patrickbdevaney/mia9 · Datasets at Hugging Face

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TR Taylor River 0.08 0.37 0.89 0.125
MK Murray Key −0.51 0.22 0.89 0.127 14.67 34.84 54.79 3.60
E146 Taylor Slough −0.18 0.39 0.80 0.143
TSH Taylor Slough Hilton −0.12 0.63 1.02 0.176
2.4. Empirical Mode Decomposition
Water level and salinity data are decomposed into Intrinsic Mode Functions (IMFs) and nonlinear
trends through Empirical Mode Decomposition (EMD) using the Hilbert–Huang transform [20,21]
as implemented in the R package hht. Application of the EMD requires uniformly-sampled data
without gaps. We reconstruct missing data in our time series by using random samples drawn from
distributions of all available data for a specific year day. For example, if 1 January 2000 is missing,
a Gaussian kernel is fit to all available data for 1 January. A random sample is then drawn from this
distribution and used as the reconstructed value. This preserves the overall distribution of the data for
a year day capturing seasonal trends, while realistically allowing for variance away from the mean on
the daily timescale.
2.5. Water Level Exceedance
Water level exceedances are computed from daily mean water levels by summing the number
of exceedance events above an elevation threshold for each year. The probability of exceedance at
a specific threshold as a function of time follows a logistic function exhibiting exponential growth
followed by a linear increase, terminating in nonlinear saturation as water levels continuously exceed
the threshold [22]. The logistic function suggests a growth model for water level exceedances as they
enter the initial growth phase:
E(t) = E0 + α(t − TL) + (1 + r)
t−TG
τ (1)
where E0 is the number of exceedances at year t = 0; α the linear rate of exceedance; r the growth rate;
TL and TG the zero-crossing time of linear and exponential growth, respectively; and τ the growth
time constant. This model is fit to yearly exceedance data with maximum likelihood estimation over a
wide parameter space of initial conditions (Table 3), and the best-fit model from the parameter search
is selected based on the minimum Akaike information criteria [23].
Table 3. Initial values and phase space search increments for the exceedance model parameters of
Equation (1).
Parameter Values Increment
E0 1 0
α 1 0
TL 1990–2010 5
TG 1995–2010 5
r 0–200 20
τ 0–60 20
To forecast the evolution of water level exceedance, we select an elevation threshold with
landscape-specific relevance. For example, at the Little Madeira Bay (LM) station, inspection of
coastal ridge elevations from the United States Geological Survey (USGS) mapping [24] finds a mean
J. Mar. Sci. Eng. 2017, 5, 31 7 of 26
elevation of 70 cm NGVD29. Daily mean water levels are then extracted from the station data for the
most recent three-year period, and yearly values of sea level rise from the low and high sea level rise
projections are added to the dataset. Each set of yearly data is then processed to sum the total number
of yearly threshold exceedances per year.
2.6. Marsh to Ocean Transformation Index
As sea levels rise, we expect a gradual transformation of freshwater coastal marshes into saltwater
marshes and eventually into submarine basins. Florida Bay is largely open to the Gulf of Mexico
to the west and relatively isolated from the Atlantic Ocean to the east by the island chain of the
Florida Keys; as such, marine conditions can be found in western Florida Bay as shown by the
tidally-dominated water levels at Buoy Key (BK) (Figure 3) and marine-like salinities at Murray Key
(MK) and Buoy Key (Figure 4). As one moves eastward, the tidal signal diminishes (LM in Figure 3)
with a transition to a terrestrial hydrologic cycle dominated by seasonal rainfall moving up Taylor
Slough (Taylor River (TR), E146 and Taylor Slough Hilton (TSH)).
To assess this change, we decompose the water level signals shown in Figure 3 using IMFs
retaining only modes with intra-annual and longer oscillatory cycles, as shown in Figure 5. These low
pass versions of water levels allow one to recognize lower amplitude ocean-dominated locations such
as Buoy Key (BK) and the higher amplitude, more variable marsh-dominated water levels exemplified
at TSH.
Figure 5. Low frequency cumulative IMFs of water level data in Florida Bay and Taylor Slough shown
in Figure 3. (a) BK; (b) LM; (c) TR; (d) E146; (e) TSH.
We next identify IMFs representing ocean-dominated and freshwater marsh-dominated locales at
BK and TSH, respectively, as shown in Figure 6, and use these IMFs as empirical basis functions
to reconstruct the low pass water level signals at the intermediate stations LM, TR and E146.
The reconstruction is based on linear combinations of weighted ocean and marsh basis functions with
the goal of comparing the relative magnitudes of the ocean and marsh basis function fit coefficients as
a metric describing the relative hydrologic influence of the marsh or ocean at a particular station.
J. Mar. Sci. Eng. 2017, 5, 31 8 of 26
Figure 6. Low frequency IMFs at the BK and TSH stations to represent ocean-dominated and
marsh-dominated hydrologic dynamics respectively. (a) Intra-annual modes; (b) annual modes;
(c) comparison of low pass water level signals at BK and TSH constructed from the addition of the
IMFs in (a) and (b).
The model is thus:
W(t) =
i=H
∑
i=L
ωi
IMFΩi + µi
IMFMi
(2)
where IMFΩ represent ocean-dominated empirical basis functions, IMFM marsh-dominated basis
functions, L the IMF mode number of the lowest frequency mode or residual, H the mode number of
the highest frequency mode and ωi and µi fit coefficients determined by a nonlinear quasi-Newton
minimization of the variance of the difference between the weighted sum of the empirical basis
functions, W(t), and the target time series (low pass signal of station LM, TR or E146 shown in
Figure 5) [25].
The resultant coefficient vectors ω and µ are summed to produce an overall metric Ω = ∑ ωi
,
M = ∑ µi representing the ocean or marsh influence. For example, with N = 3 empirical basis
functions and using the Buoy Key (BK) time series as the target, all ωi equal 1 with the result Ω = 3,
M = 0, while if TSH is the target then Ω = 0, M = 3. To construct a relative metric denoted as the
Marsh-to-Ocean Index (MOI), we normalize the difference of the two influence metrics by the number
of basis functions N:
MOI =
M − Ω
N
(3)
so that a water level signal identical with that of Buoy Key (BK) would express MOI = −1, while a

TR Taylor River 0.08 0.37 0.89 0.125

MK Murray Key −0.51 0.22 0.89 0.127 14.67 34.84 54.79 3.60

E146 Taylor Slough −0.18 0.39 0.80 0.143

TSH Taylor Slough Hilton −0.12 0.63 1.02 0.176

2.4. Empirical Mode Decomposition

Water level and salinity data are decomposed into Intrinsic Mode Functions (IMFs) and nonlinear

trends through Empirical Mode Decomposition (EMD) using the Hilbert–Huang transform [20,21]

as implemented in the R package hht. Application of the EMD requires uniformly-sampled data

without gaps. We reconstruct missing data in our time series by using random samples drawn from

distributions of all available data for a specific year day. For example, if 1 January 2000 is missing,

a Gaussian kernel is fit to all available data for 1 January. A random sample is then drawn from this

distribution and used as the reconstructed value. This preserves the overall distribution of the data for

a year day capturing seasonal trends, while realistically allowing for variance away from the mean on

the daily timescale.

2.5. Water Level Exceedance

Water level exceedances are computed from daily mean water levels by summing the number

of exceedance events above an elevation threshold for each year. The probability of exceedance at

a specific threshold as a function of time follows a logistic function exhibiting exponential growth

followed by a linear increase, terminating in nonlinear saturation as water levels continuously exceed

the threshold [22]. The logistic function suggests a growth model for water level exceedances as they

enter the initial growth phase:

E(t) = E0 + α(t − TL) + (1 + r)

t−TG

τ (1)

where E0 is the number of exceedances at year t = 0; α the linear rate of exceedance; r the growth rate;

TL and TG the zero-crossing time of linear and exponential growth, respectively; and τ the growth

time constant. This model is fit to yearly exceedance data with maximum likelihood estimation over a

wide parameter space of initial conditions (Table 3), and the best-fit model from the parameter search

is selected based on the minimum Akaike information criteria [23].

Table 3. Initial values and phase space search increments for the exceedance model parameters of

Equation (1).

Parameter Values Increment

E0 1 0

α 1 0

TL 1990–2010 5

TG 1995–2010 5

r 0–200 20

τ 0–60 20

To forecast the evolution of water level exceedance, we select an elevation threshold with

landscape-specific relevance. For example, at the Little Madeira Bay (LM) station, inspection of

coastal ridge elevations from the United States Geological Survey (USGS) mapping [24] finds a mean

J. Mar. Sci. Eng. 2017, 5, 31 7 of 26

elevation of 70 cm NGVD29. Daily mean water levels are then extracted from the station data for the

most recent three-year period, and yearly values of sea level rise from the low and high sea level rise

projections are added to the dataset. Each set of yearly data is then processed to sum the total number

of yearly threshold exceedances per year.

2.6. Marsh to Ocean Transformation Index

As sea levels rise, we expect a gradual transformation of freshwater coastal marshes into saltwater

marshes and eventually into submarine basins. Florida Bay is largely open to the Gulf of Mexico

to the west and relatively isolated from the Atlantic Ocean to the east by the island chain of the

Florida Keys; as such, marine conditions can be found in western Florida Bay as shown by the

tidally-dominated water levels at Buoy Key (BK) (Figure 3) and marine-like salinities at Murray Key

(MK) and Buoy Key (Figure 4). As one moves eastward, the tidal signal diminishes (LM in Figure 3)

with a transition to a terrestrial hydrologic cycle dominated by seasonal rainfall moving up Taylor

Slough (Taylor River (TR), E146 and Taylor Slough Hilton (TSH)).

To assess this change, we decompose the water level signals shown in Figure 3 using IMFs

retaining only modes with intra-annual and longer oscillatory cycles, as shown in Figure 5. These low

pass versions of water levels allow one to recognize lower amplitude ocean-dominated locations such

as Buoy Key (BK) and the higher amplitude, more variable marsh-dominated water levels exemplified

at TSH.

Figure 5. Low frequency cumulative IMFs of water level data in Florida Bay and Taylor Slough shown

in Figure 3. (a) BK; (b) LM; (c) TR; (d) E146; (e) TSH.

We next identify IMFs representing ocean-dominated and freshwater marsh-dominated locales at

BK and TSH, respectively, as shown in Figure 6, and use these IMFs as empirical basis functions

to reconstruct the low pass water level signals at the intermediate stations LM, TR and E146.

The reconstruction is based on linear combinations of weighted ocean and marsh basis functions with

the goal of comparing the relative magnitudes of the ocean and marsh basis function fit coefficients as

a metric describing the relative hydrologic influence of the marsh or ocean at a particular station.

J. Mar. Sci. Eng. 2017, 5, 31 8 of 26

Figure 6. Low frequency IMFs at the BK and TSH stations to represent ocean-dominated and

marsh-dominated hydrologic dynamics respectively. (a) Intra-annual modes; (b) annual modes;

(c) comparison of low pass water level signals at BK and TSH constructed from the addition of the

IMFs in (a) and (b).

The model is thus:

W(t) =

i=H

∑

i=L

ωi

IMFΩi + µi

IMFMi

(2)

where IMFΩ represent ocean-dominated empirical basis functions, IMFM marsh-dominated basis

functions, L the IMF mode number of the lowest frequency mode or residual, H the mode number of

the highest frequency mode and ωi and µi fit coefficients determined by a nonlinear quasi-Newton

minimization of the variance of the difference between the weighted sum of the empirical basis

functions, W(t), and the target time series (low pass signal of station LM, TR or E146 shown in

Figure 5) [25].

The resultant coefficient vectors ω and µ are summed to produce an overall metric Ω = ∑ ωi

,

M = ∑ µi representing the ocean or marsh influence. For example, with N = 3 empirical basis

functions and using the Buoy Key (BK) time series as the target, all ωi equal 1 with the result Ω = 3,

M = 0, while if TSH is the target then Ω = 0, M = 3. To construct a relative metric denoted as the

Marsh-to-Ocean Index (MOI), we normalize the difference of the two influence metrics by the number

of basis functions N:

MOI =

M − Ω

N

(3)

so that a water level signal identical with that of Buoy Key (BK) would express MOI = −1, while a