#
The Sequential Generation of Gaussian Random Fields for Applications in the Geospatial Sciences^{ †}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction and Motivation

#### Roadmap

## 2. A Scalar Gaussian 2D Random Field and Its Sequential Generation

#### 2.1. Statistical Characteristics

_{z}, i.e., is normally distributed N(0, σ

_{z}) for all grid locations (k, l). Its spatial correlation across the grid is separable, i.e., has the (normalized) correlation function ρ(∆k, ∆l), where ∆k and ∆l are the absolute values of the component-wise differences in the (k, l) location of two arbitrary grid points. This function is represented as:

^{−∆kδy}

^{/Ty}e

^{−∆lδx}

^{/Tx}

_{y}and T

_{x}are specifiable spatial correlation distance constants (meters) and δ

_{y}and δ

_{x}specifiable grid spacing (meters) in the y and the x directions in the horizontal plane, respectively. Note that ∆y = ∆kδ

_{y}and ∆x = ∆lδ

_{x}

_{y}= 200 m, T

_{x}= 100 m, and δ

_{y}= δ

_{x}= 10 m. The spatial correlation ρ(∆y, ∆x) is applicable to any pair of grid points within the entire grid that are separated by ∆y meters in the y-direction and ∆x meters in the x-direction. The use of two different spatial correlation distance constants allows for specification of different correlation characteristics in each of the horizontal directions.

**Figure 1.**An example of the separable spatial correlation function; in the plot of ρ(∆y, ∆x), ∆y and ∆x have signed values.

_{y}> 0, T

_{x}> 0, δ

_{y}> 0, δ

_{x}> 0, |ρ(∆k, ∆l)| ˂ 1 if ∆k ≠ 0 or ∆l ≠ 0, and ρ(∆k, ∆l) = 1 when ∆k = ∆l = 0. Section 3.0 of this paper also presents the covariance matrix associated with two or more z(k, l), each associated with a different grid point (k,l).

#### 2.2. Core Grid-Generation Equation

^{−δy/Ty}, and r = e

^{−δx/Tx}. u(k, l) is a random sample of Gaussian white noise, and is normally distributed N(0, σ

_{u}), where . That is, given a desired s, r, and σ

_{z}, a corresponding value of σ

_{u}is computed per the above.

^{∆k}r

^{∆l}

#### 2.3. Sequential Generation Algorithm for Realization over a pxq Grid

- z(1,1) = random_N(0, σ
_{z}); - z(2,1) = sz(1,1) + random_N(0, σ
_{u}); - z(1,2) = rz(1,1) + random_N(0, σ
_{u}); - z(2,2) = rz(2,1) + sz(1,2) − rsz(1,1) + random_N(0, σ
_{u}); - z(1,q) = rz(1,q-1) + random_N(0, σ
_{u}); - z(2,q) = rz(2, q-1) + sz(1, q) − rsz(1, q-1) + random_N(0, σ
_{u});

- z(3,1) = sz(2,1) + random_N(0, σ
_{u}); - z(3,2) = rz(3,1) +sz(2,2) − rsz(2,1) + random_N(0, σ
_{u}); - z(3,q) = rz(3, q − 1) + sz(2, q) − rsz(2, q − 1) + random_N(0, σ
_{u});

_{a})” corresponds to a random number (realization) from a N(0, σ

_{u}) probability distribution; for example, in matlab this is implemented as “sigma_a * randn(1,1)”.

#### 2.3.1. Grid Spacing

_{y}and δ

_{x}equivalent to approximately one-ninth or less their respective spatial correlation distance constant, insuring at least 0.9 correlation with an adjacent grid point, i.e., s = e

^{−δy/Ty}≥ e

^{−1/9}≅ 0.9 and r = e

^{−δx/Tx}≥ e

^{−1/9}≅ 0.9, or equivalently, δ

_{y}≤ T

_{y}/9 m and δ

_{x}≤ T

_{x}/9 m. Of course, this “rule-of-thumb” is application dependent. For example, if very high spatial correlation between adjacent grid points is of interest, spacing should be closer.

#### 2.3.2. Grid Buffer

_{x}/δ

_{x}. Or equivalently, if s and r equal the value 0.9, 19 grid rows and 19 grid columns. This will ensure generation of errors throughout the final grid with the desired statistical properties.

#### 2.4. Example Realizations: Surface Plots

_{z}= 10 m, and specified s = r = 0.95 (thus σ

_{u}= 0.975 m). Assuming a grid spacing in both the y and x directions of 1 m (δ

_{y}= δ

_{x}= 1), this corresponds to spatial distance constants equal to T

_{y}= T

_{x}= 19.5 m.

_{y}and T

_{x}were derived from the specified s and r, given assumed grid spacing δ

_{y}and δ

_{x}not vice versa. The spatial distance constants were computed for information only. That is, there are two basic but equivalent approaches for the specification, application, and interpretation of spatial correlation, the particular approach selected based on convenience:

_{y}and δ

_{x}(meters), the spatial distance constants T

_{y}and T

_{x}(meters) can be derived for information purposes only.

_{y}and T

_{x}(meters) and grid spacing δ

_{y}and δ

_{x}(meters), compute s and r, implement Equation (3), and then interpret location-dependent results in terms of y-x horizontal space in meters. The approach works well when the generated random field is to correspond to the a priori statistics and spatial resolution of a specific application of interest in the Geospatial Sciences.

**Figure 4.**Example 1—Realization of z-error with high spatial correlation between adjacent grid points.

_{z}= 10 m, but with s = r = 0.1 (thus σ

_{u}= 9.9 m).

_{z}= 10 m, but with s = r = 0.999 (thus σ

_{u}= 0.02 m).

_{z}= 10 m, but with s = 0.1 and r = 0.95 (thus, σ

_{u}= 3.107), i.e., different spatial correlations in the two directions.

**Figure 5.**Example 2—Realization of z-error with low spatial correlation between adjacent grid points.

**Figure 6.**Example 3—Realization of z-error with very high spatial correlation between adjacent grid points.

#### 2.5. Example Realizations: Sample Statistics

_{z}= 10 m. The first realization corresponded to a priori correlations represented by r = 0.95 and s = 0.75, and is presented in Figure 8. Correlation functions were plotted as a function of horizontal distance in the x-direction and horizontal distance in the y-direction, and are different as expected per the values of r and s.

**Figure 9.**Sample statistics corresponding to 1000 × 1000 grid with the same a priori correlations for the x and y directions, evaluated across three different directions.

**Figure 10.**Semivariograms corresponding to 200 × 200 grid with the same a priori correlations for the x and y directions, evaluated across the y-direction for five different realizations.

## 3. Multi-Grid Point Covariance Matrix

^{−∆lδy}

^{/Ty}e

^{−∆kδx}

^{/Tx}, i.e., the multi-grid point covariance matrix equals:

^{T}= [z

_{1}, …, z

_{m}] and the z

_{i}= z(k

_{i}, l

_{i}), i = 1,…,m, correspond to an ordered list of the m grid point locations. Also, ∆y

_{ij}= ∆y

_{ji}and ∆x

_{ij}= ∆x

_{ji}are the y and x distances in meters in the horizontal plane between the ordered points i, j ∈ {1,…,m}; directly multiplies each element of the mxm matrix. (Alternatively, the spatial correlation function and distances could have been written based on grid units.)

^{−∆lδy}

^{/Ty}e

^{−∆lδx}

^{/Tx}, such that the applications can then build the appropriate multi-grid point covariance matrix themselves.

#### Homogeneity and Gaussian Joint Probability Density

## 4. Interpolation into the Grid

#### Related Effects

## 5. Comparison to Alternate Generation Methods

^{−∆kδy}

^{/Ty}e

^{−∆lδx}

^{/Tx}. (Albeit, reasonably general in that the distance constants T

_{y}and T

_{x}are specifiable).

#### 5.1. Timing Comparisons among Simulation Techniques

_{x}= T

_{y}). The following describes what main computations were timed in each of the five methods, and are listed in ascending order of computational speed gain according to Figure 12.

_{z};

**r**is a n × 1 vector realization of n independent N(0,1) distributed random variables, and ϵ

_{z}is the n × 1 vector of perturbations corresponding to the random variable z or z(k, l) over the grid. Σ

_{z}was assumed to be a full, and positive definite matrix, i.e., the a priori covariance matrix corresponding to the random field at the n different points in the grid. Matlab uses the Schur decomposition technique to compute SQRTM for a general square matrix, which can be further sped up for symmetric and real matrices.

**Figure 12.**Time comparison among methods for unconditional simulation of a scalar random field z(k, l) over a 2D square grid (Note that FSS is cutoff at ~15 s due to reaching the system memory limit).

_{z}= L

**r**

_{z}= LL*, where L* is the conjugate transpose of L;

**r**is a n × 1 vector realization of n independent N(0,1) distributed random variables, and ϵ

_{z}is the n × 1 vector of perturbations corresponding to the random variable z or z(k, l) over the grid. Σ

_{z}was assumed to be a full and positive definite matrix.

_{z}= 1), and the spatial distance correlation constants (T

_{x}= T

_{y}= 10), as all described in Section 2.3 and Section 2.4 of this paper.

#### 5.2. Discussion of Timing Results

^{8}points). However, this constraint can be easily overcome by performing the computation with a local moving window versus storing the entire grid into system memory. The speed gain of FSS makes simulation of considerably denser grids more practical compared to the other methods. With this capability, our conjecture is that for those applications requiring interpolation, less expensive bilinear or nearest neighbor interpolation could be adequate for very dense grids versus more expensive Kriging in coarser grids. The variogram (correlation) model is constrained to an exponential function with FSS, which makes it less flexible than the GSTAT (Sequential Gaussian Simulation) methods. However, the tradeoff in speed gain and simplicity of implementation offers practical and useful advantages to motivate a potentially broader community of users.

## 6. Extension of FSS to a Multivariate Gaussian 2D Random Field

_{i}< 1, i = 1,..,n; diagonal , 0 < s

_{i}< 1, i = 1,..,n;

^{T}= P

_{X}, the n × n covariance matrix;

^{T}} = P

_{X}S

^{∆k}R

^{∆l}, {X(k, l)X(k ∆ ∆k, l + ∆l)

^{T}} = S

^{∆k}P

_{X}R

^{∆l},

^{T}} = R

^{∆l}P

_{X}S

^{∆k}, E{X(k, l) X(k − ∆k, l − ∆l)

^{T}} = S

^{∆k}R

^{∆l}P

_{X}, for ∆k ≥ 0 and ∆l ≥ 0;

_{U}= E{U(k, l) U(k, l)

^{T}must be a valid (symmetric and positive definite) covariance matrix which satisfies the following:

_{U}= H * P

_{X}, the Hadamard product (term by term product) of two nxn matrices, where

_{U}is a positive definite matrix is not satisfied for all possible combinations of s

_{i}, r

_{i}, and desired (valid) steady state error covariance P

_{X}, in which case Equation (9) and its statistics are no longer valid.

_{U}, is somewhat complicated and presented in Appendix D.

_{z}) is replaced by , and random_N(0, σ

_{u}) is replaced by , where the superscript 1/2 corresponds to principal matrix square root and random_v is the realization of an independent nx1 random vector with each component an independent realization of a scalar random variable that is distributed N(0,1). Of course, S replaces s, R replaces r , and X replaces z, as well.

#### 6.1. Common Spatial Correlation Subcase

_{U}is always satisfied:

_{nxn}, S = sI

_{nxn}, P

_{U}= (1 − s

^{2})(1 − r

^{2})P

_{X}.

^{T}} = ρ(∆k, ∆l)P

_{X}; that is, all n components of X(k, l) have common inter-grid (spatial) correlation via a common (scalar) spdcf ρ(∆k, ∆l) = s

^{∆k}r

^{∆l}= e

^{−∆kδy}

^{/Ty}e

^{−∆lδx}

^{/Tx}

_{X}, i.e., there can be non-zero intra-component correlations among the components of X(k, l). For example, assume that n = 3 and X

^{T}= [x y z] consists of error components x, y, and z. Furthermore, at an arbitrary grid point location (k, l) the z-component of error is correlated +0.10 with the y-component of error and the same x -component of error is correlated −0.60 with the z-component of error. Of course, all n-choose-2 (a value of 3 for the case n = 3) combinations of correlation among error components must correspond to a symmetric and positive definite P

_{X}.

#### Multi-Grid Point Covariance Matrix

_{i}= X(k

_{i}, l

_{i}), i = 1,...,m, correspond to an ordered list of the m grid point locations. The n × n cross-covariance terms ρ(∆y

_{ij}, ∆x

_{ij})P

_{X}consist of each element of P

_{X}multiplied by the scalar value ρ(∆y

_{ij}, ∆x

_{ij}), i,j = 1,...,m.

#### 6.2. Diagonal Covariance Subcase

_{X}is a diagonal matrix. This allows for any values of 0 < s

_{i}< 1 and 0 < r

_{i}< 1, i.e., different specifiable spatial correlations for each of the two directions for each of the n error components. Additionally, of course, this allows for different variances specified along the diagonal elements of P

_{X}; also, the constraint on P

_{U}is always satisfied. Note that this special case is simply equivalent to the scalar case for each of the n components applied independently.

#### 6.2.1. Example Realizations

_{1}= s

_{2}and r

_{1}= r

_{2}. Figure 13 (automatically scaled) corresponds to σ

_{1}= σ

_{2}= 10 m, and s

_{1}= s

_{2}= 0.95 and r

_{1}= r

_{2}= 0.95. Figure 14 (automatically scaled) corresponds to σ

_{1}= σ

_{2}= 10 m, and s

_{1}= s

_{2}= 0.95 and r

_{1}= r

_{2}= 0. 5.

#### 6.2.2. Multi-Grid Point Covariance Matrix

_{X}, the corresponding mn × mn covariance matrix for a collection of X(k, l) at m arbitrary grid points (k, l) has a convenient and valid representation:

_{i}= X(k

_{i}, l

_{i}), i = 1,...,m, correspond to an ordered list of the m grid point locations. Also, * corresponds to the matrix Hadamard (element by element) product, the nxn diagonal matrix , and ρ

_{v}(∆y

_{ij}, ∆x

_{ij}) corresponds to the spatial correlation function associated with component v = 1,...,m of X(k, l).

#### 6.3. General Case with Constraint Enforced

_{1}, s

_{2}, r

_{1}, r

_{2}can have any combination of values such that each is within the positive interval (0,1) and P

_{U}, a function of the desired P

_{X}and thes

_{1}, s

_{2}, r

_{1}, r

_{2}, is a symmetric and positive definite matrix.

**Figure 13.**Realization of two-dimensional multivariate errors over a 2D grid: high spatial correlation in the grid’s k or y-direction and high spatial correlation in the grid’s l or x-direction.

_{1}= s

_{2}= s and r

_{1}= r

_{2}= r, to narrow down the possible combinations; therefore,

_{1}= r

_{1}and s

_{2}= r

_{2}, therefore,

_{2}given the desired value of s

_{1}and assuming that |ρ| = 0.5; the right side assuming that |ρ| = 0.9.

**Figure 14.**Realization of two-dimensional multivariate errors over a 2D grid: high spatial correlation in the grid’s k or y-direction and lower spatial correlation in the grid’s l or x-direction.

_{2}= r

_{2}must be to s

_{1}= r

_{1}.

_{X}presented earlier.

## 7. Extension of FSS to a Non-Homogeneous 2D Random Field

_{z}, s, and r (variance and spatial correlation parameters) corresponding to z(k, l) are either explicitly or implicitly a function of grid location (k, l). There is no one “right way” to do the extension.

#### 7.1. Method 1: Convex Combination

_{z}, s, and r. Thus, after the above is performed there are n grids, each homogenous and in accordance with the σ

_{z}, s , and r specified for use with that particular grid. Each grid is uncorrelated with the others.

_{i}(k, l) for i = 1,...,n, are then combined based on a convex combination into a final grid of z(k, l). That is, at each (k, l) location in the p × q grid:

_{i}(k, l) values, also symbolized as wi

_{kl}for convenience, can be as simple or as complicated as appropriate over the locations across the p × q grid. However, their recommended values are in accordance with the following:

_{i}(k, l) across the various (k, l) in Region i of the pxq grid; hence, in this region, all wi

_{kl}= 1. In addition, let us define Region i–j as a “buffer region” from Region i into Region j. In this buffer region, wi

_{kl}varies linearly from 1–0 corresponding to the (k, l) at the start to the end of the buffer region, respectively. Furthermore, of course, wj

_{kl}= 1 − wi

_{kl}throughout Region i–j. Finally, wi

_{kl}= 0 for all locations (k, l) in Region j. See Figure 16 as an example for n = 2.

_{i}(k, l) and z

_{j}(k, l), expressed in grid unit. This ensures reasonable spatial correlation across the buffer region. If there were no buffer region, the spatial correlation between two points, one anywhere in Region i and the other anywhere in Region j, would be 0, i.e., there would be an unwanted abrupt change across the boundary of the two regions.

#### 7.1.1. Multi-Grid Point Covariance Matrix

^{T}= [z

_{1}… z

_{m}]. Such a vector can also be defined for the same ordered locations for each of the different realizations as Z

_{i}, = 1,...,n. Therefore, based on Equation (17) we have:

_{i}is an mxm diagonal matrix for i = 1,...,n with the appropriate values of down its diagonal.

_{i}equals .

_{i}as P

_{i}. (See Section 3 for how this matrix is computed given the corresponding σ

_{z}, s, and r.) The mxm multi grid point covariance matrix for Z is computed as follows:

^{T}= [z

_{1}… z

_{m}] and the z

_{i}= z(k

_{i}, l

_{i}), i = 1,...,m, correspond to the ordered list of the m grid point locations; ρ

_{j1j2}corresponds to the explicit correlation between two such points; the matrix entries “_” indicate symmetry.

#### 7.1.2. Typical Statistics

_{kl}in Region 1 (w1

_{kl}= 1), in Region 2 (w1

_{kl}= 0), and w1

_{kl}in Region 1–2 (w1

_{kl}= 1 → 0). Let us designate the a priori one-sigma and correlation functions for the homogeneous z

_{1}(k, l) and z

_{2}(k, l) across the grid as σz

_{1}and ρ

_{1}(∆k, ∆l), and σz

_{2}and ρ

_{2}(∆k, ∆l), respectively, for convenience. We have the following location-dependent statistics for the final combined z(k, l):

_{z}: a point in Region 1, σz

_{1}; a point in Region 2, σz

_{2}; a point in Region 1–2, .

_{1}(∆k, ∆l); both in Region 2, ρ

_{2}(∆k, ∆l), one in Region 1 and one in Region 2, 0; one in Region 1 and one in Region 1–2, , etc.

#### 7.1.3. Example Realizations

_{z1}= 10 m} and {r2 = s2 = 0.9, σ

_{z}

_{2}= 30 m}, respectively. Region 1 of the displayed portion of the final grid consists of k = 1–10, Region 1–2 k = 11–30, and Region 2 k = 31–60. (For each region, the corresponding l = 1–60.) Use of the typical assignment scheme for the values of w1

_{kl}(and w2

_{kl}= 1 − w1

_{kl}) was employed. Figure 17 below presents the results.

**Figure 17.**A smooth transition from Region 1 to Region 2, each with their own specified a priori statistics.

#### 7.2. Method 2: Functional Variation of a Priori Statistics

_{u}(k, l)), and where . Also, s(k, l) = e

^{−δy}

^{/Ty(k,l)}, and r(k, l) = e

^{−δx}

^{/Tx(k,l)}, that is, the spatial distance constants can be considered a function of (k, l) as well.

**Figure 18.**An abrupt transition between Region 1 and Region 2, each with their own specified a priori statistics.

_{z}, s, and r, and hence σ

_{u}, are a function of the grid location (k, l). In addition, for Method 2, they are determined by the bilinear interpolation of such specified values over a less-dense grid overlaying the grid of errors to be generated. For example, if the 2D pxq grid of errors to be generated is 900 × 1000, the grid for interpolation might be an evenly spaced 4 × 3 parameter grid overlying the denser grid. Each of the corresponding 12 parameter grid points contains the specified values for σ

_{z}, s , and r for the corresponding local region around the parameter grid point. Note that σ

_{u}is a function of the interpolated values of σ

_{z}, s, and r; hence, is also recalculated in Equation (21) for every grid location (k, l). See Figure 19 for a graphical representation of the interpolation parameter grid. Each interpolation parameter grid location contains a unique set of values for σ

_{z}, s, and r.

_{u}, s, and r values that vary with (k, l) location.

#### 7.2.1. Example Realizations

_{z}} parameters were identical except for four sets corresponding to an interior rectangle near the center of the final grid. Let us term the 45 common sets as Group 1 and the other four common sets as Group 2. In Figure 20 below, the Group 1 set contain values σ

_{z}= 10 m, s = r = 0.9. Group 2 sets contain values σ

_{z}= 50 m, s = r = 0.9.

_{z}= 10 m, s = r = 0.1. Group 2 sets contain σ

_{z}= 50 m, r = 0.95

#### 7.2.2. Statistics and Multi-Grid Point Covariance Matrix

_{z}corresponding to a specific location (k, l) is the corresponding bilinear interpolated value. The spatial correlation function corresponding to m different locations (k, l) is the average of m spatial correlation functions, each corresponding to the bilinear interpolated values for and r for that location. Of course, these statistics reflect non-homogeneity, i.e., are a function of the specific (k, l) locations of interest. The corresponding mxm approximation for the multi-grid point covariance matrix corresponding to scalar errors at m different grid locations is represented as follows:

^{T}= [z

_{1}... z

_{m}] and the z

_{i}= z(k

_{i}, l

_{i}), i = 1,...,m, correspond to an ordered list of the m grid point locations.

_{zi}can vary in value (Reference [7]).

## 8. Summary and Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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## Appendix A: Derivation of Statistics for the FSS Scalar Gaussian 2D Random Field

#### A.1. Relationship of Core Grid-Generation Equation with Underlying Random Samples

#### A.1.1. Proof of Relationship

#### A.2. Derivation of Statistics

_{z}, σ

_{u}= (1 − s

^{2})

^{1/2}(1 − r

^{2})

^{1/2}σ

_{z}; ρ(∆k, ∆l) = s

^{∆k}r

^{∆l}, or equivalently, ρ(∆k, ∆l) = ρ(∆y, ∆x) = e

^{−∆kδy}

^{/Ty}e

^{−∆lδx}

^{/Tx}

#### A.2.1. Detailed Derivations

^{2}= e

^{−2δy/Ty}< 1 and 0 < a = r

^{2}= e

^{−2δx}

^{/Tx}< 1 in the above.

#### A.3. Further Relationship of the Core Grid-Generation Equation with Underlying Random Samples

^{k}

^{ − i}r

^{l}

^{ − j}u(i, j) to z(k, l) via Equation (A2). In the following figure, i = k − 2 and j = l − 3 for specificity.

^{2}= r(sr) + s(r

^{2}) − rs(r), via applications of Equation (A2). This same three-term paradigm is also applicable to all other grid locations with an incoming “diagonal arrow”.

^{2}r

^{3}corresponds to the contribution of u(k − 2, l − 3) to z(k, l), i.e., z(k, l) = s

^{2}r

^{3}u(k − 2, l − 3) + other u terms, or more generally, z(k, l) = ∑

_{i}

_{≤}

_{k}∑

_{j}

_{≤}

_{l}s

^{k−i}r

^{l−j}u(i, j), i.e., Equation (A1).

**Figure A1.**The “route” of u(k − 2, l − 3) to z(k, l) the lower left grid location in the orange area.

#### Appendix B: Mathematical Comparison of FSS to Sequential Gaussian Simulation

_{1}as the nx1 vector of generated values and X

_{2}as the scalar value to be generated:

_{2}is the best estimate of the realization at the appropriate horizontal location. X

_{2}+ random_N(0, σ

_{X2}) is the corresponding simulated value, where random_N(0, σ

_{X2}) corresponds to a mean-zero Gaussian random number with variance, , i.e., the variance of the Kriging solution relative to the value of the true realization.

_{z}= 10 m and separable exponential spatial correlation with correlation between adjacent grid point locations r = e

^{−1/10}and s = e

^{−1/20}. A representative set of 14 grid locations (n = 13) are represented graphically as follows:

_{1}and then X

_{2}based on the FSS sequential method presented in this paper (Section 2), and obtain: [7.15 6.20 5.45 6.88 8.08 8.66 8.46 5.92 8.16 11.28 8.88 9.99 8.34] and X

_{2}= [9.81]. (Components 1–13 of X

_{1}correspond to grid points #1–13, and X

_{2}corresponds to the red point in Figure B1.) During generation of X

_{2}, the additive Gaussian random number u per Equation (3) was equal to −0.39 corresponding to the standard deviation = 1.31. (Also. using the symbology of Equation (3), X

_{2}corresponds to z(k + 1, l + 1) and u to u(k + 1, l + 1).)

_{1}generated by FSS and detailed in the previous paragraph. In addition, the same additive Gaussian random number u will be added to the Kriging solution for X

_{2}per the Sequential Gaussian Simulation procedure since, as will be demonstrated below, σ

_{X2}= σ

_{u}. In support of the Kriging solution, the a priori cross-covariance matrix between X

_{2}and X

_{1}and the a priori covariance matrix for X

_{1}are computed in accordance with the assumed statistics (σ

_{z}, r , s) of the random field presented previously. Correspondingly, both P

_{21}and P

_{11}are full (no zero elements), but the product P

_{21}P

_{11}

^{-1}is only non-zero for the three elements of X

_{1}which correspond to the nearest three grid locations 8, 9, and 13 to the point to be simulated. (It also follows that a pre-computed “compressed” 1 × 3 version of P

_{21}P

_{11}

^{−1}can actually be used as common Kriging weights for the realizations at the three nearest grid points.)

_{21}P

_{11}

^{−1}= [0 0 0 0 0 0 0 −rs s 0 0 0 r] = [0 0 0 0 0 0 0 −86 0.95 0 0 0 0.90], and the solution (with additive random number) is P

_{21}P

_{11}

^{−1}X

_{1}− 0.39 = 9.81, identical to that generated using the FSS method of this paper. In addition, σ

_{X}

_{2 }= 1.31, which is equal to σ

_{u}. These equivalences and the use of only the nearest three grid points were enabled due to both the separable exponential spatial correlation and the regular grid of realizations generated in a simple, ordered fashion. (Thus, for example, if grid point #8 were moved +0.5 grid units in the y-direction, there would be 7 instead of 3 non-zero scalar weights.)

_{21}P

_{11}

^{−1}has only non-zero weights –rs, s, and r corresponding to the three nearest grid points is relatively easy and done by direct verification that P

_{21}= WP

_{11}, where W is a 1 × n row vector consisting of all zeroes except for the non-zero scalar weights –rs, s, and r at the appropriate locations corresponding to the three nearest grid points (see Figure B1). Similarly, in order that σ

_{z}= σ

_{u}, P

_{21}P

_{11}

^{−1}P

_{12}must equal , which is easily verified by direct evaluation of P

_{21}P

_{11}

^{−1}P

_{12}= WP

_{12}.

#### Appendix C: VISIM Parameter File

#### Appendix D: Derivation of Statistics for the Multivariate Gaussian 2D Random Field

_{i≤k}∑

_{j≤l}S

^{k−i}R

^{l−j}U(i, j), a vector of assumed dimension v × 1

_{X}S

^{m}R

^{n}, dimension vxv, where

_{X}= A * P

_{U}, , p, q∈{1,…,v}.

_{U}) corresponds to

_{U }= H * P

_{X}, where . Also, P

_{U}must be positive definite in order that P

_{X}is positive definite via the earlier summation. (Note that, in the above, S

^{m}R

^{n}= R

^{n}S

^{m}, i.e., they commute since they are diagonal matrices.) Also,

^{T}} = R

^{n}P

_{X}S

^{m}

^{T}} = S

^{m}R

^{n}P

_{X}

^{T}} = E{X(k + m, l + n) (X(k, l)

^{T})

^{T}}, etc.

^{T}} = E{X(k − m, l − m) X(k, l)

^{T}}, since P

_{X}S

^{m}R

^{n}= (S

^{m}R

^{n}P

_{X})

^{T}, as required for wide-sense homogeneity (see Reference [7]).

© 2014 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

## Share and Cite

**MDPI and ACS Style**

Dolloff, J.; Doucette, P.
The Sequential Generation of Gaussian Random Fields for Applications in the Geospatial Sciences. *ISPRS Int. J. Geo-Inf.* **2014**, *3*, 817-852.
https://doi.org/10.3390/ijgi3020817

**AMA Style**

Dolloff J, Doucette P.
The Sequential Generation of Gaussian Random Fields for Applications in the Geospatial Sciences. *ISPRS International Journal of Geo-Information*. 2014; 3(2):817-852.
https://doi.org/10.3390/ijgi3020817

**Chicago/Turabian Style**

Dolloff, John, and Peter Doucette.
2014. "The Sequential Generation of Gaussian Random Fields for Applications in the Geospatial Sciences" *ISPRS International Journal of Geo-Information* 3, no. 2: 817-852.
https://doi.org/10.3390/ijgi3020817