While-drilling azimuth electromagnetic wave inversion method based on single-chain-multi-chain MCMC adaptive selection

Abstract

The invention discloses a while-drilling azimuth electromagnetic wave inversion method based on single-chain-multi-chain MCMC adaptive selection, and belongs to the field of oil and gas field development. The method comprises the following steps: establishing a multi-parameter inversion model; determining parameters needing to be inverted; constructing a target function; constructing posterior distribution according to the prior distribution and the likelihood function; performing single-chain MCMC sampling by using an MH algorithm, and selecting whether to accept the state of the next moment or not according to an acceptance rate function alpha each time; when the number of iterations meets a set value, sampling is ended, and the result of each iteration of each parameter is collected; whether the MCMC inversion result meets the expectation or not is judged, and if yes, inversion is ended; and if not, taking the result as an initial model of multi-chain parallel MCMC sampling, and further iterating to obtain a maximum posterior solution as a final inversion result. According to the method, the combination mode of the single-chain MCMC and the multi-chain MCMC is adaptively selected, so that rapid and accurate inversion of the while-drilling azimuth electromagnetic wave logging data is realized.

Description

Technical Field

[0001] The invention relates to the field of oil and gas field development, and in particular to a drilling azimuth electromagnetic wave inversion method based on single-chain-multi-chain MCMC adaptive selection. Background Art

[0002] In order to provide real-time geological guidance information, major oilfield service companies have successively launched azimuth electromagnetic wave logging while drilling technology, adding inclined / orthogonal antennas to the traditional electromagnetic wave logging while drilling instruments to enable the instruments to have azimuth detection capabilities. However, in practical applications, due to the influence of many factors such as layer thickness, there is often a large deviation between the measured resistivity and the true resistivity, and the directional signal of the interface may be superimposed by multiple layers of signals, making it difficult to accurately determine the layer interface. Therefore, it is necessary to accurately extract the true formation interface information through inversion methods.

[0003] Gradient inversion methods are often used in traditional LWD electromagnetic wave logging data processing. However, when processing more complex LWD azimuthal electromagnetic wave logging data, due to the dependence on the initial model, it is easy to fall into the local minimum and cannot find the global optimum. Therefore, the MCMC (Markov Chain Monte Carlo) random sampling method is used to overcome the local optimum problem of the gradient method. Single-chain MCMC sampling has a faster convergence speed, but it is easy to fall into the local minimum; while multi-chain parallel MCMC sampling has a stronger global optimization ability, but the computational cost is higher. Summary of the invention

[0004] Based on the above technical problems, the present invention proposes a method for azimuthal electromagnetic wave inversion while drilling based on single-chain-multi-chain MCMC adaptive selection.

[0005] The technical solution adopted by the present invention is:

[0006] A method for azimuthal electromagnetic wave inversion while drilling based on single-chain-multi-chain MCMC adaptive selection includes the following steps:

[0007] s1. Establish a multi-parameter inversion model;

[0008] s2. Analyze the response characteristics of the inversion model parameters and determine the parameters that need to be inverted;

[0009] s3. Obtain observation data and construct the objective function;

[0010] s4. Determine the initial model for single-chain MCMC sampling, determine the length and temperature of the sampling chain, and set the number of iterations;

[0011] s5. Define the range of parameters and set the prior distribution based on the established initial model;

[0012] s6. Select the error model to construct the likelihood function;

[0013] s7. Construct the posterior distribution based on the prior distribution and likelihood function;

[0014] s8. Select the Metropolis-Hastings algorithm for sampling;

[0015] s9. Select the initial value and start MCMC sampling using the Metropolis-Hastings algorithm. Each time, choose whether to accept the next state according to the acceptance rate function α;

[0016] s10. Sampling ends when the number of iterations meets the set value, and the results of each iteration of each parameter are collected;

[0017] s11. Determine whether the MCMC inversion results meet expectations. If so, end the inversion. If not, use the results as the initial model for multi-chain parallel MCMC sampling.

[0018] s12. Set up multi-chain parallel MCMC sampling, set the number of sampling chains and the temperature gradient of each chain, and repeat the same steps s6 to s8 as MCMC;

[0019] s13. Perform multi-chain parallel MCMC sampling on each chain, and choose whether to accept the next state each time according to the acceptance rate function α;

[0020] s14. After this iteration is completed, decide whether to interact between chains based on the acceptance probability;

[0021] s15. Repeat s13 to s14 until a predetermined number of iterations is reached, and the maximum a posteriori solution is obtained as the final inversion result.

[0022] Preferably, in step s2, the parameters to be inverted include formation resistivity and the relative position between the instrument and the layer interface; accordingly, in step s15, the formation resistivity and the relative position between the instrument and the layer interface finally inverted are statistically analyzed for subsequent work.

[0023] Preferably, step s3 specifically includes the following steps:

[0024] s3.1. Preprocess the azimuthal electromagnetic wave logging data while drilling, normalize the measured phase difference and amplitude ratio signals respectively, and ensure the consistency of the magnitude of the logging response;

[0025] s3.2. The residual sum of squares between the observed data f(m) and the forward response d is used to construct the inversion objective function of azimuthal electromagnetic logging while drilling:

[0026]

[0027] where w d and w m respectively represent the weights corresponding to the logging response and the model parameters, λ is the regularization parameter, m ref is the reference value of the model parameters, and m is the value of the model parameters to be inverted.

[0028] Preferably, step s9 specifically includes the following steps:

[0029] s9.1. Use the initial value selected according to the set initial model as m (0) state;

[0030] s9.2. Sample from the posterior distribution. The probability distribution of the solution m k+1 at future times depends only on the solution m k at the current time and is independent of the solutions at past times;

[0031] s9.3. Adopt the Metropolis-Hasting algorithm and introduce the acceptance rate function α, and the expression is as follows:

[0032]

[0033] Assume that the transition kernel function follows a Gaussian distribution, then Π(x,y) = Π(y,x), and the acceptance rate function is further simplified to:

[0034]

[0035] s9.4. Calculate a(x,y) using the above formula and compare it with a random number μ following a uniform distribution. If μ < a(x,y), then accept the model solution m k+1 , otherwise retain the model solution m k ;

[0036] s9.5. Repeat the above process until the set number of iterations is satisfied.

[0037] Preferably, step s11 specifically includes the following steps:

[0038] s11.1. Judge whether the inversion result meets the expectation; use the noise level contained in the observed data as the reference value for whether the inversion result meets the expectation;

[0039] s11.2. Compare the forward result of the inverted model parameters with the observed data and calculate the relative error δ:

[0040]

[0041] where δ is the relative error, d obs is the observed data, d simis the forward data calculated using the inversion model, ||.|| represents the norm of the vector;

[0042] s11.3. If the relative error is less than the reference value, the inversion result is used as the final inversion value at that point and the inversion ends;

[0043] s11.4. If the relative error is greater than the reference value, the result is used as the initial model for multi-chain parallel MCMC sampling.

[0044] Preferably, step s12 specifically includes the following steps:

[0045] s12.1. Construct K Markov chains with increasing temperatures, where the solutions of temperature chain l all satisfy the posterior distribution of different temperatures;

[0046] s12.2. A temperature parameter T is introduced into the likelihood function l Control the sampling step size:

[0047]

[0048] Where, T l is the temperature parameter of each chain, T min is defined as the chain with the lowest temperature, T max The chain with the highest temperature is defined.

[0049] Preferably, step s14 specifically includes the following steps:

[0050] s14.1. After the sampling of the previous moment is completed, a random number ρ that obeys a uniform distribution is generated. If ρ is greater than the preset temperature exchange parameter θ, the adjacent temperature chains do not interact;

[0051] s14.2. If ρ is less than the preset temperature exchange parameter θ, then the state interaction function defined by formula (6) is used to determine whether the adjacent temperature chains interact with each other. Adjacent temperature chains interact with each other, and vice versa:

[0052]

[0053] The beneficial technical effects of the present invention are as follows:

[0054] The present invention adaptively selects the combination of single-chain MCMC and multi-chain MCMC, gives full play to the rapid convergence of single-chain MCMC and the global optimization ability of multi-chain MCMC, overcomes the local optimal problem in traditional methods, and thus realizes rapid and accurate inversion of azimuthal electromagnetic wave logging data while drilling, providing important support for real-time geological guidance, reservoir evaluation and layer interface prediction of oil and gas fields. BRIEF DESCRIPTION OF THE DRAWINGS

[0055] Figure 1 It is a flow chart of the method for drilling azimuthal electromagnetic wave inversion based on single-chain-multi-chain MCMC adaptive selection according to the present invention;

[0056] Figure 2 The inversion model and the schematic diagram of the model parameterization established in the present invention;

[0057] Figure 3 Schematic diagram of the adaptive MCMC iterative process in the present invention; wherein a is a single-chain MCMC iterative process; b is a single-chain-multi-chain adaptive selection MCMC iterative process;

[0058] Figure 4 Schematic diagram of the inverted two-dimensional curtain of the dual interface model in the present invention; wherein a is a schematic diagram of the two-dimensional curtain inverted by MCMC; b is a schematic diagram of the adaptive MCMC two-dimensional curtain;

[0059] Figure 5 It is a comparison diagram of the inversion reconstruction curve of the present invention and the inversion reference curve; wherein a is an amplitude ratio apparent resistivity comparison diagram, b is a phase difference apparent resistivity comparison diagram, and c is a real part azimuth signal comparison diagram. DETAILED DESCRIPTION

[0060] Azimuth electromagnetic wave logging while drilling is an important means to achieve geosteering, real-time geological evaluation, etc. In order to convert the measurement data of azimuth electromagnetic wave logging while drilling instruments into visual formation parameters, inversion calculation is required. However, in the inversion process, the gradient inversion method has a strong dependence on the initial value and is prone to fall into the local minimum. In addition, this type of method requires the objective function to be differentiable and must be able to calculate or approximate the gradient information, which may have certain limitations in practical applications; in contrast, the random inversion method can effectively overcome the limitations of the gradient inversion method, has a strong global optimization capability, and is suitable for complex optimization problems with discontinuous or non-differentiable objective functions. Therefore, choosing an efficient and accurate inversion method is crucial to improving the effect of geosteering and real-time geological evaluation.

[0061] Based on this, the present invention proposes an inversion method based on Bayesian theory, which combines the advantages of single-chain MCMC sampling and multi-chain parallel MCMC sampling. Specifically, single-chain MCMC sampling has a fast convergence speed, while multi-chain parallel MCMC sampling is strong in global optimization. By adaptively combining the advantages of these two methods, the key parameters in the constructed model are inverted, and a method for azimuthal electromagnetic wave inversion while drilling based on single-chain-multi-chain MCMC adaptive selection is proposed, which includes the following steps: s1. Establish a multi-parameter inversion model; s2. Analyze the response characteristics of the model parameters and determine the key parameters that need to be inverted: the formation resistivity obtained by the required inversion, denoted as R; the relative position between the instrument and the layer interface, denoted as DTB; s3. Obtain observation data and construct an objective function; s4. Determine the initial model of single-chain MCMC sampling and determine the length and temperature of the sampling chain, and set the number of iterations; s5. Define the value range of the parameters in combination with the established model and set a suitable prior distribution; s6. Define model prediction and select a suitable error model to construct a likelihood function; s7. Construct a posterior distribution based on the prior distribution and the likelihood function; s8. Based on the characteristics of the sampling method and the objective function, select the Metropolis-Hastings algorithm (MH) for sampling; s9. Select Initial value, start MCMC sampling using the MH algorithm, and choose whether to accept the next moment state each time according to the acceptance rate function α; s10. Sampling ends when the number of iterations meets the set value, and the results of each iteration of each parameter are collected; s11. Determine whether the MCMC inversion result meets the expectation, and end the inversion if it does; if not, use the result as the initial model for multi-chain parallel MCMC sampling; s12. Set up multi-chain parallel MCMC sampling, set the number of sampling chains and the temperature gradient of each chain, and repeat the same steps s6 to s8 as MCMC; s13. Perform multi-chain parallel MCMC sampling on each chain, and choose whether to accept the next moment state each time according to the acceptance rate function α; s14. When this iteration ends, decide whether to interact between chains based on the acceptance probability; s15. Repeat s13 to s14 until the predetermined number of iterations is reached, and the maximum a posteriori solution is obtained as the final inversion result; s16. Statistics are performed on the formation resistivity, the relative position of the instrument and the layer interface, and other data obtained by inversion for subsequent work. The present invention involves real-time geological guidance, layer interface prediction and reservoir evaluation in the process of oil and gas field development, plays a vital role in the exploration and development of high-angle wells / horizontal wells, combines the advantages of MCMC sampling and multi-chain parallel MCMC sampling, and provides strong support for the subsequent application of azimuthal electromagnetic wave logging while drilling.

[0062] The present invention is further described in detail below with reference to the accompanying drawings and specific embodiments:

[0063] like Figure 1As shown, a method for azimuthal electromagnetic wave inversion while drilling based on single-chain-multi-chain MCMC adaptive selection includes the following steps:

[0064] s1. Establish Figure 2 The inversion model shown.

[0065] s2. Analyze the response characteristics of the inversion model established in s1 and determine the key parameters that need to be inverted: the formation resistivity required for inversion, denoted as R; the relative position between the instrument and the layer interface, denoted as DTB.

[0066] s3. Obtain observation data and construct the objective function.

[0067] s3.1. Preprocess the LWD azimuthal electromagnetic wave logging data and normalize the measured phase difference and amplitude ratio signals respectively to ensure the consistency of the magnitude of the logging response.

[0068] s3.2. The residual sum of squares between the observed data f(m) and the forward response d is used to construct the inversion objective function of azimuthal electromagnetic logging while drilling:

[0069]

[0070] In the formula, w d and w m They represent the weights corresponding to the logging response and the model parameters, λ is the regularization parameter, m ref is the reference value of the model parameter, m is the value of the model parameter to be inverted, and the second term λ It is a regularization term that reduces the impact of measurement noise on the inversion results and avoids matrix singularity and ill-conditioning problems in nonlinear inversion.

[0071] s4. Determine the initial model for single-chain MCMC sampling and determine the length and temperature of the sampling chain, and set the number of iterations.

[0072] s5. Define the range of parameters and set appropriate prior distribution based on the inversion model established in s1.

[0073] s6. Define model predictions and select an appropriate error model to construct the likelihood function.

[0074] s7. Construct the posterior distribution based on the prior distribution set in s5 and the likelihood function obtained in s6.

[0075] s8. Based on the sampling method and the characteristics of the objective function constructed in s3, the Metropolis-Hastings algorithm (MH algorithm for short) is selected for sampling.

[0076] s9. Select the initial value and start MCMC sampling using the MH algorithm. Each time, choose whether to accept the state at the next moment according to the acceptance rate function α.

[0077] s9.1. The initial value selected according to the inversion model established in s1 is used as m (0) state.

[0078] s9.2. Use the prior distribution and likelihood function established in s5-s7 to determine the posterior distribution.

[0079] s9.3. Sampling from the posterior distribution, solving m at future time k+1 The probability distribution of depends only on the solution m at the current moment k , which has nothing to do with the solution at past moments.

[0080] s9.4. Using the Metropolis-Hasting algorithm, the receiving rate function α is introduced, and the expression is as follows:

[0081]

[0082] Assuming that the transfer kernel function obeys Gaussian distribution, then Π(x, y) = Π(y, x), and the reception rate function can be further simplified as:

[0083]

[0084] s9.5. Calculate a(x,y) using the above formula and compare it with the random number μ that follows a uniform distribution. If μ≤a(x,y), then the receiving model solution m k+1 Otherwise, the model solution m is retained k .

[0085] s9.6. Repeat the above process until the set number of iterations is met.

[0086] s10. When the number of iterations meets the set value, sampling ends and the results of each iteration of each parameter are collected.

[0087] s11. Determine whether the MCMC inversion results meet expectations. If so, end the inversion; if not, use the results as the initial model for multi-chain parallel MCMC sampling.

[0088] s11.1. Select appropriate reference values ​​to determine whether the inversion results meet expectations.

[0089] s11.2. The noise level contained in the observed data is used as a reference value in s11.1 to determine whether the inversion results meet expectations. s11.3. The forward modeling results of the model parameters obtained by inversion are compared with the observed data, and the relative error δ is calculated:

[0090]

[0091] In the formula, δ is the relative error, d obsis the observed data, d sim For the forward data calculated using the inversion model, ||.|| represents the norm of the vector, and the L2 norm is usually chosen.

[0092] s11.4. If the relative error is less than the reference value, the inversion result is used as the final inversion value at that point and the inversion ends.

[0093] s11.5. If the relative error is greater than or equal to the reference value, the result is used as the initial model for multi-chain parallel MCMC sampling.

[0094] s12. Set up multi-chain parallel MCMC sampling, set the number of sampling chains and the temperature gradient of each chain, and repeat the same steps s6 to s8 as MCMC.

[0095] s12.1. Construct K Markov chains with increasing temperature, where the solutions of temperature chain l all satisfy the posterior distribution of different temperatures.

[0096] s12.2. A temperature parameter T is introduced into the likelihood function l Control the sampling step size:

[0097]

[0098] Where, T l is the temperature parameter of each chain, T min The lowest temperature chain is defined, usually at 1, T max The chain with the highest temperature is defined.

[0099] s13. Perform multi-chain parallel MCMC sampling on each chain, and choose whether to accept the next state each time according to the acceptance rate function α.

[0100] s14. After this iteration is completed, the acceptance probability is used to determine whether to interact between chains.

[0101] s14.1. After the sampling of the previous moment is completed, a random number ρ that obeys a uniform distribution is generated. If ρ is greater than the preset temperature exchange parameter θ (0≤θ≤1), there is no interaction between adjacent temperature chains.

[0102] s14.2. If ρ is less than or equal to the preset temperature exchange parameter θ (0≤θ≤1), then the state interaction function defined by formula (6) is used to determine whether the adjacent temperature chains are to interact with each other. Adjacent temperature chains interact with each other, and vice versa:

[0103]

[0104] In the formula, is the state transition probability, indicating that in the current state, two adjacent temperature chains T l and T l+1 The probability of state exchange, T l is the temperature of the lth temperature chain, T l+1 is the temperature of the l+1th temperature chain, is the energy difference function, indicating the current state of two adjacent temperature chains and The adjustment item for the corresponding target distribution energy difference.

[0105] s15. Repeat s13 to s14 until the predetermined number of iterations is reached, and the maximum a posteriori solution is obtained as the final inversion result. Figure 3 For example, at a certain point, the single-chain MCMC algorithm falls into a local minimum and is difficult to converge, but after sampling using the adaptive algorithm, it jumps out of the local minimum and seeks global optimization.

[0106] s16. Figure 4 In order to utilize MCMC inversion and adaptive MCMC two-dimensional curtain schematic, the inverted formation resistivity, relative position of the instrument and the layer interface and other data are statistically analyzed for subsequent work. Figure 4 The two-dimensional curtain diagram inverted by the single-chain MCMC is compared with the two-dimensional curtain diagram of the adaptive selection method. It can be seen that the inversion result of the adaptive selection algorithm is far better than that of the single-chain MCMC algorithm. Figure 5 The feasibility of the inversion method of the present invention is demonstrated by comparing the reference value with the result of curve reconstruction.

[0107] The parts not mentioned in the above methods can be realized by adopting or drawing on existing technologies.

[0108] The above-described embodiments are merely descriptions of preferred embodiments of the present invention and are not intended to limit the scope of the present invention. Without departing from the spirit of the present invention, various modifications and improvements made to the technical solutions of the present invention by ordinary technicians in this field should fall within the scope of protection determined by the claims of the present invention.

Claims

1. A method for azimuthal electromagnetic wave inversion while drilling based on single-chain-multi-chain MCMC adaptive selection, characterized in that The following steps are involved: s1. Establish a multi-parameter inversion model; s2. Analyze the response characteristics of the inversion model parameters and determine the parameters that need to be inverted; s3. Obtain observation data and construct the objective function; s4. Determine the initial model for single-chain MCMC sampling, determine the length and temperature of the sampling chain, and set the number of iterations; s5. Define the range of parameters and set the prior distribution based on the established initial model; s6. Select the error model to construct the likelihood function; s7. Construct the posterior distribution based on the prior distribution and likelihood function; s8. Select the Metropolis-Hastings algorithm for sampling; s9. Select the initial value and start MCMC sampling using the Metropolis-Hastings algorithm. Each time, choose whether to accept the next state according to the acceptance rate function α; s10. Sampling ends when the number of iterations meets the set value, and the results of each iteration of each parameter are collected; s11. Determine whether the MCMC inversion results meet expectations. If so, end the inversion. If not, use the results as the initial model for multi-chain parallel MCMC sampling. s12. Set up multi-chain parallel MCMC sampling, set the number of sampling chains and the temperature gradient of each chain, and repeat the same steps s6 to s8 as MCMC; s13. Perform multi-chain parallel MCMC sampling on each chain, and choose whether to accept the next state each time according to the acceptance rate function α; s14. After this iteration is completed, decide whether to interact between chains based on the acceptance probability; s15. Repeat s13 to s14 until a predetermined number of iterations is reached, and the maximum a posteriori solution is obtained as the final inversion result.

2. The method for azimuthal electromagnetic wave inversion while drilling based on single-chain-multi-chain MCMC adaptive selection according to claim 1, characterized in that: In step s2, the parameters to be inverted include the formation resistivity and the relative position between the instrument and the layer interface; accordingly, in step s15, the formation resistivity and the relative position between the instrument and the layer interface finally inverted are statistically analyzed for subsequent work.

3. The method for azimuthal electromagnetic wave inversion while drilling based on single-chain-multi-chain MCMC adaptive selection according to claim 1 is characterized in that The step s3 specifically comprises the following steps: s3.

1. Preprocess the azimuthal electromagnetic wave logging data while drilling, normalize the measured phase difference and amplitude ratio signals respectively, and ensure the consistency of the magnitude of the logging response; s3.

2. The residual sum of squares between the observed data f(m) and the forward response d is used to construct the inversion objective function of azimuthal electromagnetic logging while drilling: In the formula, and They represent the weights corresponding to the logging response and the model parameters, λ is the regularization parameter, m ref is the reference value of the model parameter, and m is the value of the model parameter to be inverted.

4. The method for azimuthal electromagnetic wave inversion while drilling based on single-chain-multi-chain MCMC adaptive selection according to claim 1 is characterized in that The step s9 specifically includes the following steps: s9.

1. The initial value selected according to the initial model setting is used as m (0) state; s9.

2. Sampling from the posterior distribution, solving m at future moments k+1 The probability distribution of depends only on the solution m at the current moment k , which has nothing to do with the solution at the past moment; s9.

3. Using the Metropolis-Hasting algorithm, the receiving rate function α is introduced, and the expression is as follows: Assuming that the transfer kernel function obeys Gaussian distribution, then Π(x,y)=Π(y,x), and the reception rate function is further simplified to: s9.

4. Calculate a(x,y) using the above formula and compare it with the random number μ that follows a uniform distribution. If μ < a(x,y), then accept the model solution m k+1 , otherwise retain the model solution m k ; s9.

5. Repeat the above process until the set number of iterations is met.

5. The method for azimuthal electromagnetic wave inversion while drilling based on single-chain-multi-chain MCMC adaptive selection according to claim 1 is characterized in that The step s11 specifically includes the following steps: s11.

1. Determine whether the inversion results meet expectations; use the noise level contained in the observation data as a reference value for whether the inversion results meet expectations; s11.

2. Compare the forward modeling results of the inverted model parameters with the observed data and calculate the relative error δ: In the formula, δ is the relative error, d obs is the observed data, d sim is the forward data calculated using the inversion model, ||.|| represents the norm of the vector; s11.

3. If the relative error is less than the reference value, the inversion result is used as the final inversion value at that point and the inversion ends; s11.

4. If the relative error is greater than the reference value, the result is used as the initial model for multi-chain parallel MCMC sampling.

6. The method for azimuthal electromagnetic wave inversion while drilling based on single-chain-multi-chain MCMC adaptive selection according to claim 1 is characterized in that The step s12 specifically includes the following steps: s12.

1. Construct K Markov chains with increasing temperatures, where the solutions of temperature chain l all satisfy the posterior distribution of different temperatures; s12.

2. A temperature parameter T is introduced into the likelihood function l Control the sampling step size: Where, T l is the temperature parameter of each chain, T min is defined as the chain with the lowest temperature, T max The chain with the highest temperature is defined.

7. The method for azimuthal electromagnetic wave inversion while drilling based on single-chain-multi-chain MCMC adaptive selection according to claim 1 is characterized in that The step s14 specifically includes the following steps: s14.

1. After the sampling of the previous moment is completed, a random number ρ that obeys a uniform distribution is generated. If ρ is greater than the preset temperature exchange parameter θ, the adjacent temperature chains do not interact; s14.

2. If ρ is less than the preset temperature exchange parameter θ, then the state interaction function defined by formula (6) is used to determine whether the adjacent temperature chains interact with each other. Adjacent temperature chains interact with each other, and vice versa:

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