In hrvg/statisticalRoughness: Estimation of roughness exponents across scale for large areas
knitr::opts_chunk$set(
collapse = TRUE,
comment = "#>",
fig.width = 7,
fig.height = 7
)
### LIBRARIES ###
library("ggplot2")
library("cowplot")
library("viridis")
library("viridisLite")
library("knitr")
library("kableExtra")
library("dplyr")
library("statisticalRoughness")
Self-Affine Scaling of Surfaces
The width of a surface $S$ of vertical dimension $z$ and lateral dimension $(x,y)$ is defined as:
[\left\langle w \right\rangle_R = \sqrt{\left\langle \left\langle u^2(x,y) \right\rangle_R \right\rangle_s} ]
with $u_R$ the deviation function:
[u_R(x,y) = z(x,y) - \left\langle z(x,y) \right\rangle_R ]
Notice that $\left\langle w \right\rangle_R$ is an average over all regions the surface $S$ with lateral dimensions $R$ and corresponds to the average over $S$ of the standard deviations over $R$.
[\left\langle w \right\rangle_R \sim R^H]
$H$ depends on the size $L_S$ of the surface $S$ and the lateral dimensions $R$.
In the following, I drop the subscript and equate $L_s$ with $L$.
When dealing with data on a grid, it is obviously convenient if the lateral dimensions $R$ corresponds to factors of $L$.
A factor $b\in\mathbb{N}$ of $a\in\mathbb{N}$ is such that the remainder of $a/b$ is zero and $b \notin \left{1;a\right}$.
@Guillon2020's approach
The original set of scales was defined from 5 seeds $s = \left{ 2;3;5;7;9 \right}$ (the prime numbers between 2 and 10) and used as multiplier for the power of 2 from 0 to 13:
[ \left{R_i\right} = \left{s_i \cdot 2^{k}\right}_{i = 1, k = 0}^{i = 5, k = 13}]
```{warning = FALSE, message = FALSE, include = TRUE, echo = TRUE}
colnames(.Rs) <- seeds
rownames(.Rs) <- 2**(0:13)
.Rs %>%
kable("html") %>%
add_header_above(header = c("power of two" = 1, "seed" = 5)) %>%
kable_styling(bootstrap_options = c("striped", "hover"),
position = "center",
full_width = FALSE)
As some of the scale failed to compute, they were filtered to be lower than $10^4$ (or $10^5$ m in map units):
```r
seeds <- c(2,3,5,7,9)
Rs <- lapply(seeds, function(seed) 2**(0:13) * seed)
.Rs <- do.call(cbind, Rs) %>% as.data.frame()
Lmax <- 8192
.Rs <- .Rs %>%
mutate_all(~ ifelse(. > Lmax, NA, .))
colnames(.Rs) <- seeds
rownames(.Rs) <- 2**(0:13)
.Rs %>%
kable("html") %>%
add_header_above(header = c("power of two" = 1, "seed" = 5)) %>%
kable_styling(bootstrap_options = c("striped", "hover"),
position = "center",
full_width = FALSE) %>%
row_spec(1:5, color = "white", background = "steelblue")
# Rs <- lapply(Rs, function(R) R[R < Lmax])
# print(Rs)
The size of the surfaces $L$ over which the scaling is computed were defined as having at least 5 factors:
Ls <- lapply(Rs, function(R) tail(R, -5))
print(Ls)
The run with the largest scale ($L = 9216$) failed such that the final set of surface sizes $L$ is:
Ls <- head(sort(unlist(Ls)), -1)
print(Ls)
The following table displays the set of 32 surface size $\left{L_i\right}{i = 1}^{n_S}$ and the associated set of lateral dimensions $\left{R_i\right}{i = 1}^{n_R}$ with $n_L = 32$ and $n_R = 5$ and where $R_i$ are the largest $n_R$ factors for a given $S$.
Ls_factors <- t(sapply(Ls, function(L) tail(get_factors(L), 6)))
df2017 <- data.frame(Ls_factors)
colnames(df2017) <- c(paste0("R.", 1:5), "L")
print(df2017)
statisticalRoughness's approach
However, there are $5369$ numbers that satisfy the conditions @Guillon2020 used to select $\left{L_i\right}{i = 1}^{n_S}$ and $\left{R_i\right}{i = 1}^{n_R}$ with $n_R = 5$ and 6 of them are smaller than the minimum surface size. This is important as it potentially leverages information at a finer spatial scale.
nR <- 5
RL <- get_all_R_L(Lmax, nR, logfilter = FALSE)
print(length(RL$allL))
print(head(RL$allL[RL$allL < min(Ls)]))
Comparison of Factor Distributions
So why use the 32 surface sizes defined in @Guillon2020? What is so peculiar about them?
They have been initially defined as:
[ \left{R_i\right} = \left{s_i \cdot 2^{k}\right}_{i = 1, k = 0}^{i = 5, k = 13}]
Because of the $2^{k}$ part of this definition, the factors $R$ are perfectly logarithmically distributed on the number line for $s = 2$ and well-behaved for the other seeds.
Is this true for the approach implemented in statisticalRoughness?
Factor Distribution from @Guillon2020
The following plots identify the number $L$ and its 5 largest factors on the number line.
numberline <- 1:Lmax
par(mfrow = c(1,1))
for (k in seq_along(Ls)){
par(pty = "s")
plot(numberline, log = "xy", lty = 1, ylab = "Numbers", type = "l", main = paste(nR, "largest Factors for", Ls[k]))
grid()
points(numberline[numberline %in% df2017[k, 1:5]], numberline[numberline %in% df2017[k, 1:5]], col = "blue")
points(numberline[numberline %in% df2017[k, 6]], numberline[numberline %in% df2017[k, 6]], col = "blue", pch=19)
}
Factor Distribution from statisticalRoughness
The following plots identify the number $L$ and all its factors on the number line.
The factors appears similarly well-behaved for @Guillon2020 and statisticalRoughness approaches.
numberline <- 1:Lmax
RL$allL[RL$allL < min(Ls)]
par(mfrow = c(1,1))
for (k in seq_along(RL$allL[RL$allL < min(Ls)])){
par(pty = "s")
plot(numberline, log = "xy", lty = 1, ylab = "Numbers", type = "l", main = paste("All factors for", RL$allL[[k]]))
grid()
points(numberline[numberline %in% RL$allR[[k]]], numberline[numberline %in% RL$allR[[k]]], col = "red")
points(numberline[numberline %in% RL$allL[[k]]], numberline[numberline %in% RL$allL[[k]]], col = "red", pch=19)
}
Distribution of Surface Sizes $L$
Here I compare the distribution of the surface sizes $L$ on the number line.
When indicated, statisticalRoughness approach is filtered so that $L$ are distributed logarithmically along the number line.
numberline <- 1:Lmax
par(pty = "s")
plot(numberline, log = "xy", lty = 1, ylab = "Numbers", type = "l", main = "Original")
grid()
points(numberline[numberline %in% df2017[, 6]], numberline[numberline %in% df2017[, 6]], col = "blue")
plot(numberline, log = "xy", lty = 1, ylab = "Numbers", type = "l", main = "New")
grid()
points(numberline[numberline %in% unlist(RL$allL)], numberline[numberline %in% unlist(RL$allL)], col = "red")
RL <- get_all_R_L(Lmax, nR, logfilter = TRUE, len = nrow(df2017))
plot(numberline, log = "xy", lty = 1, ylab = "Numbers", type = "l", main = "New (filtered)")
grid()
points(numberline[numberline %in% unlist(RL$allL)], numberline[numberline %in% unlist(RL$allL)], col = "red")
par(mfrow = c(1, 2))
plot(numberline, log = "xy", lty = 1, ylab = "Numbers", type = "l", main = "Original")
grid()
points(numberline[numberline %in% df2017[, 6]], numberline[numberline %in% df2017[, 6]], col = "blue")
plot(numberline, log = "xy", lty = 1, ylab = "Numbers", type = "l", main = "New (filtered)")
grid()
points(numberline[numberline %in% unlist(RL$allL)], numberline[numberline %in% unlist(RL$allL)], col = "red")
abline(v = numberline[numberline %in% df2017[1, 6]], col = "blue", pch = 19)
Choosing $n_R$
All Numbers
This graph displays the number of factors for numbers between 2 and $L_\textrm{max}$.
numbers <- 2:Lmax
facts <- sapply(numbers, function(i) length(get_factors(i)) - 2)
df <- data.frame(Numbers = numbers, Factors = facts)
p_fact <- ggplot(df) +
geom_point(aes(x = Factors, y = Numbers, color = Factors)) +
geom_segment(aes(x = 0, xend = Factors, y = Numbers, yend = Numbers, color = Factors)) +
scale_colour_viridis(alpha = 1, begin = 0, end = .8, direction = -1, option = "magma", aesthetics = "colour") +
theme_minimal()
p_fact
This graph displays the number of factors for numbers between 2 and 64. 24 is the first number with more than 5 factors but there are several numbers smaller than 24 that have 4 factors.
p_fact + ylim(c(2,64)) + xlim(c(0 , 20))
Multiples of 6 and 10
Of the numbers between 2 and 10, 6 and 10 have the greatest number of factors.
In consequence, all multiple of these numbers also have a higher number of factors.
In the following, I display factors such that $a \in \left{6;10\right}$.
This graph displays the number of factors for numbers between 2 and $L_\textrm{max}$.
numbers <- 2:Lmax
facts <- sapply(numbers, function(i) length(get_factors(i)) - 2)
df <- data.frame(Numbers = numbers, Factors = facts)
df <- df[which(df$Numbers %% 6 == 0 | df$Numbers %% 10 == 0), ]
p_fact <- ggplot(df) +
geom_point(aes(x = Factors, y = Numbers, color = Factors)) +
geom_segment(aes(x = 0, xend = Factors, y = Numbers, yend = Numbers, color = Factors)) +
scale_colour_viridis(alpha = 1, begin = 0, end = .8, direction = -1, option = "magma", aesthetics = "colour") +
theme_minimal()
p_fact
This graph displays the number of factors for numbers between 2 and 64.
p_fact + ylim(c(2,64)) + xlim(c(0 , 20))
Multiples of 12
Interestingly as 12 is the first number with at least 4 factors, all subsequent multiples of 12 have at least 4 factors.
In the following, I display factors such that: $a \in \left{6;10;12\right}$.
numbers <- 2:Lmax
facts <- sapply(numbers, function(i) length(get_factors(i)) - 2)
df <- data.frame(Numbers = numbers, Factors = facts)
df <- df[which(df$Numbers %% 6 == 0 | df$Numbers %% 10 == 0), ]
df$multiple <- as.factor(ifelse(df$Numbers %% 6 == 0, ifelse(df$Numbers %% 12 == 0, "Twelve", "Six"), "Ten"))
p_fact <- ggplot(df) +
geom_point(aes(x = Factors, y = Numbers, color = multiple)) +
geom_segment(aes(x = 0, xend = Factors, y = Numbers, yend = Numbers, color = multiple)) +
theme_minimal()
p_fact + ylim(c(2,64)) + xlim(c(0 , 20))
In the following, I display factors such that: $a = 12$.
numberline <- 1:Lmax
RL <- get_all_R_L(Lmax, 4, only = 12, logfilter = TRUE, len = 32)
par(mfrow = c(1,1))
for (k in seq_along(RL$allL[RL$allL < min(Ls)])){
par(pty = "s")
plot(numberline, log = "xy", lty = 1, ylab = "Numbers", type = "l", main = paste("All factors for", RL$allL[[k]]))
grid()
points(numberline[numberline %in% RL$allR[[k]]], numberline[numberline %in% RL$allR[[k]]], col = "red")
points(numberline[numberline %in% RL$allL[[k]]], numberline[numberline %in% RL$allL[[k]]], col = "red", pch=19)
}
par(mfrow = c(1, 2))
plot(numberline, log = "xy", lty = 1, ylab = "Numbers", type = "l", main = "Original")
grid()
points(numberline[numberline %in% df2017[, 6]], numberline[numberline %in% df2017[, 6]], col = "blue")
plot(numberline, log = "xy", lty = 1, ylab = "Numbers", type = "l", main = "New (filtered, multiples of 12)")
grid()
points(numberline[numberline %in% unlist(RL$allL)], numberline[numberline %in% unlist(RL$allL)], col = "red")
abline(v = numberline[numberline %in% df2017[1, 6]], col = "blue", pch = 19)
print(RL$allL)
print(RL$allL[which(RL$allL %in% df2017[, 6])])
Of these surface sizes, two appeared in @Guillon2020: 192 and 1536.
Importantly, using this new approach divides by 5.3 the smaller scale at which fractal dimension was computed in @Guillon2020 (120-m vs 640-m).
References
hrvg/statisticalRoughness documentation built on March 12, 2021, 4:55 p.m.
R Package Documentation
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knitr::opts_chunk$set( collapse = TRUE, comment = "#>", fig.width = 7, fig.height = 7 )
### LIBRARIES ### library("ggplot2") library("cowplot") library("viridis") library("viridisLite") library("knitr") library("kableExtra") library("dplyr") library("statisticalRoughness")
Self-Affine Scaling of Surfaces
The width of a surface $S$ of vertical dimension $z$ and lateral dimension $(x,y)$ is defined as:
[\left\langle w \right\rangle_R = \sqrt{\left\langle \left\langle u^2(x,y) \right\rangle_R \right\rangle_s} ] with $u_R$ the deviation function: [u_R(x,y) = z(x,y) - \left\langle z(x,y) \right\rangle_R ]
Notice that $\left\langle w \right\rangle_R$ is an average over all regions the surface $S$ with lateral dimensions $R$ and corresponds to the average over $S$ of the standard deviations over $R$.
[\left\langle w \right\rangle_R \sim R^H]
$H$ depends on the size $L_S$ of the surface $S$ and the lateral dimensions $R$. In the following, I drop the subscript and equate $L_s$ with $L$.
When dealing with data on a grid, it is obviously convenient if the lateral dimensions $R$ corresponds to factors of $L$. A factor $b\in\mathbb{N}$ of $a\in\mathbb{N}$ is such that the remainder of $a/b$ is zero and $b \notin \left{1;a\right}$.
@Guillon2020's approach
The original set of scales was defined from 5 seeds $s = \left{ 2;3;5;7;9 \right}$ (the prime numbers between 2 and 10) and used as multiplier for the power of 2 from 0 to 13:
[ \left{R_i\right} = \left{s_i \cdot 2^{k}\right}_{i = 1, k = 0}^{i = 5, k = 13}]
```{warning = FALSE, message = FALSE, include = TRUE, echo = TRUE} colnames(.Rs) <- seeds rownames(.Rs) <- 2**(0:13) .Rs %>% kable("html") %>% add_header_above(header = c("power of two" = 1, "seed" = 5)) %>% kable_styling(bootstrap_options = c("striped", "hover"), position = "center", full_width = FALSE)
As some of the scale failed to compute, they were filtered to be lower than $10^4$ (or $10^5$ m in map units): ```r seeds <- c(2,3,5,7,9) Rs <- lapply(seeds, function(seed) 2**(0:13) * seed) .Rs <- do.call(cbind, Rs) %>% as.data.frame() Lmax <- 8192 .Rs <- .Rs %>% mutate_all(~ ifelse(. > Lmax, NA, .)) colnames(.Rs) <- seeds rownames(.Rs) <- 2**(0:13) .Rs %>% kable("html") %>% add_header_above(header = c("power of two" = 1, "seed" = 5)) %>% kable_styling(bootstrap_options = c("striped", "hover"), position = "center", full_width = FALSE) %>% row_spec(1:5, color = "white", background = "steelblue") # Rs <- lapply(Rs, function(R) R[R < Lmax]) # print(Rs)
The size of the surfaces $L$ over which the scaling is computed were defined as having at least 5 factors:
Ls <- lapply(Rs, function(R) tail(R, -5)) print(Ls)
The run with the largest scale ($L = 9216$) failed such that the final set of surface sizes $L$ is:
Ls <- head(sort(unlist(Ls)), -1) print(Ls)
The following table displays the set of 32 surface size $\left{L_i\right}{i = 1}^{n_S}$ and the associated set of lateral dimensions $\left{R_i\right}{i = 1}^{n_R}$ with $n_L = 32$ and $n_R = 5$ and where $R_i$ are the largest $n_R$ factors for a given $S$.
Ls_factors <- t(sapply(Ls, function(L) tail(get_factors(L), 6))) df2017 <- data.frame(Ls_factors) colnames(df2017) <- c(paste0("R.", 1:5), "L") print(df2017)
statisticalRoughness's approach
However, there are $5369$ numbers that satisfy the conditions @Guillon2020 used to select $\left{L_i\right}{i = 1}^{n_S}$ and $\left{R_i\right}{i = 1}^{n_R}$ with $n_R = 5$ and 6 of them are smaller than the minimum surface size. This is important as it potentially leverages information at a finer spatial scale.
nR <- 5 RL <- get_all_R_L(Lmax, nR, logfilter = FALSE) print(length(RL$allL)) print(head(RL$allL[RL$allL < min(Ls)]))
Comparison of Factor Distributions
So why use the 32 surface sizes defined in @Guillon2020? What is so peculiar about them?
They have been initially defined as: [ \left{R_i\right} = \left{s_i \cdot 2^{k}\right}_{i = 1, k = 0}^{i = 5, k = 13}]
Because of the $2^{k}$ part of this definition, the factors $R$ are perfectly logarithmically distributed on the number line for $s = 2$ and well-behaved for the other seeds.
Is this true for the approach implemented in statisticalRoughness?
Factor Distribution from @Guillon2020
The following plots identify the number $L$ and its 5 largest factors on the number line.
numberline <- 1:Lmax par(mfrow = c(1,1)) for (k in seq_along(Ls)){ par(pty = "s") plot(numberline, log = "xy", lty = 1, ylab = "Numbers", type = "l", main = paste(nR, "largest Factors for", Ls[k])) grid() points(numberline[numberline %in% df2017[k, 1:5]], numberline[numberline %in% df2017[k, 1:5]], col = "blue") points(numberline[numberline %in% df2017[k, 6]], numberline[numberline %in% df2017[k, 6]], col = "blue", pch=19) }
Factor Distribution from statisticalRoughness
The following plots identify the number $L$ and all its factors on the number line.
The factors appears similarly well-behaved for @Guillon2020 and statisticalRoughness approaches.
numberline <- 1:Lmax RL$allL[RL$allL < min(Ls)] par(mfrow = c(1,1)) for (k in seq_along(RL$allL[RL$allL < min(Ls)])){ par(pty = "s") plot(numberline, log = "xy", lty = 1, ylab = "Numbers", type = "l", main = paste("All factors for", RL$allL[[k]])) grid() points(numberline[numberline %in% RL$allR[[k]]], numberline[numberline %in% RL$allR[[k]]], col = "red") points(numberline[numberline %in% RL$allL[[k]]], numberline[numberline %in% RL$allL[[k]]], col = "red", pch=19) }
Distribution of Surface Sizes $L$
Here I compare the distribution of the surface sizes $L$ on the number line.
When indicated, statisticalRoughness approach is filtered so that $L$ are distributed logarithmically along the number line.
numberline <- 1:Lmax par(pty = "s") plot(numberline, log = "xy", lty = 1, ylab = "Numbers", type = "l", main = "Original") grid() points(numberline[numberline %in% df2017[, 6]], numberline[numberline %in% df2017[, 6]], col = "blue") plot(numberline, log = "xy", lty = 1, ylab = "Numbers", type = "l", main = "New") grid() points(numberline[numberline %in% unlist(RL$allL)], numberline[numberline %in% unlist(RL$allL)], col = "red") RL <- get_all_R_L(Lmax, nR, logfilter = TRUE, len = nrow(df2017)) plot(numberline, log = "xy", lty = 1, ylab = "Numbers", type = "l", main = "New (filtered)") grid() points(numberline[numberline %in% unlist(RL$allL)], numberline[numberline %in% unlist(RL$allL)], col = "red") par(mfrow = c(1, 2)) plot(numberline, log = "xy", lty = 1, ylab = "Numbers", type = "l", main = "Original") grid() points(numberline[numberline %in% df2017[, 6]], numberline[numberline %in% df2017[, 6]], col = "blue") plot(numberline, log = "xy", lty = 1, ylab = "Numbers", type = "l", main = "New (filtered)") grid() points(numberline[numberline %in% unlist(RL$allL)], numberline[numberline %in% unlist(RL$allL)], col = "red") abline(v = numberline[numberline %in% df2017[1, 6]], col = "blue", pch = 19)
Choosing $n_R$
All Numbers
This graph displays the number of factors for numbers between 2 and $L_\textrm{max}$.
numbers <- 2:Lmax facts <- sapply(numbers, function(i) length(get_factors(i)) - 2) df <- data.frame(Numbers = numbers, Factors = facts) p_fact <- ggplot(df) + geom_point(aes(x = Factors, y = Numbers, color = Factors)) + geom_segment(aes(x = 0, xend = Factors, y = Numbers, yend = Numbers, color = Factors)) + scale_colour_viridis(alpha = 1, begin = 0, end = .8, direction = -1, option = "magma", aesthetics = "colour") + theme_minimal() p_fact
This graph displays the number of factors for numbers between 2 and 64. 24 is the first number with more than 5 factors but there are several numbers smaller than 24 that have 4 factors.
p_fact + ylim(c(2,64)) + xlim(c(0 , 20))
Multiples of 6 and 10
Of the numbers between 2 and 10, 6 and 10 have the greatest number of factors. In consequence, all multiple of these numbers also have a higher number of factors. In the following, I display factors such that $a \in \left{6;10\right}$.
This graph displays the number of factors for numbers between 2 and $L_\textrm{max}$.
numbers <- 2:Lmax facts <- sapply(numbers, function(i) length(get_factors(i)) - 2) df <- data.frame(Numbers = numbers, Factors = facts) df <- df[which(df$Numbers %% 6 == 0 | df$Numbers %% 10 == 0), ] p_fact <- ggplot(df) + geom_point(aes(x = Factors, y = Numbers, color = Factors)) + geom_segment(aes(x = 0, xend = Factors, y = Numbers, yend = Numbers, color = Factors)) + scale_colour_viridis(alpha = 1, begin = 0, end = .8, direction = -1, option = "magma", aesthetics = "colour") + theme_minimal() p_fact
This graph displays the number of factors for numbers between 2 and 64.
p_fact + ylim(c(2,64)) + xlim(c(0 , 20))
Multiples of 12
Interestingly as 12 is the first number with at least 4 factors, all subsequent multiples of 12 have at least 4 factors. In the following, I display factors such that: $a \in \left{6;10;12\right}$.
numbers <- 2:Lmax facts <- sapply(numbers, function(i) length(get_factors(i)) - 2) df <- data.frame(Numbers = numbers, Factors = facts) df <- df[which(df$Numbers %% 6 == 0 | df$Numbers %% 10 == 0), ] df$multiple <- as.factor(ifelse(df$Numbers %% 6 == 0, ifelse(df$Numbers %% 12 == 0, "Twelve", "Six"), "Ten")) p_fact <- ggplot(df) + geom_point(aes(x = Factors, y = Numbers, color = multiple)) + geom_segment(aes(x = 0, xend = Factors, y = Numbers, yend = Numbers, color = multiple)) + theme_minimal() p_fact + ylim(c(2,64)) + xlim(c(0 , 20))
In the following, I display factors such that: $a = 12$.
numberline <- 1:Lmax RL <- get_all_R_L(Lmax, 4, only = 12, logfilter = TRUE, len = 32) par(mfrow = c(1,1)) for (k in seq_along(RL$allL[RL$allL < min(Ls)])){ par(pty = "s") plot(numberline, log = "xy", lty = 1, ylab = "Numbers", type = "l", main = paste("All factors for", RL$allL[[k]])) grid() points(numberline[numberline %in% RL$allR[[k]]], numberline[numberline %in% RL$allR[[k]]], col = "red") points(numberline[numberline %in% RL$allL[[k]]], numberline[numberline %in% RL$allL[[k]]], col = "red", pch=19) } par(mfrow = c(1, 2)) plot(numberline, log = "xy", lty = 1, ylab = "Numbers", type = "l", main = "Original") grid() points(numberline[numberline %in% df2017[, 6]], numberline[numberline %in% df2017[, 6]], col = "blue") plot(numberline, log = "xy", lty = 1, ylab = "Numbers", type = "l", main = "New (filtered, multiples of 12)") grid() points(numberline[numberline %in% unlist(RL$allL)], numberline[numberline %in% unlist(RL$allL)], col = "red") abline(v = numberline[numberline %in% df2017[1, 6]], col = "blue", pch = 19)
print(RL$allL) print(RL$allL[which(RL$allL %in% df2017[, 6])])
Of these surface sizes, two appeared in @Guillon2020: 192 and 1536. Importantly, using this new approach divides by 5.3 the smaller scale at which fractal dimension was computed in @Guillon2020 (120-m vs 640-m).
References
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