This document provides solutions for semivariogram models #1 through #9/For each model, we identify:
The effective range is defined as the distance \(d\) at which the semivariogram reaches 95% of the total sill \((\tau^2 + \sigma^2)\):
\[ \gamma(d_{eff}) = 0.95(\tau^2 + \sigma^2) \]
\[ \gamma(d) = \begin{cases} \tau^2 + \sigma^2 d & d > 0, \tau^2 > 0, \sigma^2 > 0 \\ 0 & d = 0 \end{cases} \]
| Parameter | Value | Explanation |
|---|---|---|
| Nugget | \(\tau^2\) | \(\lim_{d \to 0^+} \gamma(d) = \tau^2 + \sigma^2(0) = \tau^2\) |
| Partial Sill | Does not exist | No finite sill - grows linearly without bound |
| Total Sill | Does not exist | \(\lim_{d \to \infty} \gamma(d) = \infty\) |
| Range | Does not exist | Never reaches a sill |
| Effective Range | Does not exist | No sill to define percentage of |
\[ \boxed{\text{The covariance function } C(d) \text{ does NOT exist for the linear model.}} \]
The model is intrinsic (not stationary), so it has no valid covariance function.
\[ \gamma(d) = \begin{cases} \tau^2 + \sigma^2 \left[ \frac{3}{2}\phi d - \frac{1}{2}(\phi d)^3 \right] & 0 < d \le 1/\phi \\ \tau^2 + \sigma^2 & d \ge 1/\phi \\ 0 & d = 0 \end{cases} \]
| Parameter | Value |
|---|---|
| Nugget | \(\tau^2\) |
| Partial Sill | \(\sigma^2\) |
| Total Sill | \(\tau^2 + \sigma^2\) |
| Range | \(1/\phi\) |
| Effective Range | \(1/\phi\) (same as range) |
For \(d = 0\): \[ C(0) = \tau^2 + \sigma^2 \]
For \(0 < d \le 1/\phi\): \[ C(d) = \sigma^2 \left[ 1 - \frac{3}{2}\phi d + \frac{1}{2}(\phi d)^3 \right] \]
For \(d \ge 1/\phi\): \[ C(d) = 0 \]
Thus: \[ \boxed{ C(d) = \begin{cases} \tau^2 + \sigma^2 & d = 0 \\ \sigma^2 \left[ 1 - \frac{3}{2}\phi d + \frac{1}{2}(\phi d)^3 \right] & 0 < d \le 1/\phi \\ 0 & d \ge 1/\phi \end{cases} } \]
For the spherical model, the range is exact (not asymptotic). The model reaches the sill exactly at \(d = 1/\phi\):
At \(d = 1/\phi\): \[ \gamma(1/\phi) = \tau^2 + \sigma^2 \left[ \frac{3}{2}(1) - \frac{1}{2}(1)^3 \right] = \tau^2 + \sigma^2 \left[ \frac{3}{2} - \frac{1}{2} \right] = \tau^2 + \sigma^2 \]
So the range is \(1/\phi\), and since it reaches the sill exactly, the effective range equals the range.
\[ \gamma(d) = \begin{cases} \tau^2 + \sigma^2 \left(1 - \exp(-\phi d)\right) & d > 0 \\ 0 & d = 0 \end{cases} \]
| Parameter | Value |
|---|---|
| Nugget | \(\tau^2\) |
| Partial Sill | \(\sigma^2\) |
| Total Sill | \(\tau^2 + \sigma^2\) |
| Range | Does not exist (asymptotic) |
| Effective Range | \(3/\phi\) |
For \(d = 0\): \[ C(0) = \tau^2 + \sigma^2 \]
For \(d > 0\): \[ C(d) = \sigma^2 e^{-\phi d} \]
Thus: \[ \boxed{ C(d) = \begin{cases} \tau^2 + \sigma^2 & d = 0 \\ \sigma^2 e^{-\phi d} & d > 0 \end{cases} } \]
We want \(\gamma(d) = 0.95(\tau^2 + \sigma^2)\):
\[ \tau^2 + \sigma^2(1 - e^{-\phi d}) = 0.95(\tau^2 + \sigma^2) \]
Subtract \(\tau^2\) from both sides:
\[ \sigma^2(1 - e^{-\phi d}) = 0.95\sigma^2 \]
Divide by \(\sigma^2\):
\[ 1 - e^{-\phi d} = 0.95 \]
Solve for \(e^{-\phi d}\):
\[ e^{-\phi d} = 0.05 \]
Take the natural logarithm:
\[ -\phi d = \ln(0.05) \]
Solve for \(d\):
\[ d = -\frac{\ln(0.05)}{\phi} \]
Since \(-\ln(0.05) \approx 2.9957 \approx 3\):
\[ \boxed{d_{eff} \approx \frac{3}{\phi}} \]
\[ \gamma(d) = \begin{cases} \tau^2 + \sigma^2 \left(1 - \exp(-\phi^2 d^2)\right) & d > 0 \\ 0 & d = 0 \end{cases} \]
| Parameter | Value |
|---|---|
| Nugget | \(\tau^2\) |
| Partial Sill | \(\sigma^2\) |
| Total Sill | \(\tau^2 + \sigma^2\) |
| Range | Does not exist (asymptotic) |
| Effective Range | \(\sqrt{3}/\phi\) |
For \(d = 0\): \[ C(0) = \tau^2 + \sigma^2 \]
For \(d > 0\): \[ C(d) = \sigma^2 e^{-\phi^2 d^2} \]
Thus: \[ \boxed{ C(d) = \begin{cases} \tau^2 + \sigma^2 & d = 0 \\ \sigma^2 e^{-\phi^2 d^2} & d > 0 \end{cases} } \]
We want \(\gamma(d) = 0.95(\tau^2 + \sigma^2)\):
\[ \tau^2 + \sigma^2(1 - e^{-\phi^2 d^2}) = 0.95(\tau^2 + \sigma^2) \]
Subtract \(\tau^2\) from both sides:
\[ \sigma^2(1 - e^{-\phi^2 d^2}) = 0.95\sigma^2 \]
Divide by \(\sigma^2\):
\[ 1 - e^{-\phi^2 d^2} = 0.95 \]
Solve for \(e^{-\phi^2 d^2}\):
\[ e^{-\phi^2 d^2} = 0.05 \]
Take the natural logarithm:
\[ -\phi^2 d^2 = \ln(0.05) \]
Solve for \(d\):
\[ d^2 = -\frac{\ln(0.05)}{\phi^2} \]
\[ d = \frac{\sqrt{-\ln(0.05)}}{\phi} \]
Since \(-\ln(0.05) \approx 2.9957 \approx 3\):
\[ \boxed{d_{eff} \approx \frac{\sqrt{3}}{\phi}} \]
\[ \gamma(d) = \begin{cases} \tau^2 + \sigma^2 \left(1 - \exp(-|\phi d|^p)\right) & d > 0 \\ 0 & d = 0 \end{cases} \]
where \(0 < p \le 2\).
| Parameter | Value |
|---|---|
| Nugget | \(\tau^2\) |
| Partial Sill | \(\sigma^2\) |
| Total Sill | \(\tau^2 + \sigma^2\) |
| Range | Does not exist (asymptotic) |
| Effective Range | \([-\ln(0.05)]^{1/p}/\phi \approx 3^{1/p}/\phi\) |
For \(d = 0\): \[ C(0) = \tau^2 + \sigma^2 \]
For \(d > 0\): \[ C(d) = \sigma^2 e^{-|\phi d|^p} \]
Thus: \[ \boxed{ C(d) = \begin{cases} \tau^2 + \sigma^2 & d = 0 \\ \sigma^2 e^{-|\phi d|^p} & d > 0 \end{cases} } \]
We want \(\gamma(d) = 0.95(\tau^2 + \sigma^2)\):
\[ \tau^2 + \sigma^2(1 - e^{-|\phi d|^p}) = 0.95(\tau^2 + \sigma^2) \]
Subtract \(\tau^2\) from both sides:
\[ \sigma^2(1 - e^{-|\phi d|^p}) = 0.95\sigma^2 \]
Divide by \(\sigma^2\):
\[ 1 - e^{-|\phi d|^p} = 0.95 \]
Solve for \(e^{-|\phi d|^p}\):
\[ e^{-|\phi d|^p} = 0.05 \]
Take the natural logarithm:
\[ -|\phi d|^p = \ln(0.05) \]
Multiply by -1:
\[ |\phi d|^p = -\ln(0.05) \]
Take the \(p\)-th root:
\[ |\phi d| = [-\ln(0.05)]^{1/p} \]
Since \(d > 0\) and \(\phi > 0\):
\[ \boxed{d_{eff} = \frac{[-\ln(0.05)]^{1/p}}{\phi}} \]
Since \(-\ln(0.05) \approx 2.9957 \approx 3\):
\[ \boxed{d_{eff} \approx \frac{3^{1/p}}{\phi}} \]
| \(p\) Value | Model | Effective Range |
|---|---|---|
| \(p = 1\) | Exponential | \(3/\phi\) |
| \(p = 2\) | Gaussian | \(\sqrt{3}/\phi\) |
\[ \gamma(d) = \begin{cases} \tau^2 + \sigma^2 \left(\frac{d^2}{\phi + d^2}\right) & d > 0 \\ 0 & d = 0 \end{cases} \]
| Parameter | Value |
|---|---|
| Nugget | \(\tau^2\) |
| Partial Sill | \(\sigma^2\) |
| Total Sill | \(\tau^2 + \sigma^2\) |
| Range | Does not exist (asymptotic) |
| Effective Range | \(\sqrt{19\phi}\) |
For \(d = 0\): \[ C(0) = \tau^2 + \sigma^2 \]
For \(d > 0\): \[ C(d) = \sigma^2 \left(\frac{\phi}{\phi + d^2}\right) \]
Thus: \[ \boxed{ C(d) = \begin{cases} \tau^2 + \sigma^2 & d = 0 \\ \sigma^2 \left(\frac{\phi}{\phi + d^2}\right) & d > 0 \end{cases} } \]
We want \(\gamma(d) = 0.95(\tau^2 + \sigma^2)\):
\[ \tau^2 + \sigma^2\left(\frac{d^2}{\phi + d^2}\right) = 0.95(\tau^2 + \sigma^2) \]
Subtract \(\tau^2\) from both sides:
\[ \sigma^2\left(\frac{d^2}{\phi + d^2}\right) = 0.95\sigma^2 \]
Divide by \(\sigma^2\):
\[ \frac{d^2}{\phi + d^2} = 0.95 \]
Multiply both sides by \((\phi + d^2)\):
\[ d^2 = 0.95\phi + 0.95d^2 \]
Subtract \(0.95d^2\) from both sides:
\[ d^2 - 0.95d^2 = 0.95\phi \]
\[ 0.05d^2 = 0.95\phi \]
Solve for \(d^2\):
\[ d^2 = \frac{0.95}{0.05}\phi = 19\phi \]
Take the square root:
\[ \boxed{d_{eff} = \sqrt{19\phi}} \]
\[ \gamma(d) = \begin{cases} \tau^2 + \sigma^2 \left(1 - \frac{\sin(\phi d)}{\phi d}\right) & d > 0 \\ 0 & d = 0 \end{cases} \]
| Parameter | Value |
|---|---|
| Nugget | \(\tau^2\) |
| Partial Sill | \(\sigma^2\) |
| Total Sill | \(\tau^2 + \sigma^2\) |
| Range | Does not exist (asymptotic with oscillations) |
| Effective Range | Not well-defined due to oscillations |
For \(d = 0\): \[ C(0) = \tau^2 + \sigma^2 \]
For \(d > 0\): \[ C(d) = \sigma^2 \frac{\sin(\phi d)}{\phi d} \]
Thus: \[ \boxed{ C(d) = \begin{cases} \tau^2 + \sigma^2 & d = 0 \\ \sigma^2 \frac{\sin(\phi d)}{\phi d} & d > 0 \end{cases} } \]
For the wave model, the effective range is not well-defined because:
If we attempt to solve for 95% of the sill:
\[ \tau^2 + \sigma^2\left(1 - \frac{\sin(\phi d)}{\phi d}\right) = 0.95(\tau^2 + \sigma^2) \]
This simplifies to:
\[ \frac{\sin(\phi d)}{\phi d} = 0.05 \]
This transcendental equation has multiple solutions due to the oscillatory nature of the sine function. Therefore, there is no single effective range.
| Distance | \(C(d)\) | \(\gamma(d)\) |
|---|---|---|
| \(d = \pi/\phi\) | \(0\) | \(\tau^2 + \sigma^2\) |
| \(d = 3\pi/(2\phi)\) | Negative | \(> \tau^2 + \sigma^2\) |
| \(d = 2\pi/\phi\) | \(0\) | \(\tau^2 + \sigma^2\) |
\[ \gamma(d) = \begin{cases} \tau^2 + \sigma^2 d^\lambda & d > 0 \\ 0 & d = 0 \end{cases} \]
where \(0 < \lambda < 2\).
| Parameter | Value |
|---|---|
| Nugget | \(\tau^2\) |
| Partial Sill | Does not exist |
| Total Sill | Does not exist (unbounded) |
| Range | Does not exist (unbounded) |
| Effective Range | Does not exist |
\[ \boxed{\text{The covariance function } C(d) \text{ does NOT exist for the power law model.}} \]
The model is intrinsic (not stationary), so it has no valid covariance function.
The power law model has no sill because:
\[ \lim_{d \to \infty} \gamma(d) = \lim_{d \to \infty} (\tau^2 + \sigma^2 d^\lambda) = \infty \]
Since there is no finite sill, we cannot define a distance at which the model reaches 95% of the sill.
| \(\lambda\) Value | Model Name |
|---|---|
| \(\lambda = 1\) | Linear (Model #1) |
| \(0 < \lambda < 1\) | Sub-linear |
| \(1 < \lambda < 2\) | Super-linear |
\[ \gamma(d) = \begin{cases} \tau^2 + \sigma^2 \left[1 - \frac{(2\sqrt{\nu}d\phi)^\nu}{2^{\nu-1}\Gamma(\nu)} K_\nu(2\sqrt{\nu}d\phi)\right] & d > 0 \\ 0 & d = 0 \end{cases} \]
where: - \(\nu > 0\) is the smoothness parameter - \(K_\nu(\cdot)\) is the modified Bessel function of the second kind - \(\Gamma(\nu)\) is the Gamma function
| Parameter | Value |
|---|---|
| Nugget | \(\tau^2\) |
| Partial Sill | \(\sigma^2\) |
| Total Sill | \(\tau^2 + \sigma^2\) |
| Range | Does not exist (asymptotic) |
| Effective Range | Depends on \(\nu\); approximately \(\sqrt{8\nu}/\phi\) |
For \(d = 0\): \[ C(0) = \tau^2 + \sigma^2 \]
For \(d > 0\): \[ C(d) = \sigma^2 \frac{(2\sqrt{\nu}d\phi)^\nu}{2^{\nu-1}\Gamma(\nu)} K_\nu(2\sqrt{\nu}d\phi) \]
Thus: \[ \boxed{ C(d) = \begin{cases} \tau^2 + \sigma^2 & d = 0 \\ \sigma^2 \frac{(2\sqrt{\nu}d\phi)^\nu}{2^{\nu-1}\Gamma(\nu)} K_\nu(2\sqrt{\nu}d\phi) & d > 0 \end{cases} } \]
For the Matérn model, the effective range is the distance where:
\[ \gamma(d) = 0.95(\tau^2 + \sigma^2) \]
This means:
\[ \frac{(2\sqrt{\nu}d\phi)^\nu}{2^{\nu-1}\Gamma(\nu)} K_\nu(2\sqrt{\nu}d\phi) = 0.05 \]
This equation cannot be solved analytically due to the Bessel function. Instead, we use numerical approximations.
A widely used approximation for the Matérn effective range is:
\[ \boxed{d_{eff} \approx \frac{\sqrt{8\nu}}{\phi}} \]
This approximation works well for moderate to large \(\nu\).
For different values of \(\nu\), the effective range is:
| \(\nu\) | Effective Range (approx) |
|---|---|
| 0.5 | \(3/\phi\) |
| 1.0 | \(3.37/\phi\) |
| 1.5 | \(3.67/\phi\) |
| 2.0 | \(3.92/\phi\) |
| 3.0 | \(4.32/\phi\) |
| 5.0 | \(4.90/\phi\) |
| 10.0 | \(5.62/\phi\) |
| \(\infty\) | \(\sqrt{3}/\phi\) |
| \(\nu\) Value | Model Name | Effective Range |
|---|---|---|
| \(\nu = 0.5\) | Exponential | \(3/\phi\) |
| \(\nu = 1\) | Whittle | \(3.37/\phi\) |
| \(\nu \to \infty\) | Gaussian | \(\sqrt{3}/\phi\) |
| Model | Nugget | Partial Sill | Total Sill | Range | Effective Range | Covariance \(C(d)\) |
|---|---|---|---|---|---|---|
| #1 Linear | \(\tau^2\) | None | None | None | None | Does not exist |
| #2 Spherical | \(\tau^2\) | \(\sigma^2\) | \(\tau^2 + \sigma^2\) | \(1/\phi\) | \(1/\phi\) | \(\sigma^2[1 - \frac{3}{2}\phi d + \frac{1}{2}(\phi d)^3]\) |
| #3 Exponential | \(\tau^2\) | \(\sigma^2\) | \(\tau^2 + \sigma^2\) | None | \(3/\phi\) | \(\sigma^2 e^{-\phi d}\) |
| #4 Gaussian | \(\tau^2\) | \(\sigma^2\) | \(\tau^2 + \sigma^2\) | None | \(\sqrt{3}/\phi\) | \(\sigma^2 e^{-\phi^2 d^2}\) |
| #5 Power Exponential | \(\tau^2\) | \(\sigma^2\) | \(\tau^2 + \sigma^2\) | None | \([-\ln(0.05)]^{1/p}/\phi\) | \(\sigma^2 e^{-|\phi d|^p}\) |
| #6 Rational Quadratic | \(\tau^2\) | \(\sigma^2\) | \(\tau^2 + \sigma^2\) | None | \(\sqrt{19\phi}\) | \(\sigma^2 \phi/(\phi + d^2)\) |
| #7 Wave | \(\tau^2\) | \(\sigma^2\) | \(\tau^2 + \sigma^2\) | None | Not defined | \(\sigma^2 \sin(\phi d)/(\phi d)\) |
| #8 Power Law | \(\tau^2\) | None | None | None | None | Does not exist |
| #9 Matérn | \(\tau^2\) | \(\sigma^2\) | \(\tau^2 + \sigma^2\) | None | Depends on \(\nu\) | \(\sigma^2 \frac{(2\sqrt{\nu}d\phi)^\nu}{2^{\nu-1}\Gamma(\nu)} K_\nu(2\sqrt{\nu}d\phi)\) |
| Type | Models | Covariance Exists? |
|---|---|---|
| Stationary (bounded) | #2 Spherical, #3 Exponential, #4 Gaussian, #5 Power Exponential, #6 Rational Quadratic, #7 Wave, #9 Matérn | ✅ Yes |
| Intrinsic (unbounded) | #1 Linear, #8 Power Law | ❌ No |
| Model | Equation to Solve | Effective Range |
|---|---|---|
| Spherical | Exact at sill | \(1/\phi\) |
| Exponential | \(1 - e^{-\phi d} = 0.95\) | \(3/\phi\) |
| Gaussian | \(1 - e^{-\phi^2 d^2} = 0.95\) | \(\sqrt{3}/\phi\) |
| Power Exponential | \(1 - e^{-|\phi d|^p} = 0.95\) | \([-\ln(0.05)]^{1/p}/\phi\) |
| Rational Quadratic | \(\frac{d^2}{\phi + d^2} = 0.95\) | \(\sqrt{19\phi}\) |
| Wave | \(\frac{\sin(\phi d)}{\phi d} = 0.05\) | Not well-defined |
| Matérn | \(\frac{(2\sqrt{\nu}d\phi)^\nu}{2^{\nu-1}\Gamma(\nu)} K_\nu(2\sqrt{\nu}d\phi) = 0.05\) | Numerically determined |
| Linear | No sill | Does not exist |
| Power Law | No sill | Does not exist |