Introduction
Non-asymptotic numerical simulation of transient heat transfer requires knowledge of initial conditions. If the heat transfer medium is soil, they typically use temperature logs (depth-temperature tables) of monitoring wells for the initial time point. Then one may either use scattered-data interpolation [1] or solve a steady heat transfer problem with Dirichlet boundary conditions at temperature measurement points. If you choose the second way and the problem is nonlinear, there might be some convergence issues.
Figure 1: Soil Temperature Interpolation in Frost 3D Universal
We use scattered-data interpolation [1] and our method provides a series of useful features, including the following:
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In case of single monitoring well, isothermal surfaces follow the terrain, or more precisely, its smoothed (Gaussian blurred [2]) versions. The blur radius is depth-proportional.
-
In the multi-well case, isothermal surfaces adapt to the terrain. In order to achieve this, we use a straightforward piecewise-linear triangulation-based interpolation [1] along isosurfaces of the smoothed depth field. The smoothed depths of soil points are initially computed as the differences between their smoothed (with Gaussian blur [2] of depth-proportional intensity) elevations and altitudes. Then, if needed, the smoothed depth field is locally corrected to become strictly decreasing with respect to altitude.
-
Globally, our interpolation is range-restricted. This guarantees that the resulting temperature won't go below the absolute zero. Also, if all measured temperatures are above (below) the freezing point, then the entire interpolation domain will be thawed (frozen).
-
Along the monitoring wells, the interpolation is shape-preserving, i.e. preserves monotonicity and convexity [3].
Statement of Problem
Let ![\Pi=[x_0,x_1]\times[y_0,y_1]\neq\emptyset](http://blog.simmakers.com/wp-content/ql-cache/quicklatex.com-621e331b22e5ed3435d8bf38c70ff2ee_l3.png)
and let 
be the elevation map determining the part 
of the surface 
of the infinitely extended and infinitely deep soil 
.
We investigate the problem of interpolating the unknown temperature field 
of points) from the temperature logs
![\[\left[\begin{matrix}T\left(A_i,\mathrm{z}_{i1}\right) \\ T\left(A_i,\mathrm{z}_{i2}\right) \\ \vdots \\ T\left(A_i,\mathrm{z}_{im_i}\right) \end{matrix}\right]=\left[\begin{matrix}T_{i1} \\ T_{i2} \\ \vdots \\ T_{im_i} \end{matrix}\right]\!\!,~i=1,2,\ldots,n,\]](http://blog.simmakers.com/wp-content/ql-cache/quicklatex.com-898a3b89c0adea45482ccc5b0b44b50e_l3.png)
where 
are altitudes of the measurement points 
in the 
th monitoring well ![W_i=\left\{A_i\right\}\times\left(-\infty,\hbar(A_i)\right]](http://blog.simmakers.com/wp-content/ql-cache/quicklatex.com-8ae00fafde2964d9da97222a300cdb75_l3.png)
, 
, 
. Here 
is the "reasonable" continuation (see stage 3, below) of the function to the entire plane 
. The exact definitions of the infinitely extended and deep soil 
and its surface 
are the following:
![\[S=\bigcup\limits_{A\in\mathbb{R}^2}\{A\}\times\left(-\infty,\hbar(A)\right],~\partial S=\left\{\left.\left(A,h(A)\right)\right|A\in\mathbb{R}^2\right\}.\]](http://blog.simmakers.com/wp-content/ql-cache/quicklatex.com-c98e8961d8946ebc7bc2017c86548662_l3.png)
Note that 
.
Algorithm
One may cope with the given interpolation problem in several stages. Let's list them.
-
Merge the duplicate wells and remove the empty (containing no measurement points) ones. If there are no wells now, stop. Renumber the remaining wells, redefine
![n]()
. -
Merge the duplicate measurement points (for multiple temperature values, use their arithmetic mean as the new temperature). Renumber the updated measurement points in
![W_i]()
and redefine ![m_i]()
, ![i=1,2,\ldots,n]()
. -
Construct such a function
![\hbar:\mathbb{R}^2\rightarrow\mathrm{Conv}\left(h(\Pi)\right)]()
that for all points ![A\in\mathbb{R}^2]()
, one has![\[\hbar(A)=h(A_\Pi),\]]()
where
![A_\Pi]()
is the closest to ![A\in\mathbb{R}^2]()
point of ![\Pi]()
. In particular, if ![A\in\Pi]()
, then ![A_\Pi=A]()
and ![\hbar(A)=h(A)]()
. -
Construct the depth field
![d:S\rightarrow\left[0,\infty\right)]()
,![\[d(A,\mathrm{z})\equiv\hbar(A)-\mathrm{z}.\]]()
Obviously,
![d\left(\partial S\right)=\{0\}]()
. -
For each nonempty 2-dimensional soil layer
![S_\mathrm{z}=\left\{\left.A\in\mathbb{R}^2\right|(A,\mathrm{z})\in S\right\}]()
( , obtain a smoothed (with depth-proportional intensity) version![\left(\mathbb{R}^2\times\{\mathrm{z}\}\right)\cap S\neq\emptyset]()
)![\tilde{\hbar}_\mathrm{z}:S_\mathrm{z}\rightarrow\mathrm{Conv}(h(\Pi))]()
,![\[\tilde{\hbar}_\mathrm{z}=B_{\lambda d(A,\mathrm{z})}\hbar\]]()
of the extended elevation field
![\hbar]()
. Here![\[\left(B_{\sigma}f\right)(A)\equiv\begin{cases} \frac{1}{2\pi\sigma^2}\displaystyle{\int\limits_{\mathbb{R}^2}}e^{-\frac{\left(|X-A|/\sigma\right)^2}{2}}f(X)dX, & \sigma>0; \\ f(A), & \sigma=0;\end{cases}\]]()
![B_\sigma]()
is Gaussian blur [2] with the standard deviation (radius) ![\sigma\geq0]()
. We take![\[\sigma=\lambda d\left(A,\mathrm{z}\right),~\left(A,\mathrm{z}\right)\in S,\]]()
where the physically dimensionless constant
![\lambda>0]()
doesn't depend on ![\left(A,\mathrm{z}\right)]()
and is derived from empirical considerations.Note that for all
![\left(A,\mathrm{z}\right)\in\partial S]()
![\[\tilde{\hbar}_\mathrm{z}(A)=(B_0\hbar)(A)=\hbar(A).\]]()
-
Construct the smoothed depth field
![\tilde{d}:S\rightarrow\mathbb{R}]()
,![\[\tilde{d}\left(A,\mathrm{z}\right)\equiv\hbar_\mathrm{z}(A)-\mathrm{z}.\]]()
For all
![\left(A,\mathrm{z}\right)\in\partial S]()
![\[\tilde{d}\left(A,\mathrm{z}\right)=\tilde{\hbar}_\mathrm{z}(A)-\hbar(A)=\hbar(A)-\hbar(A)=0,\]]()
i.e.
![\tilde{d}\left(\partial S\right)=\{0\}]()
, just as for ![d]()
. Where needed, correct the field ![\tilde{d}]()
in such a way that the strict decreasing of ![\tilde d\left(A,\mathrm{z}\right)]()
with respect to ![\mathrm{z}\in\left(-\infty,\hbar(A)\right]]()
would hold for all ![A\in\mathbb{R}^2]()
, but ![\tilde{d}\left(\partial S\right)=\{0\}]()
would remain true. From here we obtain that ![\tilde{d}\geq0]()
. -
Compute the smoothed depths
![\[\tilde{d}_{ij}=\tilde{d}\left(A_i,\mathrm{z}_{ij}\right)\geq0,~j=1,2,\ldots,m_i,~i=1,2,\ldots,n\]]()
of the measurement points. These depths are pairwise different within the monitoring well
![W_i]()
due to the strict monotonicity of ![\tilde{d}\left(A_i,\mathrm{z}\right)]()
with respect to ![\mathrm{z}\in\left(-\infty,\hbar(A_i)\right]]()
, ![i=1,2,\ldots,n]()
.Construct such linear interpolation (for extrapolation, the nearest neighbor method is used) splines
![T_i:\left[0,\infty\right)\rightarrow\mathrm{Conv}\left\{T_{i1},T_{i2,},\ldots,T_{im_i}\right\}]()
that![\[\left[\begin{matrix}T_i\left(\tilde{d}_{i1}) \\ T_i\left(\tilde{d}_{i2}) \\ \vdots \\ T_i\left(\tilde{d}_{im_i}) \end{matrix}\right]=\left[\begin{matrix} T_{i1} \\ T_{i2} \\ \vdots \\ T_{im_i} \end{matrix}\right]\!\!,~i=1,2,\ldots,n.\]]()
-
If
![n=1]()
, put![\[T\left(A,\mathrm{z}\right)\equiv T_1\left(\tilde{d}\left(A,\mathrm{z}\right)\right)\]]()
and stop.
-
Construct the convex hull
![C=\mathrm{Conv}\left\{A_1,A_2,\ldots,A_n\right\}]()
and the Delaunay triangulation of the same set of points ![\left\{A_1,A_2,\ldots,A_n\right\}]()
. -
In each fixed point
![\left(A^*,\mathrm{z}^*\right)\in S]()
(let's denote its smoothed depth ![\tilde{d}\left(A^*,\mathrm{z}^*\right)]()
with ![\tilde{d}^*]()
)![T^*]()
using the following algorithm. Let ![A_C^*]()
be the closest to ![A^*]()
point of the convex hull ![C]()
. In particular, if ![A^*\in C]()
, then ![A_C^*=A^*]()
. If the point ![A_C^*]()
lies in some non-degenerate Delaunay triangle ![A_iA_jA_k]()
, ![i,j,k\in\left\{1,2,\ldots,n\right\}]()
and therefore there exist such uniquely determined scalars (barycentric coordinates) ![\alpha,\beta,\gamma\geq0]()
that ![A_C^*=\alpha A_i+\beta A_j+\gamma A_k]()
and ![\alpha+\beta+\gamma=1]()
, then put![\[T^*=\alpha T_i\left(\tilde{d^*}\right)+\beta T_j\left(\tilde{d^*}\right)+\gamma T_k\left(\tilde{d^*}\right).\]]()
There's also the case when the Delaunay triangulation is empty (contains no non-degenerate triangles). In this situation (keep in mind that
![n\geq2]()
), the point ![A_C^*]()
lies on some non-degenerate segment ![A_iA_j]()
( , i.e.![i,j\in\{1,2,\ldots,n\}]()
)![A_C^*=\alpha A_i+\beta A_j]()
, ![\alpha+\beta=1]()
, where the choice of the scalars ![\alpha,\beta\geq0]()
is unique. Put![\[T^*=\alpha T_i\left(\tilde{d^*}\right)+\beta T_j\left(\tilde{d^*}\right).\]]()
.
and redefine 
, 
, one has![\[\hbar(A)=h(A_\Pi),\]](http://blog.simmakers.com/wp-content/ql-cache/quicklatex.com-d824673581594a9337db8850eebb94a3_l3.png)
is the closest to 
. In particular, if 
, then 
and 
.
,![\[d(A,\mathrm{z})\equiv\hbar(A)-\mathrm{z}.\]](http://blog.simmakers.com/wp-content/ql-cache/quicklatex.com-a403d8eb9dcac62068b9469ae6d0fbc9_l3.png)
.
)
,![\[\tilde{\hbar}_\mathrm{z}=B_{\lambda d(A,\mathrm{z})}\hbar\]](http://blog.simmakers.com/wp-content/ql-cache/quicklatex.com-b8db104937edb4722dca00817deaf376_l3.png)
. Here![\[\left(B_{\sigma}f\right)(A)\equiv\begin{cases} \frac{1}{2\pi\sigma^2}\displaystyle{\int\limits_{\mathbb{R}^2}}e^{-\frac{\left(|X-A|/\sigma\right)^2}{2}}f(X)dX, & \sigma>0; \\ f(A), & \sigma=0;\end{cases}\]](http://blog.simmakers.com/wp-content/ql-cache/quicklatex.com-c34cc4c9457da875afa259cc3e76e4d7_l3.png)
is Gaussian blur [2] with the standard deviation (radius) 
. We take![\[\sigma=\lambda d\left(A,\mathrm{z}\right),~\left(A,\mathrm{z}\right)\in S,\]](http://blog.simmakers.com/wp-content/ql-cache/quicklatex.com-1016ae9583a1da372cece93886ee65fc_l3.png)
doesn't depend on 
and is derived from empirical considerations.
![\[\tilde{\hbar}_\mathrm{z}(A)=(B_0\hbar)(A)=\hbar(A).\]](http://blog.simmakers.com/wp-content/ql-cache/quicklatex.com-1a8076ae24c83839e56dd91234878677_l3.png)
,![\[\tilde{d}\left(A,\mathrm{z}\right)\equiv\hbar_\mathrm{z}(A)-\mathrm{z}.\]](http://blog.simmakers.com/wp-content/ql-cache/quicklatex.com-8fbc420f37e203577164f99a867cbbb3_l3.png)
![\[\tilde{d}\left(A,\mathrm{z}\right)=\tilde{\hbar}_\mathrm{z}(A)-\hbar(A)=\hbar(A)-\hbar(A)=0,\]](http://blog.simmakers.com/wp-content/ql-cache/quicklatex.com-6c3fb1c8d87c286da50fd84e3511a43d_l3.png)
, just as for 
in such a way that the strict decreasing of 
with respect to ![\mathrm{z}\in\left(-\infty,\hbar(A)\right]](http://blog.simmakers.com/wp-content/ql-cache/quicklatex.com-f9522c07e7c713e35e64b68abec9f2c2_l3.png)
would hold for all 
.![\[\tilde{d}_{ij}=\tilde{d}\left(A_i,\mathrm{z}_{ij}\right)\geq0,~j=1,2,\ldots,m_i,~i=1,2,\ldots,n\]](http://blog.simmakers.com/wp-content/ql-cache/quicklatex.com-888cbc87e8a95bf9c6660b7723606e9c_l3.png)
with respect to ![\mathrm{z}\in\left(-\infty,\hbar(A_i)\right]](http://blog.simmakers.com/wp-content/ql-cache/quicklatex.com-4ec8fa7fec4b8f862f2133ca058d591b_l3.png)
, 
that![\[\left[\begin{matrix}T_i\left(\tilde{d}_{i1}) \\ T_i\left(\tilde{d}_{i2}) \\ \vdots \\ T_i\left(\tilde{d}_{im_i}) \end{matrix}\right]=\left[\begin{matrix} T_{i1} \\ T_{i2} \\ \vdots \\ T_{im_i} \end{matrix}\right]\!\!,~i=1,2,\ldots,n.\]](http://blog.simmakers.com/wp-content/ql-cache/quicklatex.com-6c8c50aff7964ac2ccc91455e034f3c0_l3.png)
, put![\[T\left(A,\mathrm{z}\right)\equiv T_1\left(\tilde{d}\left(A,\mathrm{z}\right)\right)\]](http://blog.simmakers.com/wp-content/ql-cache/quicklatex.com-4e64564e211f6c833d147b3e0537a60f_l3.png)
and the Delaunay triangulation of the same set of points 
.
(let's denote its smoothed depth 
with 
)
using the following algorithm. Let 
be the closest to 
point of the convex hull 
. In particular, if 
, then 
. If the point 
, 
and therefore there exist such uniquely determined scalars (barycentric coordinates) 
that 
and 
, then put![\[T^*=\alpha T_i\left(\tilde{d^*}\right)+\beta T_j\left(\tilde{d^*}\right)+\gamma T_k\left(\tilde{d^*}\right).\]](http://blog.simmakers.com/wp-content/ql-cache/quicklatex.com-52c7664f577093330866b8ffed0fdda6_l3.png)
), the point 
)
, 
, where the choice of the scalars 
is unique. Put![\[T^*=\alpha T_i\left(\tilde{d^*}\right)+\beta T_j\left(\tilde{d^*}\right).\]](http://blog.simmakers.com/wp-content/ql-cache/quicklatex.com-42aed6f08ed4d46bd8820b94b4857ede_l3.png)
It is clear from construction that isosurfaces of the resulting temperature field 
adapt to the terrain, but, as the depth 
increases, this adaptation weakens monotonically.
References
-
I. Amidror. Scattered data interpolation methods for electronic imaging systems: a survey. J. Electron. Imaging, 11(2), 157–176 (April 2002).
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Wikipedia contributors. Gaussian blur. Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Gaussian_blur&oldid=672686407 (accessed August 3, 2015).
-
F. N. Fritsch and R. E. Carlson. Monotone Piecewise Cubic Interpolation. SIAM J. Numer. Anal., 17, 238–246 (1980).
I. Amidror. Scattered data interpolation methods for electronic imaging systems: a survey. J. Electron. Imaging, 11(2), 157–176 (April 2002).
Wikipedia contributors. Gaussian blur. Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Gaussian_blur&oldid=672686407 (accessed August 3, 2015).
F. N. Fritsch and R. E. Carlson. Monotone Piecewise Cubic Interpolation. SIAM J. Numer. Anal., 17, 238–246 (1980).