# Full Information Numerical Simulation of Two-Dimensional Steady Ground Temperature Field
## 2 Principle of Methods
### Wavenumber Sampling Rules
#### 2.3.2 Inverse Transform Sampling Rule  
Inverse transform sampling is related to the distribution of the spectrum; the division is flexible, and it has a relationship with the spatial domain sampling. According to the sampling theorem, the maximum value of the wavenumber domain is  
\[k_{max}=\frac{\pi}{\Delta x_{min}}, \tag{13}\]  
\(k_{max}\), also known as cutoff frequency, sampling within the cutoff frequency can ensure that all spectrum information can be sampled. When the spectrum changes sharply, the maximum wavenumber needs to be reached, and when the spectrum changes slowly and the energy is relatively concentrated, the maximum wavenumber range can not be reached, thus reducing the number of sampling points to improve the calculation efficiency.  
Sampling in the wavenumber domain can be divided into equal interval sampling and non-equal interval sampling. According to the sampling theorem of discrete Fourier transform, when sampling at equal intervals, wavenumber is arranged as  
\[k_{j}=p\Delta k,j=1,2,\cdots,M. \tag{14}\]  
where \(k_{j}\) is the wavenumber arrangement, \(j\) is the serial number of the sampling point, \(\Delta k\) is the sampling interval, and \(M\) is the number of sampling points in the wavenumber domain.  
When the number of sampling points \(M\) is even, \(p=-\frac{M}{2},-\frac{M}{2}+1,\cdots,-1,0,1,\cdots,\)\(\frac{M}{2}\ -\ 1\); when the number of sample points \(M\) is odd, \(p=-\frac{M-1}{2},-\frac{M-1}{2}+1,\cdots,-1,0,1,\)\(\cdots,\frac{M-1}{2}\).  
In equal-interval sampling, two common strategies are segmented uniform sampling and logarithmic-interval sampling. Segmented uniform sampling is based on uniform division, in the spectrum changes sharply and energy strong several interval encryption, in the spectrum changes slowly interval sparse sampling. Logarithmic interval sampling is suitable for spectrum concentration and strong energy in the small wavenumber region, and the spectrum decays rapidly with the wavenumber energy or the large wavenumber energy concentration, and the energy decays rapidly when the wavenumber decreases.  
For logarithmic interval sampling, let the wavenumber selection range be \([-k_{max},k_{max}]\), the wavenumber domain sampling number is \(2M+1\), in the logarithmic domain sampling equal interval sampling, the sampling interval is  
\[\Delta k=\frac{lg(k_{max})-lg(k_{min})}{M}. \tag{15}\]where \(k_{min}\) is a decimal number, generally \(10^{-6}\sim 10^{-3}\).  
Wavenumber is arranged on \([-k_{max},0]\) as  
\[-k_{max},\ \sim-10^{g(k_{min})+(M-1)\Delta k}\ \sim-10^{g(k_{min})+(M-2)\Delta k} \sim\cdots\ \sim-10^{g(k_{min})+\Delta k}\ \sim 0. \tag{16}\]  
Wavenumber arranged on \([0,k_{max}]\) as  
\[0\ \sim 10^{Ig(k_{min})+\Delta k}\ \sim 10^{Ig(k_{min})+2\Delta k}\ \sim\ \cdots\ \sim 10^{Ig(k_{min})+(M-1)\Delta k}\ \sim k_{max}. \tag{17}\]  
For frequency spectrum changes that are slow and violent, the equal interval and non-equal interval can be combined for sampling. According to the logarithmic function change law, the sampling with fast energy change is encrypted, and the sampling where the energy transformation is slow is sparse, and the sampling is flexible, taking into account the calculation accuracy and efficiency.