0029] A method for calculating freezing depth of surrounding rock of tunnel in cold regions based on freezing index comprises the following steps:

[0030] S1. Select at least one cold zone tunnel as a test tunnel, and take the cold zone where the test tunnel is located as the study area; the study area is a cold area with a higher altitude or higher latitude. The study area can be a province with lower temperatures in winter, or a plateau area with a higher altitude. Tunnels in the study area are called cold zone tunnels due to the geological conditions of low temperature and frozen soil. More than half of my country's land belongs to the cold zone, which mainly includes permafrost areas and seasonal frozen soil areas, accounting for 20% and 55% of the land area respectively.

[0031] S2. Conduct continuous longitudinal temperature tests in the test tunnels within the study area, especially set multiple monitoring points at longitudinal intervals in the negative temperature sections at both ends of the test tunnel entrance. The negative temperature section in the tunnel refers to the section where negative temperature often occurs in the tunnel. Calculate the annual freezing index at all monitoring points; the annual freezing index refers to the daily cumulative value of the average daily temperature of negative degrees (℃) in a full year. The more test tunnels selected and the more monitoring points set, the higher the reference value of the annual freezing index obtained;

[0032] S3, monitoring and obtaining the maximum radial freezing depth of all monitoring points; in this step, the maximum radial freezing depth of all monitoring points is measured by actual measurement;

[0033] S4. Carry out correlation fitting analysis on the annual freezing index and the maximum radial freezing depth at all monitoring points, and obtain the correlation formula or relationship curve of the annual freezing index and the maximum radial freezing depth in the tunnel in the study area; fitting is to connect a series of points on the plane with a smooth curve; because this curve has countless possibilities, there are various fitting methods; the fitted curve can generally be represented by a function, and there are different fitting names according to the different functions; commonly used fitting methods include least squares curve fitting method, etc., and polyfit can also be used to fit polynomials in MATLAB; fitting, interpolation and approximation are the three basic tools of numerical analysis. In a popular sense, their differences are: fitting is a known point series, approaching them as a whole; interpolation is a known point series and completely passes through the point series; approximation is a known curve, or point series, and through approximation, the constructed function is infinitely close to them; if the function to be determined is linear, it is called linear fitting or linear regression (mainly in statistics), otherwise it is called nonlinear fitting or nonlinear regression; the expression can also be a piecewise function, in which case it is called spline fitting;

[0034] S5. Monitor and obtain the annual freezing index of the proposed tunnel entrance or the existing tunnel in the study area, and obtain the freezing depth of the surrounding rock of the proposed tunnel or the existing tunnel based on the above-mentioned relevant formula or relationship curve.

[0035] Specifically, in step S2, the calculation formula of the annual freezing index at each monitoring point is: Where FI is the freezing index, unit is ℃·d; t i is the negative value of the average daily temperature on the ith day, in degrees Celsius (℃); n is the total number of days with negative average daily temperatures throughout the year, in days (d).

[0036] Furthermore, each monitoring point is provided with a temperature sensor that can automatically store temperature data. Each temperature sensor detects the temperature data of the corresponding monitoring point once every day at an interval. The detection time of all temperature sensors is at least one full year. By calculating the daily average temperature value, all negative daily average temperature values ​​that occur in a full year at all monitoring points can be selected; the detection frequency of each temperature sensor is 1 time / 2h. The daily average temperature refers to the average temperature of 24 hours a day. The methods for calculating the daily average temperature value include the four-time average method, the daily maximum and minimum average method, and the 24-hour average method. Among them, the more commonly used methods are the four-time average method and the daily maximum and minimum average method. 1. Four-time average method: This method involves four specific time points in a day, namely 2 a.m., 8 a.m., 2 p.m., and 8 p.m. Take the temperature values ​​at these four time points, add them up and divide by 4, the average value is the daily average temperature; 2. Daily maximum and minimum temperature average method: This method is relatively simple. You only need to add the maximum and minimum temperatures of the day, and then divide by 2. The average value is the daily average temperature; 3. 24-hour average method: This method requires adding up all the temperature values ​​of 24 hours in a day, and then dividing by 24. The average value is the daily average temperature. The above methods are widely used in meteorological observations because of their simplicity and representativeness; and the daily maximum and minimum average method is also often used in agriculture and daily life because of its simplicity.

[0037] Furthermore, in step S3, multiple temperature monitoring holes are opened at intervals along the circumference of the tunnel at each monitoring point, and temperature data of all temperature monitoring holes at different depths throughout the year are collected; first, the freezing depths of all temperature monitoring holes at each monitoring point are determined, and then the maximum radial freezing depth at each monitoring point is obtained. Among them, those skilled in the art should only know that multiple temperature monitoring holes are opened at the side walls and top walls inside the tunnel; the maximum value of the freezing depths monitored by all temperature monitoring holes at each monitoring point is the maximum radial freezing depth at the current monitoring point.

[0038] Furthermore, the depth of each temperature monitoring hole is greater than the historical maximum freezing depth of the strata in the study area. This design can prevent the situation where the 0°C position cannot be monitored due to insufficient opening depth of the temperature monitoring hole, which is conducive to improving the accuracy of the monitoring results, and further improve the accuracy of the relevant formula or relationship curve of the annual freezing index and the maximum radial freezing depth in the tunnel in the study area.

[0039] Furthermore, when determining the freezing depth of each temperature monitoring hole, a plurality of temperature sensors are arranged at intervals in each temperature monitoring hole, a temperature sensor is arranged near the opening of the temperature monitoring hole and at the deepest part of the temperature monitoring hole, at least one temperature sensor is arranged at equal intervals in the middle of the temperature monitoring hole, and the spacing between two adjacent temperature sensors along the depth direction of the corresponding temperature monitoring hole is set to 30 cm;

[0040] Based on the temperature data and depth data of the temperature sensor in each temperature monitoring hole, the maximum depth position of 0℃ in the corresponding temperature monitoring hole is calculated and determined by linear interpolation. The maximum depth position of 0℃ in each temperature monitoring hole is the freezing depth in the corresponding temperature monitoring hole. Linear interpolation is to connect two known points (x1, y1) and (x2, y2), where x1 and x2 are the temperature data measured by the two temperature sensors in each temperature monitoring hole, and x1 is less than x2, x1 is less than 0℃, and x2 is greater than 0℃. y1 and y2 are the depth position data of the two temperature sensors before and after the 0℃ position in each temperature monitoring hole, and y1 is less than y2. This straight line is used to estimate the y value corresponding to any point x between the two points. The formula is: y=(y2-y1) / (x2-x1)*(x-x1)+y1, where x=0℃ is substituted into the above formula, and the obtained y value is the depth position data of the 0℃ point in the corresponding temperature monitoring hole.

[0041] Interpolation is also called interpolation. Interpolation is a mathematical method for estimating an intermediate value between two known values. Interpolation is often used in data analysis and numerical calculations, and is widely used in the fields of physics, engineering, and finance. Interpolation estimates the position of an unknown data point by estimating the curve or straight line between known data points. Interpolation can be divided into linear interpolation and nonlinear interpolation. Linear interpolation is a method of estimating an intermediate value using a straight line, while nonlinear interpolation uses a curve to estimate the intermediate value. Linear interpolation is usually used between simpler data points, such as when calculating the position of a point on a straight line. Nonlinear interpolation is used between more complex data points, such as in weather forecasting. The linear interpolation method used in this embodiment is linear interpolation.

[0042] The advantage of interpolation is that it can predict the location of intermediate points when there is no information. This is very useful when dealing with real-world problems, such as predicting market trends in investment analysis, estimating unknown data in a laboratory environment in physics, or predicting population growth trends in urban planning.

[0043] Although interpolation has many advantages, it also has some limitations. First, interpolation can only be estimated within the range of known data points. If the data point to be estimated is not within the range of known data, this method cannot be used for prediction. In this embodiment, since the depth of each temperature monitoring hole is greater than the historical maximum freezing depth of the formation in the study area, each temperature monitoring hole contains a part where 0°C is located. By measuring the temperature data inside each temperature monitoring hole, the linear interpolation method can be used to more accurately predict and calculate the depth position of 0°C in the temperature monitoring hole.

[0044] Example 1

[0045] In this embodiment, the tunnels in the extremely cold area in Jilin Province (40°50′~46°19′ north latitude, 121°38′~131°19′ east longitude) are taken as an example. This area is used as the study area of ​​this embodiment, and several existing operating tunnels in this area are selected for testing.

[0046] Specifically, multiple monitoring points are set at different locations away from the tunnel entrance to obtain the annual temperature data at all monitoring points in the tunnel. Based on the negative value of the daily average temperature obtained at each monitoring point, the annual freezing index at different longitudinal positions in each tunnel is obtained through the annual freezing index calculation formula.

[0047] Combined with the temperature monitoring data in the cave, multiple temperature monitoring holes are set along the circumferential direction for the surrounding rock at each monitoring point in the negative temperature section of the cave entrance. By collecting the temperature monitoring data of all the temperature monitoring holes at all the monitoring points throughout the year, the freezing depth of all the temperature monitoring holes at each monitoring point is first determined, and then the maximum radial freezing depth at each monitoring point is obtained, and then the maximum radial freezing depth of the surrounding rock at each monitoring point is selected.

[0048] The maximum radial freezing depth and annual freezing index of the surrounding rock at different longitudinal positions in the tunnel of the study area were statistically summarized (see Table 1), and regression analysis was performed on the maximum radial freezing depth and annual freezing index of the surrounding rock at all monitoring points.

[0049] Table 1 Statistics of annual freezing index and maximum radial freezing depth at different longitudinal positions in the tunnel of the study area

[0050]

[0051] Among them, the lining thickness inside the tunnel is 0.5m.

[0052] By fitting the data in Table 1, the relationship curve between freezing depth and freezing index is obtained. Figure 1Based on the lining thickness of 0.5m, the critical freezing index of the tunnel surrounding rock in Jilin area (40°50′~46°19′N, 121°38′~131°19′E) is inversely calculated to be 285℃.d, that is, the surrounding rock will freeze only when the freezing index is greater than 285℃.d.

[0053] Based on the relationship curve between freezing depth and freezing index, the above formula can be used to calculate the freezing depth of surrounding rocks of planned tunnels or existing tunnels in Jilin Province (40°50′~46°19′N, 121°38′~131°19′E), so as to quickly conduct safety assessment of tunnel structures in cold regions.

[0054] Example 2

[0055] Taking the cold zone tunnel in the western Sichuan plateau (97°21′~106°3′ east longitude, 27°10′~34°19′ north latitude) as an example, this area is used as the study area in this embodiment. The G317 and G318 high-altitude tunnels along the study area are selected for the in-tunnel temperature test.

[0056] Specifically, multiple monitoring points are set under different working conditions at different distances from the tunnel entrance to obtain the annual temperature data at all monitoring points in the tunnel. Based on the negative value of the daily average temperature obtained at each monitoring point, the annual freezing index at different longitudinal positions in each tunnel is calculated using the annual freezing index calculation formula.

[0057] Combined with the temperature monitoring data in the cave, multiple temperature monitoring holes are set along the circumferential direction for the surrounding rock at each monitoring point in the negative temperature section of the cave entrance. By collecting the temperature monitoring data of all the temperature monitoring holes at all the monitoring points throughout the year, the freezing depth of all the temperature monitoring holes at each monitoring point is first determined, and then the maximum radial freezing depth at each monitoring point is obtained, and then the maximum radial freezing depth of the surrounding rock at each monitoring point is selected.

[0058] The maximum radial freezing depth and annual freezing index of the surrounding rock at different longitudinal positions in the tunnel of the study area are statistically summarized, see Table 2, and regression analysis is performed on the maximum radial freezing depth and annual freezing index of the surrounding rock at all monitoring points.

[0059] Table 2 Statistics of annual freezing index and maximum radial freezing depth under different longitudinal conditions in the tunnel of the study area

[0060] Calculation conditions Distance from the cave entrance / m Annual freezing index / ℃·d Depth of surrounding rock freezing / m Condition 1 250 761.04 3.97 Condition 2 500 694.89 3.74 Condition 3 750 507.70 2.57 Condition 4 1000 375.35 1.85 Condition 5 1500 121.25 0.25

[0061] Among them, the lining thickness inside the tunnel is 0.5m.

[0062] By fitting and analyzing the data of multiple groups of maximum radial freezing depth and annual freezing index in Table 2, the relationship curve between the maximum radial freezing depth and annual freezing index of the tunnel surrounding rock in the study area was obtained. Figure 2 .

[0063] Based on the lining thickness of 0.5m, it is concluded that the critical freezing index of the tunnel surrounding rock in the western Sichuan Plateau (97°21′~106°3′ east longitude, 27°10′~34°19′ north latitude) is 165℃.d, that is, the surrounding rock will freeze only when the freezing index is greater than 165℃.d. Based on the relationship curve between freezing depth and freezing index, the freezing depth of the surrounding rock of the planned or existing tunnel in the western Sichuan Plateau (97°21′~106°3′ east longitude, 27°10′~34°19′ north latitude) can be calculated at the same time, so as to quickly conduct a safety assessment of the tunnel structure in the cold region.

[0064] Finally, it should be noted that the above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Although the present invention has been described in detail with reference to the aforementioned embodiments, those skilled in the art can still modify the technical solutions described in the aforementioned embodiments or replace some of the technical features therein by equivalents. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention shall be included in the protection scope of the present invention.

Claims

1. A method for calculating the freezing depth of surrounding rock of tunnels in cold regions based on freezing index, characterized in that: The following steps are involved: S1. Select at least one cold region tunnel as a test tunnel, and take the cold region where the test tunnel is located as a research area; S2, setting a plurality of monitoring points at longitudinal intervals in the negative temperature section of the test tunnel, and obtaining the annual freezing index at all the monitoring points; S3, monitoring and obtaining the maximum radial freezing depth of all monitoring points; S4. Perform correlation fitting analysis on the annual freezing index and the maximum radial freezing depth at all monitoring points to obtain the correlation formula or relationship curve between the annual freezing index and the maximum radial freezing depth in the tunnels in the study area; S5. Monitor and obtain the annual freezing index of the proposed tunnel entrance or the existing tunnel in the study area, and obtain the freezing depth of the surrounding rock of the proposed or existing tunnel based on the relevant formula or relationship curve.

2. The method for calculating the freezing depth of surrounding rock of tunnels in cold regions based on freezing index according to claim 1 is characterized in that: In step S2, the calculation formula of the annual freezing index at each monitoring point is: Among them, FI is the annual freezing index, t i is the negative value of the average daily temperature, and n is the total number of days with negative average daily temperature throughout the year.

3. The method for calculating the freezing depth of surrounding rock of tunnels in cold regions based on freezing index according to claim 2 is characterized in that: Each monitoring point is equipped with a temperature sensor that can automatically store temperature data. Each temperature sensor monitors temperature data multiple times at even intervals every day, and calculates the daily average temperature value of each day. The negative values ​​of the daily average temperature of all monitoring points in a full year can be obtained.

4. The method for calculating freezing depth of surrounding rock of tunnel in cold regions based on freezing index according to claim 1 is characterized in that: In step S3, a plurality of the temperature monitoring holes are opened at intervals along the circumference of the tunnel at each monitoring point; the freezing depths of all the temperature monitoring holes at each monitoring point are first determined, and then the maximum radial freezing depth at each monitoring point is obtained.

5. The method for calculating freezing depth of surrounding rock of tunnel in cold regions based on freezing index according to claim 4 is characterized in that: The depth of each temperature monitoring hole is greater than the historical maximum freezing depth of the strata in the study area.

6. The method for calculating freezing depth of surrounding rock of tunnel in cold regions based on freezing index according to claim 5 is characterized in that: When determining the freezing depth of each temperature monitoring hole, at least three temperature measuring sensors are evenly spaced in each temperature monitoring hole along the hole depth direction, and a temperature measuring sensor is arranged near the opening of each temperature monitoring hole and at the deepest part; Based on the temperature measurement data and depth data of the temperature sensor in each temperature monitoring hole, the depth position of 0°C inside the corresponding temperature monitoring hole is calculated and determined by linear interpolation. The depth position of 0°C inside each temperature monitoring hole is the freezing depth in the corresponding temperature monitoring hole.

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