This page was generated from docs/source/examples/example_2D_MITGCM_Conditional.ipynb.

2D Structure Function / LLC4320 Data

This example will guide you through each step necessary to compute scalar-based structure functions from a 2D dycoms dataset

\(\textbf{General procedure}:\)

1 - Load LLC4320 Subset dataset, the dataset can be accessed upon request

2 - Format dataset to match pyturbo_sf inputs

3 - Calculate second-order Isotropic structure function as a function of separation distances

4 - Plot the Isotropic SF with errors and number of bootstraps and slope

5 - Compute conditional (Sea-Ice concentration) Second Order SF

6 - Plot Conditional SF

\(\textbf{Motivation:}\)

This example demonstrates the computation of second order structure functions from 2D MITGCM Arctic data. It also shows how to compute conditional Structure function based on some variable, here Sea Ice Concentration

\(\textbf{Note:}\)

This is Part of Ashfaq Ahmed’s PhD work. Please feel free to reach out to him through email for more questions (ashfaq_ahmed@brown.edu)

import xarray as xr
import matplotlib.pyplot as plt
from matplotlib.colors import LogNorm, SymLogNorm
import matplotlib.ticker as ticker
import pyturbo_sf as psf
import pyproj
import numpy as np
from scipy import stats
import matplotlib.colors as mcolors
from matplotlib.pyplot import cm
linewidth = 2
fontsize = 12
plt.rcParams['xtick.labelsize'] = fontsize
plt.rcParams['ytick.labelsize'] = fontsize
plt.rcParams['xtick.major.width'] = 2
plt.rcParams['xtick.minor.width'] = 2
plt.rcParams['ytick.major.width'] = 2
plt.rcParams['ytick.minor.width'] = 2
plt.rcParams['xtick.major.size'] = 10
plt.rcParams['xtick.minor.size'] = 5
plt.rcParams['ytick.major.size'] = 10
plt.rcParams['ytick.minor.size'] = 5
plt.rcParams['savefig.dpi'] = 150
plt.rc('font', family='serif')
import gc

Load Dataset

idt = 330
dsg = xr.open_dataset('/home/aayouche/Downloads/FRAM_grid.nc')
dsu = xr.open_dataset('/home/aayouche/Downloads/FRAM_u.nc').isel(time=idt)
dsv = xr.open_dataset('/home/aayouche/Downloads/FRAM_v.nc').isel(time=idt)
dsm = xr.open_dataset('/home/aayouche/Downloads/FRAM_sic.nc').isel(time=idt)
u = dsu.U.rename({'j':'y','i':'x'})
v = dsv.V.rename({'j':'y','i':'x'})
sic = dsm.SIarea.rename({'j':'y','i':'x'})
lon = dsg.XC
lat = dsg.YC


# Create the projection
proj = pyproj.Proj("EPSG:3995")

x, y = proj(lon.values, lat.values)


y_dim, x_dim = x.shape

X = xr.DataArray(
    x,
    dims=['y', 'x'],
    coords={
        'y': np.arange(y_dim),
        'x': np.arange(x_dim)
    }
)

Y = xr.DataArray(
    y,
    dims=['y', 'x'],
    coords={
        'y': np.arange(y_dim),
        'x': np.arange(x_dim)
    }
)

Make a quick plot

u.plot()

<matplotlib.collections.QuadMesh at 0x7a0e94ce0980>

Prepare Dataset for Pyturbo_SF

ds = xr.Dataset(
    data_vars={
        'u': u,
        'v': v,
        'sic': sic,
    },
    coords={
        'x': X,
        'y': Y,
    }
)

ds

<xarray.Dataset> Size: 158MB
Dimensions:  (y: 2820, x: 2000)
Coordinates:
    y        (y, x) float64 45MB -3.539e+05 -3.54e+05 ... 1.968e+06 1.968e+06
    x        (y, x) float64 45MB 1.999e+06 1.999e+06 ... -9.46e+04 -9.55e+04
    face     int64 8B 6
    k        int64 8B 1
    time     datetime64[ns] 8B 2012-08-08
Data variables:
    u        (y, x) float32 23MB ...
    v        (y, x) float32 23MB ...
    sic      (y, x) float32 23MB ...

xarray.Dataset

• Dimensions:
  • y: 2820
  • x: 2000
• Coordinates: (5)
  • y
    (y, x)
    float64
    -3.539e+05 -3.54e+05 ... 1.968e+06

    array([[-353937.90918586, -353973.43313391, -354009.65816908, ...,
            -534438.34234091, -534540.6807326 , -534643.0148121 ],
           [-353036.90956543, -353072.54823077, -353108.88983503, ...,
            -533587.96660409, -533690.3024705 , -533792.63403714],
           [-352135.90257089, -352171.65596923, -352208.11415852, ...,
            -532737.65958235, -532839.99293221, -532942.3219947 ],
           ...,
           [1950322.19646347, 1950288.64302676, 1950265.82994278, ...,
            1965972.69747253, 1965880.83002748, 1965788.84998196],
           [1950443.06538133, 1950421.04808452, 1950410.61558347, ...,
            1966902.22593981, 1966810.36534128, 1966718.39201052],
           [1950542.49776041, 1950539.84198774, 1950545.89571739, ...,
            1967815.86875521, 1967724.01487893, 1967632.04814089]])

  • x
    (y, x)
    float64
    1.999e+06 1.999e+06 ... -9.55e+04

    array([[1999468.49247573, 1998575.1165792 , 1997666.2056603 , ...,
             202344.91060701,  201515.31105105,  200685.72547682],
           [1999432.90556017, 1998539.45236006, 1997630.46280833, ...,
             202240.03158589,  201410.43422214,  200580.85084843],
           [1999397.20530291, 1998503.67496073, 1997594.60694042, ...,
             202135.16331971,  201305.56813555,  200475.98694973],
           ...,
           [1523989.24665389, 1523858.19091545, 1523714.66080149, ...,
             -93510.17977941,  -94408.80751244,  -95307.4239226 ],
           [1523952.87159171, 1523835.93050461, 1523702.07316213, ...,
             -93605.20290703,  -94503.94867074,  -95402.68310285],
           [1523930.81739072, 1523829.00318414, 1523702.9246657 , ...,
             -93698.58296965,  -94597.44486439,  -95496.29541892]])

  • face
    ()
    int64
    6

    array(6)

  • k
    ()
    int64
    1

    array(1)

  • time
    ()
    datetime64[ns]
    2012-08-08

    array('2012-08-08T00:00:00.000000000', dtype='datetime64[ns]')

• Data variables: (3)
  • u
    (y, x)
    float32
    ...

    [5640000 values with dtype=float32]

  • v
    (y, x)
    float32
    ...

    [5640000 values with dtype=float32]

  • sic
    (y, x)
    float32
    ...

    [5640000 values with dtype=float32]

Define Keywords for Isotropic/Conditional Second Order SF

nbins = 11
# Fewer bins = more shifts per bin
bin_r = np.logspace(np.log10(1e3), np.log10(1e6), nbins+1)  # Start at 5 km
bins = {'r': bin_r}

sf_kwargs_is = dict(
    ds=ds,
    variables_names=['u','v'],
    bootsize={'y': 88//4, 'x': 62//4},
    bins=bins,
    fun='default_vel',
    order=2,
    initial_nbootstrap=5000,
    max_nbootstrap=5000,
    step_nbootstrap=0,
    convergence_eps=0.001,
    confidence_interval=0.95,
    n_jobs=-1,
    backend='loky',
    seed=42
)

sf_kwargs_cond = dict(
    ds=ds,
    variables_names=['u','v'],
    bootsize={'y': 88//4, 'x': 62//4},
    bins=bins,
    fun='default_vel',
    order=2,
    initial_nbootstrap=5000,
    max_nbootstrap=5000,
    step_nbootstrap=0,
    convergence_eps=0.001,
    confidence_interval=0.95,
    conditioning_var='sic',
    conditioning_bins=np.linspace(-0.01,1.01,11),#[0,0.15] or [0.15,0.8] or [0.8,1.0]
    n_jobs=-1,
    backend='loky',
    seed=42
)

Compute Second Order SF & Conditional

%%time
iso_sf = psf.get_isotropic_sf_2d(**sf_kwargs_is)
gc.collect()
iso_cond = psf.get_isotropic_sf_2d(**sf_kwargs_cond)

Dimensions ('y', 'x') are already in the expected order
Using bootsize: {'y': 22, 'x': 15}
Bootstrappable dimensions: ['y', 'x']
Two bootstrappable dimensions. Available spacings: [1, 2, 4, 8, 16, 32, 64, 128]

============================================================
STARTING ISOTROPIC_SF WITH FUNCTION TYPE: default_vel
Variables: ['u', 'v'], Order: 2
Confidence level: 0.95
============================================================


INITIAL BOOTSTRAP PHASE
Processing spacing 1 with 625 bootstraps
Processing spacing 2 with 625 bootstraps
Processing spacing 4 with 625 bootstraps
Processing spacing 8 with 625 bootstraps
Processing spacing 16 with 625 bootstraps
Processing spacing 32 with 625 bootstraps
Processing spacing 64 with 625 bootstraps
Processing spacing 128 with 625 bootstraps

CALCULATING BIN DENSITIES
Total points collected: 70309354
Bins with points: 11/11
Marked 6 bins as converged (converged_eps)
Marked 5 bins as converged (max_bootstraps)

STARTING ADAPTIVE CONVERGENCE LOOP
All bins have converged or reached max bootstraps!

FINAL CONVERGENCE STATISTICS:
  Total bins with data (>10 points): 11
  Converged bins: 11
  Unconverged bins: 0
  Bins at max bootstraps: 11

Creating output dataset...
ISOTROPIC SF COMPLETED SUCCESSFULLY!
============================================================

============================================================
MULTI-BIN CONDITIONING: 10 bins for sic
Bin edges: [-0.01   0.092  0.194  0.296  0.398  0.5    0.602  0.704  0.806  0.908
  1.01 ]
Using seed 42 for reproducible bootstrap sampling
============================================================


--- Processing conditioning bin 1/10: [-0.01, 0.092) ---
Dimensions ('y', 'x') are already in the expected order
Using bootsize: {'y': 22, 'x': 15}
Bootstrappable dimensions: ['y', 'x']
Two bootstrappable dimensions. Available spacings: [1, 2, 4, 8, 16, 32, 64, 128]

============================================================
STARTING ISOTROPIC_SF WITH FUNCTION TYPE: default_vel
Variables: ['u', 'v'], Order: 2
Confidence level: 0.95
Conditioning: sic in [np.float64(-0.01), np.float64(0.09200000000000001)]
============================================================


INITIAL BOOTSTRAP PHASE
Processing spacing 1 with 625 bootstraps
Processing spacing 2 with 625 bootstraps
Processing spacing 4 with 625 bootstraps
Processing spacing 8 with 625 bootstraps
Processing spacing 16 with 625 bootstraps
Processing spacing 32 with 625 bootstraps
Processing spacing 64 with 625 bootstraps
Processing spacing 128 with 625 bootstraps

CALCULATING BIN DENSITIES
Total points collected: 36899281
Bins with points: 11/11
Marked 6 bins as converged (converged_eps)
Marked 5 bins as converged (max_bootstraps)

STARTING ADAPTIVE CONVERGENCE LOOP
All bins have converged or reached max bootstraps!

FINAL CONVERGENCE STATISTICS:
  Total bins with data (>10 points): 11
  Converged bins: 11
  Unconverged bins: 0
  Bins at max bootstraps: 11

Creating output dataset...
ISOTROPIC SF COMPLETED SUCCESSFULLY!
============================================================

--- Processing conditioning bin 2/10: [0.092, 0.194) ---
Dimensions ('y', 'x') are already in the expected order
Using bootsize: {'y': 22, 'x': 15}
Bootstrappable dimensions: ['y', 'x']
Two bootstrappable dimensions. Available spacings: [1, 2, 4, 8, 16, 32, 64, 128]

============================================================
STARTING ISOTROPIC_SF WITH FUNCTION TYPE: default_vel
Variables: ['u', 'v'], Order: 2
Confidence level: 0.95
Conditioning: sic in [np.float64(0.09200000000000001), np.float64(0.194)]
============================================================


INITIAL BOOTSTRAP PHASE
Processing spacing 1 with 625 bootstraps
Processing spacing 2 with 625 bootstraps
Processing spacing 4 with 625 bootstraps
Processing spacing 8 with 625 bootstraps
Processing spacing 16 with 625 bootstraps
Processing spacing 32 with 625 bootstraps
Processing spacing 64 with 625 bootstraps
Processing spacing 128 with 625 bootstraps

CALCULATING BIN DENSITIES
Total points collected: 1005052
Bins with points: 11/11
Marked 1 bins as converged (converged_eps)
Marked 10 bins as converged (max_bootstraps)

STARTING ADAPTIVE CONVERGENCE LOOP
All bins have converged or reached max bootstraps!

FINAL CONVERGENCE STATISTICS:
  Total bins with data (>10 points): 11
  Converged bins: 11
  Unconverged bins: 0
  Bins at max bootstraps: 11

Creating output dataset...
ISOTROPIC SF COMPLETED SUCCESSFULLY!
============================================================

--- Processing conditioning bin 3/10: [0.194, 0.296) ---
Dimensions ('y', 'x') are already in the expected order
Using bootsize: {'y': 22, 'x': 15}
Bootstrappable dimensions: ['y', 'x']
Two bootstrappable dimensions. Available spacings: [1, 2, 4, 8, 16, 32, 64, 128]

============================================================
STARTING ISOTROPIC_SF WITH FUNCTION TYPE: default_vel
Variables: ['u', 'v'], Order: 2
Confidence level: 0.95
Conditioning: sic in [np.float64(0.194), np.float64(0.29600000000000004)]
============================================================


INITIAL BOOTSTRAP PHASE
Processing spacing 1 with 625 bootstraps
Processing spacing 2 with 625 bootstraps
Processing spacing 4 with 625 bootstraps
Processing spacing 8 with 625 bootstraps
Processing spacing 16 with 625 bootstraps
Processing spacing 32 with 625 bootstraps
Processing spacing 64 with 625 bootstraps
Processing spacing 128 with 625 bootstraps

CALCULATING BIN DENSITIES
Total points collected: 1340681
Bins with points: 11/11
Marked 1 bins as converged (converged_eps)
Marked 10 bins as converged (max_bootstraps)

STARTING ADAPTIVE CONVERGENCE LOOP
All bins have converged or reached max bootstraps!

FINAL CONVERGENCE STATISTICS:
  Total bins with data (>10 points): 11
  Converged bins: 11
  Unconverged bins: 0
  Bins at max bootstraps: 11

Creating output dataset...
ISOTROPIC SF COMPLETED SUCCESSFULLY!
============================================================

--- Processing conditioning bin 4/10: [0.296, 0.398) ---
Dimensions ('y', 'x') are already in the expected order
Using bootsize: {'y': 22, 'x': 15}
Bootstrappable dimensions: ['y', 'x']
Two bootstrappable dimensions. Available spacings: [1, 2, 4, 8, 16, 32, 64, 128]

============================================================
STARTING ISOTROPIC_SF WITH FUNCTION TYPE: default_vel
Variables: ['u', 'v'], Order: 2
Confidence level: 0.95
Conditioning: sic in [np.float64(0.29600000000000004), np.float64(0.398)]
============================================================


INITIAL BOOTSTRAP PHASE
Processing spacing 1 with 625 bootstraps
Processing spacing 2 with 625 bootstraps
Processing spacing 4 with 625 bootstraps
Processing spacing 8 with 625 bootstraps
Processing spacing 16 with 625 bootstraps
Processing spacing 32 with 625 bootstraps
Processing spacing 64 with 625 bootstraps
Processing spacing 128 with 625 bootstraps

CALCULATING BIN DENSITIES
Total points collected: 1754001
Bins with points: 11/11
Marked 1 bins as converged (converged_eps)
Marked 10 bins as converged (max_bootstraps)

STARTING ADAPTIVE CONVERGENCE LOOP
All bins have converged or reached max bootstraps!

FINAL CONVERGENCE STATISTICS:
  Total bins with data (>10 points): 11
  Converged bins: 11
  Unconverged bins: 0
  Bins at max bootstraps: 11

Creating output dataset...
ISOTROPIC SF COMPLETED SUCCESSFULLY!
============================================================

--- Processing conditioning bin 5/10: [0.398, 0.5) ---
Dimensions ('y', 'x') are already in the expected order
Using bootsize: {'y': 22, 'x': 15}
Bootstrappable dimensions: ['y', 'x']
Two bootstrappable dimensions. Available spacings: [1, 2, 4, 8, 16, 32, 64, 128]

============================================================
STARTING ISOTROPIC_SF WITH FUNCTION TYPE: default_vel
Variables: ['u', 'v'], Order: 2
Confidence level: 0.95
Conditioning: sic in [np.float64(0.398), np.float64(0.5)]
============================================================


INITIAL BOOTSTRAP PHASE
Processing spacing 1 with 625 bootstraps
Processing spacing 2 with 625 bootstraps
Processing spacing 4 with 625 bootstraps
Processing spacing 8 with 625 bootstraps
Processing spacing 16 with 625 bootstraps
Processing spacing 32 with 625 bootstraps
Processing spacing 64 with 625 bootstraps
Processing spacing 128 with 625 bootstraps

CALCULATING BIN DENSITIES
Total points collected: 2054064
Bins with points: 11/11
Marked 2 bins as converged (converged_eps)
Marked 9 bins as converged (max_bootstraps)

STARTING ADAPTIVE CONVERGENCE LOOP
All bins have converged or reached max bootstraps!

FINAL CONVERGENCE STATISTICS:
  Total bins with data (>10 points): 11
  Converged bins: 11
  Unconverged bins: 0
  Bins at max bootstraps: 11

Creating output dataset...
ISOTROPIC SF COMPLETED SUCCESSFULLY!
============================================================

--- Processing conditioning bin 6/10: [0.5, 0.602) ---
Dimensions ('y', 'x') are already in the expected order
Using bootsize: {'y': 22, 'x': 15}
Bootstrappable dimensions: ['y', 'x']
Two bootstrappable dimensions. Available spacings: [1, 2, 4, 8, 16, 32, 64, 128]

============================================================
STARTING ISOTROPIC_SF WITH FUNCTION TYPE: default_vel
Variables: ['u', 'v'], Order: 2
Confidence level: 0.95
Conditioning: sic in [np.float64(0.5), np.float64(0.6020000000000001)]
============================================================


INITIAL BOOTSTRAP PHASE
Processing spacing 1 with 625 bootstraps
Processing spacing 2 with 625 bootstraps
Processing spacing 4 with 625 bootstraps
Processing spacing 8 with 625 bootstraps
Processing spacing 16 with 625 bootstraps
Processing spacing 32 with 625 bootstraps
Processing spacing 64 with 625 bootstraps
Processing spacing 128 with 625 bootstraps

CALCULATING BIN DENSITIES
Total points collected: 2123942
Bins with points: 11/11
Marked 2 bins as converged (converged_eps)
Marked 9 bins as converged (max_bootstraps)

STARTING ADAPTIVE CONVERGENCE LOOP
All bins have converged or reached max bootstraps!

FINAL CONVERGENCE STATISTICS:
  Total bins with data (>10 points): 11
  Converged bins: 11
  Unconverged bins: 0
  Bins at max bootstraps: 11

Creating output dataset...
ISOTROPIC SF COMPLETED SUCCESSFULLY!
============================================================

--- Processing conditioning bin 7/10: [0.602, 0.704) ---
Dimensions ('y', 'x') are already in the expected order
Using bootsize: {'y': 22, 'x': 15}
Bootstrappable dimensions: ['y', 'x']
Two bootstrappable dimensions. Available spacings: [1, 2, 4, 8, 16, 32, 64, 128]

============================================================
STARTING ISOTROPIC_SF WITH FUNCTION TYPE: default_vel
Variables: ['u', 'v'], Order: 2
Confidence level: 0.95
Conditioning: sic in [np.float64(0.6020000000000001), np.float64(0.7040000000000001)]
============================================================


INITIAL BOOTSTRAP PHASE
Processing spacing 1 with 625 bootstraps
Processing spacing 2 with 625 bootstraps
Processing spacing 4 with 625 bootstraps
Processing spacing 8 with 625 bootstraps
Processing spacing 16 with 625 bootstraps
Processing spacing 32 with 625 bootstraps
Processing spacing 64 with 625 bootstraps
Processing spacing 128 with 625 bootstraps

CALCULATING BIN DENSITIES
Total points collected: 3669126
Bins with points: 11/11
Marked 3 bins as converged (converged_eps)
Marked 8 bins as converged (max_bootstraps)

STARTING ADAPTIVE CONVERGENCE LOOP
All bins have converged or reached max bootstraps!

FINAL CONVERGENCE STATISTICS:
  Total bins with data (>10 points): 11
  Converged bins: 11
  Unconverged bins: 0
  Bins at max bootstraps: 11

Creating output dataset...
ISOTROPIC SF COMPLETED SUCCESSFULLY!
============================================================

--- Processing conditioning bin 8/10: [0.704, 0.806) ---
Dimensions ('y', 'x') are already in the expected order
Using bootsize: {'y': 22, 'x': 15}
Bootstrappable dimensions: ['y', 'x']
Two bootstrappable dimensions. Available spacings: [1, 2, 4, 8, 16, 32, 64, 128]

============================================================
STARTING ISOTROPIC_SF WITH FUNCTION TYPE: default_vel
Variables: ['u', 'v'], Order: 2
Confidence level: 0.95
Conditioning: sic in [np.float64(0.7040000000000001), np.float64(0.806)]
============================================================


INITIAL BOOTSTRAP PHASE
Processing spacing 1 with 625 bootstraps
Processing spacing 2 with 625 bootstraps
Processing spacing 4 with 625 bootstraps
Processing spacing 8 with 625 bootstraps
Processing spacing 16 with 625 bootstraps
Processing spacing 32 with 625 bootstraps
Processing spacing 64 with 625 bootstraps
Processing spacing 128 with 625 bootstraps

CALCULATING BIN DENSITIES
Total points collected: 8087840
Bins with points: 11/11
Marked 9 bins as converged (converged_eps)
Marked 2 bins as converged (max_bootstraps)

STARTING ADAPTIVE CONVERGENCE LOOP
All bins have converged or reached max bootstraps!

FINAL CONVERGENCE STATISTICS:
  Total bins with data (>10 points): 11
  Converged bins: 11
  Unconverged bins: 0
  Bins at max bootstraps: 11

Creating output dataset...
ISOTROPIC SF COMPLETED SUCCESSFULLY!
============================================================

--- Processing conditioning bin 9/10: [0.806, 0.908) ---
Dimensions ('y', 'x') are already in the expected order
Using bootsize: {'y': 22, 'x': 15}
Bootstrappable dimensions: ['y', 'x']
Two bootstrappable dimensions. Available spacings: [1, 2, 4, 8, 16, 32, 64, 128]

============================================================
STARTING ISOTROPIC_SF WITH FUNCTION TYPE: default_vel
Variables: ['u', 'v'], Order: 2
Confidence level: 0.95
Conditioning: sic in [np.float64(0.806), np.float64(0.908)]
============================================================


INITIAL BOOTSTRAP PHASE
Processing spacing 1 with 625 bootstraps
Processing spacing 2 with 625 bootstraps
Processing spacing 4 with 625 bootstraps
Processing spacing 8 with 625 bootstraps
Processing spacing 16 with 625 bootstraps
Processing spacing 32 with 625 bootstraps
Processing spacing 64 with 625 bootstraps
Processing spacing 128 with 625 bootstraps

CALCULATING BIN DENSITIES
Total points collected: 9951799
Bins with points: 11/11
Marked 11 bins as converged (converged_eps)

STARTING ADAPTIVE CONVERGENCE LOOP
All bins have converged or reached max bootstraps!

FINAL CONVERGENCE STATISTICS:
  Total bins with data (>10 points): 11
  Converged bins: 11
  Unconverged bins: 0
  Bins at max bootstraps: 11

Creating output dataset...
ISOTROPIC SF COMPLETED SUCCESSFULLY!
============================================================

--- Processing conditioning bin 10/10: [0.908, 1.01) ---
Dimensions ('y', 'x') are already in the expected order
Using bootsize: {'y': 22, 'x': 15}
Bootstrappable dimensions: ['y', 'x']
Two bootstrappable dimensions. Available spacings: [1, 2, 4, 8, 16, 32, 64, 128]

============================================================
STARTING ISOTROPIC_SF WITH FUNCTION TYPE: default_vel
Variables: ['u', 'v'], Order: 2
Confidence level: 0.95
Conditioning: sic in [np.float64(0.908), np.float64(1.01)]
============================================================


INITIAL BOOTSTRAP PHASE
Processing spacing 1 with 625 bootstraps
Processing spacing 2 with 625 bootstraps
Processing spacing 4 with 625 bootstraps
Processing spacing 8 with 625 bootstraps
Processing spacing 16 with 625 bootstraps
Processing spacing 32 with 625 bootstraps
Processing spacing 64 with 625 bootstraps
Processing spacing 128 with 625 bootstraps

CALCULATING BIN DENSITIES
Total points collected: 3423568
Bins with points: 11/11
Marked 10 bins as converged (converged_eps)
Marked 1 bins as converged (max_bootstraps)

STARTING ADAPTIVE CONVERGENCE LOOP
All bins have converged or reached max bootstraps!

FINAL CONVERGENCE STATISTICS:
  Total bins with data (>10 points): 11
  Converged bins: 11
  Unconverged bins: 0
  Bins at max bootstraps: 11

Creating output dataset...
ISOTROPIC SF COMPLETED SUCCESSFULLY!
============================================================

============================================================
MULTI-BIN CONDITIONING COMPLETE
Output dimensions: {'theta': 36, 'r': 11, 'cond_bin': 10, 'r_subset': 9, 'r_edge': 12, 'theta_edge': 37}
============================================================

CPU times: user 2min 15s, sys: 38.2 s, total: 2min 53s
Wall time: 4min 51s

/home/aayouche/Documents/postdoc/coding/pyturbo_sf/src/pyturbo_sf/two_dimensional.py:271: FutureWarning: In a future version of xarray the default value for data_vars will change from data_vars='all' to data_vars=None. This is likely to lead to different results when multiple datasets have matching variables with overlapping values. To opt in to new defaults and get rid of these warnings now use `set_options(use_new_combine_kwarg_defaults=True) or set data_vars explicitly.
  ds_combined = xr.concat(datasets, dim='cond_bin')

gc.collect()

0

Plot Second Order SF

# Create separate figures for better spacing and clarity
plt.figure(figsize=(10, 6))
# Structure Function with Confidence Interval
# Add longitudinal structure function with error bars
plt.loglog(iso_sf.r/1e3, iso_sf.sf, 'b--', linewidth=1.5, label='Second Order SF')
plt.fill_between(iso_sf.r/1e3,iso_sf.ci_lower,iso_sf.ci_upper,color='b',alpha=0.3)
plt.axvline(1,linestyle='--',alpha=0.5,color='k')
plt.axvline(10,linestyle='--',alpha=0.5,color='k')

cond = np.max(np.where(iso_sf.r<=10.0e3)) # Compute the actual slope from the data (excluding first and last points)

log_r = np.log10(iso_sf.r[:cond+1])
log_sf = np.log10(iso_sf.sf[:cond+1])
slope, intercept, r_value, p_value, std_err = stats.linregress(log_r, log_sf)

# Plot the fitted line
r_fit = iso_sf.r/1e3
fitted_line = 10**intercept * (r_fit*1e3)**slope
plt.loglog(r_fit, fitted_line, 'g-', alpha=0.7, linewidth=2,
           label=f'Fitted slope: {slope:.2f} (R²={r_value**2:.3f})')

# Add Kolmogorov 2/3 slope reference line
r_ref = iso_sf.r/1e3  # in km
mid_idx = len(r_ref)//2
amplitude = iso_sf.sf[mid_idx] / (r_ref[mid_idx]**(2/3))
kolmogorov_line = amplitude * r_ref**(2/3)
plt.loglog(r_ref, kolmogorov_line, 'k--', alpha=0.7, label=r'$r^{2/3}$ (Kolmogorov)')


# Formatting
plt.xlabel(r'$r$ [km]', fontsize=12, fontweight='bold')
plt.ylabel(r'$[m^{2}~s^{-2}]$', fontsize=12, fontweight='bold')
plt.title('Second Order Structure Function with Confidence Interval',
         fontsize=14, fontweight='bold', pad=10)
plt.legend(fontsize=11, frameon=True, facecolor='white', framealpha=0.9)
plt.grid(True, which='both', linestyle='--', alpha=0.6)
plt.yscale('log')
plt.xscale('log')

# Customize spines
ax = plt.gca()
for spine in ax.spines.values():
    spine.set_linewidth(1.5)

# Customize tick parameters
ax.tick_params(which='both', direction='in', width=1.2, length=5,
              labelsize=10, top=True, right=True)
ax.xaxis.set_major_formatter(ticker.LogFormatterSciNotation())

plt.tight_layout()

# --------------------------------------------------
# Errors of Isotropy and Homogeneity
# --------------------------------------------------
plt.figure(figsize=(12, 5))

# Error of isotropy
plt.subplot(121)
plt.plot(iso_sf.r, iso_sf.error_isotropy, 'b-', linewidth=2.5,label='Second Order SF')

plt.xlabel(r'$r$ [m]', fontsize=12, fontweight='bold')
plt.ylabel('Error of Isotropy'+' '+r'$[m^{2}~s^{-2}]$', fontsize=12, fontweight='bold')
plt.title('Error of Isotropy (Variation with Angle)',
         fontsize=14, fontweight='bold', pad=10)
plt.grid(True, which='both', linestyle='--', alpha=0.6)
plt.xscale('log')
plt.legend(fontsize=11, frameon=True, facecolor='white', framealpha=0.9)

# Customize spines and ticks
ax = plt.gca()
for spine in ax.spines.values():
    spine.set_linewidth(1.5)
ax.tick_params(which='both', direction='in', width=1.2, length=5,
              labelsize=10, top=True, right=True)
ax.xaxis.set_major_formatter(ticker.LogFormatterSciNotation())

# Error of homogeneity
plt.subplot(122)
plt.plot(iso_sf.r_subset, iso_sf.error_homogeneity, 'b-', linewidth=2.5,label='Second Order SF')
plt.xlabel(r'$r$ [m]', fontsize=12, fontweight='bold')
plt.ylabel('Error of Homogeneity'+' '+r'$[m^{2}~s^{-2}]$', fontsize=12, fontweight='bold')
plt.title('Error of Homogeneity (Variation with Radius)',
         fontsize=14, fontweight='bold', pad=10)
plt.grid(True, which='both', linestyle='--', alpha=0.6)
plt.xscale('log')
plt.legend(fontsize=11, frameon=True, facecolor='white', framealpha=0.9)

# Customize spines and ticks
ax = plt.gca()
for spine in ax.spines.values():
    spine.set_linewidth(1.5)
ax.tick_params(which='both', direction='in', width=1.2, length=5,
              labelsize=10, top=True, right=True)
ax.xaxis.set_major_formatter(ticker.LogFormatterSciNotation())

plt.tight_layout()

plt.show()

iso_cond

<xarray.Dataset> Size: 44kB
Dimensions:            (theta: 36, r: 11, cond_bin: 10, r_subset: 9,
                        r_edge: 12, theta_edge: 37)
Coordinates:
  * theta              (theta) float64 288B -3.054 -2.88 -2.705 ... 2.88 3.054
  * r                  (r) float64 88B 1.369e+03 2.565e+03 ... 7.305e+05
  * cond_bin           (cond_bin) float64 80B 0.041 0.143 0.245 ... 0.857 0.959
  * r_subset           (r_subset) float64 72B 1.369e+03 2.565e+03 ... 2.081e+05
Dimensions without coordinates: r_edge, theta_edge
Data variables: (12/13)
    sf_polar           (theta, r, cond_bin) float64 32kB 0.000438 ... 0.01002
    sf                 (r, cond_bin) float64 880B 0.0004967 ... 0.01651
    error_isotropy     (r, cond_bin) float64 880B 0.001452 nan nan ... nan nan
    std_error          (r, cond_bin) float64 880B 5.593e-05 ... 0.0003796
    ci_upper           (r, cond_bin) float64 880B 0.0006063 0.001525 ... 0.01725
    ci_lower           (r, cond_bin) float64 880B 0.000387 0.0001191 ... 0.01577
    ...                 ...
    n_bootstrap        (r, cond_bin) int64 880B 5000 5000 5000 ... 5000 5000
    bin_density        (r, cond_bin) float32 440B 1.0 1.0 ... 7.936e-06
    point_counts       (r, cond_bin) int64 880B 287662 7030 ... 506873 121564
    converged          (r, cond_bin) bool 110B True True True ... True True True
    r_bins             (cond_bin, r_edge) float64 960B 1e+03 1.874e+03 ... 1e+06
    theta_bins         (cond_bin, theta_edge) float64 3kB -3.142 ... 3.142
Attributes: (12/15)
    order:                      2
    function_type:              default_vel
    window_size_theta:          12
    window_size_r:              3
    convergence_eps:            0.001
    max_nbootstrap:             5000
    ...                         ...
    confidence_level:           0.95
    bootstrappable_dimensions:  y,x
    backend:                    loky
    conditioning_variable:      sic
    conditioning_bin_edges:     [np.float64(-0.01), np.float64(0.092000000000...
    conditioning_bin_idx:       0

xarray.Dataset

• Dimensions:
  • theta: 36
  • r: 11
  • cond_bin: 10
  • r_subset: 9
  • r_edge: 12
  • theta_edge: 37
• Coordinates: (4)
  • theta
    (theta)
    float64
    -3.054 -2.88 -2.705 ... 2.88 3.054

    array([-3.054326, -2.879793, -2.70526 , -2.530727, -2.356194, -2.181662,
           -2.007129, -1.832596, -1.658063, -1.48353 , -1.308997, -1.134464,
           -0.959931, -0.785398, -0.610865, -0.436332, -0.261799, -0.087266,
            0.087266,  0.261799,  0.436332,  0.610865,  0.785398,  0.959931,
            1.134464,  1.308997,  1.48353 ,  1.658063,  1.832596,  2.007129,
            2.181662,  2.356194,  2.530727,  2.70526 ,  2.879793,  3.054326])

  • r
    (r)
    float64
    1.369e+03 2.565e+03 ... 7.305e+05

    array([  1368.87451 ,   2565.020906,   4806.380863,   9006.280202,
            16876.124758,  31622.776602,  59255.309755, 111033.631817,
           208056.753822, 389860.370255, 730527.154266])

  • cond_bin
    (cond_bin)
    float64
    0.041 0.143 0.245 ... 0.857 0.959

    array([0.041, 0.143, 0.245, 0.347, 0.449, 0.551, 0.653, 0.755, 0.857, 0.959])

  • r_subset
    (r_subset)
    float64
    1.369e+03 2.565e+03 ... 2.081e+05

    array([  1368.87451 ,   2565.020906,   4806.380863,   9006.280202,
            16876.124758,  31622.776602,  59255.309755, 111033.631817,
           208056.753822])

• Data variables: (13)
  • sf_polar
    (theta, r, cond_bin)
    float64
    0.000438 0.0009142 ... 0.01002

    array([[[4.37951219e-04, 9.14159220e-04, 5.08229344e-04, ...,
             3.70984543e-05, 4.20294911e-05, 1.42775256e-05],
            [1.25465537e-03, 2.11917271e-03, 1.47091810e-03, ...,
             2.22429576e-04, 1.01680160e-04, 6.26584190e-05],
            [3.50931169e-03, 4.70275876e-03, 6.01078974e-03, ...,
             6.37154721e-04, 4.29255931e-04, 1.34197524e-04],
            ...,
            [5.67990383e-02, 6.44159744e-02, 6.58439740e-02, ...,
             1.03807097e-02, 6.59501892e-03, 2.83379815e-03],
            [5.91535590e-02, 6.05004256e-02, 7.01822884e-02, ...,
             1.38626128e-02, 6.94967188e-03, 1.66090620e-03],
            [4.49120494e-02, 3.67873753e-02, 5.96313463e-02, ...,
             1.34225785e-02, 1.19722666e-02, 1.16924013e-02]],
    
           [[8.13795260e-05, 9.38238078e-05, 1.25147852e-04, ...,
             2.75872410e-05, 7.55491866e-06, 1.13735896e-05],
            [3.83058079e-04, 2.21583083e-04, 3.21883274e-04, ...,
             5.48007996e-05, 1.28292719e-05, 3.53035541e-05],
            [1.51383126e-03, 1.32041463e-03, 1.71983832e-03, ...,
             1.76478605e-04, 4.42586159e-05, 8.51550457e-05],
    ...
            [6.00018615e-02, 6.04870938e-02, 6.22855182e-02, ...,
             1.07726651e-02, 6.29988219e-03, 2.47057184e-03],
            [6.37508183e-02, 5.60575427e-02, 6.96512696e-02, ...,
             1.46211123e-02, 7.31906415e-03, 1.50403896e-03],
            [5.17157624e-02, 5.41604199e-02, 6.80076591e-02, ...,
             1.94279648e-02, 1.39347346e-02, 1.39576960e-02]],
    
           [[3.12796743e-03,            nan,            nan, ...,
                        nan,            nan,            nan],
            [1.75836128e-03, 3.17248047e-03, 3.99601721e-03, ...,
             5.53297241e-04, 3.65966076e-04, 6.16735791e-05],
            [3.86062360e-03, 7.30470872e-03, 8.38476260e-03, ...,
             9.51474556e-04, 7.06829809e-04, 1.26599675e-04],
            ...,
            [6.12112839e-02, 6.89061634e-02, 6.90828604e-02, ...,
             1.06575853e-02, 6.44409896e-03, 2.62896929e-03],
            [6.41829108e-02, 5.82938622e-02, 6.96138027e-02, ...,
             1.55361651e-02, 7.77553694e-03, 1.54134637e-03],
            [4.79977202e-02, 4.33581996e-02, 5.71445124e-02, ...,
             1.53846057e-02, 1.07926454e-02, 1.00190083e-02]]])

  • sf
    (r, cond_bin)
    float64
    0.0004967 0.0008222 ... 0.01651

    array([[4.96679675e-04, 8.22203042e-04, 6.44180920e-04, 3.52675788e-04,
            2.29719822e-04, 2.56896971e-04, 1.16221000e-04, 4.44044256e-05,
            3.29369732e-05, 1.55920504e-05],
           [1.52925493e-03, 2.85964676e-03, 2.80398887e-03, 1.70863996e-03,
            9.57989679e-04, 1.06759493e-03, 4.93337208e-04, 2.09967711e-04,
            1.23254798e-04, 5.56501593e-05],
           [4.08699136e-03, 9.15404384e-03, 8.13258144e-03, 5.55094455e-03,
            2.87795269e-03, 3.07001981e-03, 1.56096036e-03, 6.99677248e-04,
            4.03001354e-04, 1.26229693e-04],
           [1.03523255e-02, 2.39258315e-02, 1.62044098e-02, 1.30912959e-02,
            6.36005151e-03, 7.53982296e-03, 3.89211402e-03, 2.01132108e-03,
            7.15848787e-04, 3.18222424e-04],
           [2.02608653e-02, 3.36691304e-02, 2.80402664e-02, 2.39875318e-02,
            1.76306223e-02, 2.18939039e-02, 1.14330286e-02, 3.85935172e-03,
            1.40662979e-03, 7.92934657e-04],
           [3.13548083e-02, 5.00318225e-02, 4.21367504e-02, 3.63398581e-02,
            3.14820931e-02, 4.67228767e-02, 1.99991590e-02, 5.09631500e-03,
            2.73174760e-03, 1.65310555e-03],
           [4.19160653e-02, 5.95211807e-02, 5.03746359e-02, 4.81019448e-02,
            4.11679341e-02, 5.36157981e-02, 2.01034781e-02, 7.41938734e-03,
            3.94103066e-03, 2.71558678e-03],
           [4.91214103e-02, 6.40605427e-02, 5.82650844e-02, 5.01007077e-02,
            3.92450221e-02, 5.10016261e-02, 2.33119880e-02, 9.60500922e-03,
            5.59247922e-03, 3.98303761e-03],
           [6.14729331e-02, 6.18070498e-02, 5.99644519e-02, 5.37604495e-02,
            4.31452578e-02, 6.29401913e-02, 2.92233838e-02, 1.21181396e-02,
            7.00771980e-03, 6.94524006e-03],
           [6.56829131e-02, 5.72678449e-02, 6.15746133e-02, 5.20940707e-02,
            4.05280101e-02, 4.97639597e-02, 2.38621304e-02, 1.02403521e-02,
            7.38954167e-03, 1.37977185e-02],
           [5.03137257e-02, 4.20551506e-02, 4.81594948e-02, 4.05339543e-02,
            2.74459024e-02, 4.00322742e-02, 3.08024972e-02, 1.71016727e-02,
            1.82129994e-02, 1.65095359e-02]])

  • error_isotropy
    (r, cond_bin)
    float64
    0.001452 nan nan ... nan nan nan

    array([[0.0014519 ,        nan,        nan,        nan,        nan,
                   nan,        nan,        nan,        nan,        nan],
           [0.00252693,        nan,        nan,        nan,        nan,
                   nan,        nan,        nan,        nan,        nan],
           [0.00822136,        nan,        nan,        nan,        nan,
                   nan,        nan,        nan,        nan,        nan],
           [0.01518869,        nan,        nan,        nan,        nan,
                   nan,        nan,        nan,        nan,        nan],
           [0.02393926,        nan,        nan,        nan,        nan,
                   nan,        nan,        nan,        nan,        nan],
           [0.05471241,        nan,        nan,        nan,        nan,
                   nan,        nan,        nan,        nan,        nan],
           [0.01434204,        nan,        nan,        nan,        nan,
                   nan,        nan,        nan,        nan,        nan],
           [       nan,        nan,        nan,        nan,        nan,
                   nan,        nan,        nan,        nan,        nan],
           [       nan,        nan,        nan,        nan,        nan,
                   nan,        nan,        nan,        nan,        nan],
           [       nan,        nan,        nan,        nan,        nan,
                   nan,        nan,        nan,        nan,        nan],
           [       nan,        nan,        nan,        nan,        nan,
                   nan,        nan,        nan,        nan,        nan]])

  • std_error
    (r, cond_bin)
    float64
    5.593e-05 0.0003587 ... 0.0003796

    array([[5.59347860e-05, 3.58711036e-04, 4.03435665e-04, 1.99941962e-04,
            9.18132091e-05, 9.68925270e-05, 4.89980437e-05, 1.77803946e-05,
            2.00989492e-05, 2.71061559e-06],
           [1.40497792e-04, 1.35231856e-03, 1.85097247e-03, 1.02428155e-03,
            3.96439448e-04, 4.33516434e-04, 2.40793065e-04, 1.18447181e-04,
            7.69180532e-05, 9.50592843e-06],
           [3.58412895e-04, 4.22745506e-03, 4.15038895e-03, 2.71254043e-03,
            1.08684136e-03, 1.21500598e-03, 7.94769210e-04, 3.55961383e-04,
            3.05690324e-04, 2.00840351e-05],
           [7.93039103e-04, 8.53234716e-03, 6.03325443e-03, 3.95561137e-03,
            2.26829703e-03, 2.73348038e-03, 1.43793327e-03, 7.95213852e-04,
            4.14392933e-04, 4.06792986e-05],
           [1.11925667e-03, 6.13258589e-03, 5.81825232e-03, 5.19566827e-03,
            4.78471260e-03, 6.37604397e-03, 2.79429093e-03, 1.17821341e-03,
            5.04873316e-04, 9.06700769e-05],
           [1.38441510e-03, 5.78947465e-03, 6.18499278e-03, 6.12237951e-03,
            5.26046314e-03, 8.65132298e-03, 4.77472193e-03, 1.25396762e-03,
            6.41926562e-04, 1.45707362e-04],
           [1.68058006e-03, 5.21411607e-03, 4.77422458e-03, 5.28402200e-03,
            5.22955629e-03, 7.68710665e-03, 3.14771689e-03, 9.31380752e-04,
            6.14009787e-04, 1.94431108e-04],
           [1.64362970e-03, 3.23276024e-03, 3.69588103e-03, 4.11860526e-03,
            4.09853844e-03, 5.41414342e-03, 2.23251690e-03, 9.12528283e-04,
            6.48804383e-04, 3.57515516e-04],
           [1.51859834e-03, 2.44544541e-03, 2.73123800e-03, 3.37159356e-03,
            3.23444447e-03, 4.34813447e-03, 1.98237240e-03, 8.62294713e-04,
            4.53105182e-04, 9.73811207e-04],
           [9.37343637e-04, 2.06598616e-03, 2.96060480e-03, 3.23176833e-03,
            2.94126286e-03, 3.77189528e-03, 1.74512326e-03, 8.97359260e-04,
            3.27080402e-04, 1.17950731e-03],
           [4.82232220e-04, 1.11046391e-03, 1.21499297e-03, 1.25938106e-03,
            1.20699312e-03, 1.84759905e-03, 1.13661358e-03, 2.97494831e-04,
            2.34228550e-04, 3.79641178e-04]])

  • ci_upper
    (r, cond_bin)
    float64
    0.0006063 0.001525 ... 0.01725

    array([[6.06309841e-04, 1.52526375e-03, 1.43490029e-03, 7.44554832e-04,
            4.09670405e-04, 4.46802834e-04, 2.12255401e-04, 7.92533586e-05,
            7.23301897e-05, 2.09047593e-05],
           [1.80462554e-03, 5.51014244e-03, 6.43182824e-03, 3.71619492e-03,
            1.73499672e-03, 1.91727153e-03, 9.65282943e-04, 4.42119919e-04,
            2.74011412e-04, 7.42814366e-05],
           [4.78946773e-03, 1.74397035e-02, 1.62671943e-02, 1.08674261e-02,
            5.00812261e-03, 5.45138777e-03, 3.11867939e-03, 1.39734874e-03,
            1.00214338e-03, 1.65593679e-04],
           [1.19066536e-02, 4.06489246e-02, 2.80293712e-02, 2.08441517e-02,
            1.08058320e-02, 1.28973461e-02, 6.71041144e-03, 3.56991159e-03,
            1.52804401e-03, 3.97952384e-04],
           [2.24545681e-02, 4.56887779e-02, 3.94438314e-02, 3.41708545e-02,
            2.70084867e-02, 3.43907204e-02, 1.69097381e-02, 6.16860758e-03,
            2.39616331e-03, 9.70644742e-04],
           [3.40682120e-02, 6.13789843e-02, 5.42591135e-02, 4.83395014e-02,
            4.17924114e-02, 6.36791582e-02, 2.93574420e-02, 7.55404638e-03,
            3.98990055e-03, 1.93868673e-03],
           [4.52099417e-02, 6.97406604e-02, 5.97319441e-02, 5.84584376e-02,
            5.14176761e-02, 6.86822503e-02, 2.62728898e-02, 9.24486007e-03,
            5.14446772e-03, 3.09666475e-03],
           [5.23428653e-02, 7.03966363e-02, 6.55088781e-02, 5.81730257e-02,
            4.72780099e-02, 6.16131522e-02, 2.76876407e-02, 1.13935318e-02,
            6.86411244e-03, 4.68375514e-03],
           [6.44493311e-02, 6.66000347e-02, 6.53175800e-02, 6.03686514e-02,
            4.94846525e-02, 7.14623782e-02, 3.31087623e-02, 1.38082062e-02,
            7.89578964e-03, 8.85387495e-03],
           [6.75200729e-02, 6.13171034e-02, 6.73772921e-02, 5.84282202e-02,
            4.62927794e-02, 5.71567386e-02, 2.72825091e-02, 1.19991439e-02,
            8.03060748e-03, 1.61095104e-02],
           [5.12588835e-02, 4.42316199e-02, 5.05408372e-02, 4.30022958e-02,
            2.98115655e-02, 4.36535018e-02, 3.30302189e-02, 1.76847518e-02,
            1.86720789e-02, 1.72536189e-02]])

  • ci_lower
    (r, cond_bin)
    float64
    0.000387 0.0001191 ... 0.01577

    array([[ 3.87049509e-04,  1.19142331e-04, -1.46538453e-04,
            -3.92032572e-05,  4.97692386e-05,  6.69911073e-05,
             2.01865988e-05,  9.55549268e-06, -6.45624332e-06,
             1.02793414e-05],
           [ 1.25388432e-03,  2.09151078e-04, -8.23850496e-04,
            -2.98914998e-04,  1.80982639e-04,  2.17918338e-04,
             2.13914730e-05, -2.21844973e-05, -2.75018155e-05,
             3.70188819e-05],
           [ 3.38451500e-03,  8.68384178e-04, -2.03141342e-06,
             2.34463002e-04,  7.47782763e-04,  6.88651850e-04,
             3.24133374e-06,  2.00575768e-06, -1.96140671e-04,
             8.68657078e-05],
           [ 8.79799746e-03,  7.20273834e-03,  4.37944837e-03,
             5.33844008e-03,  1.91427101e-03,  2.18229987e-03,
             1.07381660e-03,  4.52730569e-04, -9.63464373e-05,
             2.38492464e-04],
           [ 1.80671626e-02,  2.16494830e-02,  1.66367014e-02,
             1.38042091e-02,  8.25275797e-03,  9.39708736e-03,
             5.95631898e-03,  1.55009586e-03,  4.17096272e-04,
             6.15224572e-04],
    ...
           [ 3.86221889e-02,  4.93017010e-02,  4.10173277e-02,
             3.77454520e-02,  3.09181921e-02,  3.85493459e-02,
             1.39340663e-02,  5.59391461e-03,  2.73759359e-03,
             2.33450881e-03],
           [ 4.58999553e-02,  5.77244490e-02,  5.10212907e-02,
             4.20283897e-02,  3.12120344e-02,  4.03901000e-02,
             1.89363352e-02,  7.81648665e-03,  4.32084600e-03,
             3.28232007e-03],
           [ 5.84965350e-02,  5.70140649e-02,  5.46113237e-02,
             4.71522475e-02,  3.68058631e-02,  5.44180043e-02,
             2.53380053e-02,  1.04280730e-02,  6.11964996e-03,
             5.03660516e-03],
           [ 6.38457534e-02,  5.32185865e-02,  5.57719345e-02,
             4.57599212e-02,  3.47632408e-02,  4.23711808e-02,
             2.04417517e-02,  8.48156026e-03,  6.74847586e-03,
             1.14859267e-02],
           [ 4.93685679e-02,  3.98786814e-02,  4.57781523e-02,
             3.80656128e-02,  2.50802394e-02,  3.64110466e-02,
             2.85747756e-02,  1.65185935e-02,  1.77539198e-02,
             1.57654529e-02]])

  • error_homogeneity
    (r_subset, cond_bin)
    float64
    0.002173 0.003331 ... 0.003913

    array([[2.17257876e-03, 3.33086804e-03, 2.79859948e-03, 1.86993970e-03,
            9.51310913e-04, 1.00492145e-03, 5.38381410e-04, 2.37999683e-04,
            1.34170243e-04, 4.12187807e-05],
           [4.61182090e-03, 7.28771525e-03, 4.48396190e-03, 4.18374917e-03,
            2.09955218e-03, 2.48969994e-03, 1.22254078e-03, 6.62886820e-04,
            1.79341258e-04, 9.97031699e-05],
           [7.57737063e-03, 7.87073499e-03, 7.11363033e-03, 6.81845536e-03,
            6.26540692e-03, 7.16411362e-03, 3.64770061e-03, 1.04125286e-03,
            3.52866068e-04, 2.71039360e-04],
           [1.29597078e-02, 1.00566894e-02, 9.31511239e-03, 8.39813661e-03,
            8.85172406e-03, 1.38712646e-02, 5.42797046e-03, 1.02808726e-03,
            7.23213813e-04, 5.12638434e-04],
           [9.45239285e-03, 1.04647515e-02, 8.81128235e-03, 8.58987177e-03,
            8.49602068e-03, 1.20395365e-02, 4.42827955e-03, 1.28027824e-03,
            8.47761897e-04, 6.07249421e-04],
           [1.31723220e-03, 5.84271754e-03, 5.95361788e-03, 5.93397505e-03,
            4.13483790e-03, 2.36640540e-03, 2.04917318e-03, 1.40434320e-03,
            8.77905466e-04, 6.57663213e-04],
           [5.77295223e-03, 1.58270995e-03, 3.18822419e-03, 1.75817654e-03,
            1.27687266e-03, 5.98084455e-03, 5.31236469e-03, 1.30054071e-03,
            8.70696503e-04, 1.22609584e-03],
           [5.83841190e-03, 2.55672265e-03, 3.18028083e-04, 1.01196630e-03,
            1.35412412e-03, 6.74941890e-03, 4.60054019e-03, 8.86859740e-04,
            6.14392886e-04, 4.10978619e-03],
           [5.72592658e-03, 8.36321053e-03, 3.83710913e-03, 1.44956449e-03,
            4.05025529e-03, 6.98486049e-03, 4.54835822e-03, 3.09418187e-03,
            4.46829968e-03, 3.91268925e-03]])

  • n_bootstrap
    (r, cond_bin)
    int64
    5000 5000 5000 ... 5000 5000 5000

    array([[5000, 5000, 5000, 5000, 5000, 5000, 5000, 5000, 5000, 5000],
           [5000, 5000, 5000, 5000, 5000, 5000, 5000, 5000, 5000, 5000],
           [5000, 5000, 5000, 5000, 5000, 5000, 5000, 5000, 5000, 5000],
           [5000, 5000, 5000, 5000, 5000, 5000, 5000, 5000, 5000, 5000],
           [5000, 5000, 5000, 5000, 5000, 5000, 5000, 5000, 5000, 5000],
           [5000, 5000, 5000, 5000, 5000, 5000, 5000, 5000, 5000, 5000],
           [5000, 5000, 5000, 5000, 5000, 5000, 5000, 5000, 5000, 5000],
           [5000, 5000, 5000, 5000, 5000, 5000, 5000, 5000, 5000, 5000],
           [5000, 5000, 5000, 5000, 5000, 5000, 5000, 5000, 5000, 5000],
           [5000, 5000, 5000, 5000, 5000, 5000, 5000, 5000, 5000, 5000],
           [5000, 5000, 5000, 5000, 5000, 5000, 5000, 5000, 5000, 5000]])

  • bin_density
    (r, cond_bin)
    float32
    1.0 1.0 1.0 ... 1.581e-05 7.936e-06

    array([[1.00000000e+00, 1.00000000e+00, 1.00000000e+00, 1.00000000e+00,
            1.00000000e+00, 1.00000000e+00, 1.00000000e+00, 1.00000000e+00,
            1.00000000e+00, 1.00000000e+00],
           [6.62926614e-01, 6.39653504e-01, 6.48920119e-01, 6.11033797e-01,
            6.04052424e-01, 5.80103993e-01, 6.01634800e-01, 5.98793507e-01,
            5.98181367e-01, 5.67499638e-01],
           [4.90435243e-01, 4.41483647e-01, 4.35595125e-01, 3.94964725e-01,
            3.88016164e-01, 3.75093699e-01, 3.95756066e-01, 3.89718533e-01,
            3.70924920e-01, 3.57447118e-01],
           [3.19079816e-01, 3.16622496e-01, 2.81038493e-01, 2.57224143e-01,
            2.44806543e-01, 2.65835822e-01, 2.61675984e-01, 2.63171494e-01,
            2.53740668e-01, 2.42724240e-01],
           [1.11376852e-01, 1.25391036e-01, 8.89985263e-02, 7.19465911e-02,
            6.72401935e-02, 9.86229926e-02, 7.66606480e-02, 8.40198174e-02,
            7.73399770e-02, 7.84737989e-02],
           [3.12918648e-02, 4.18186486e-02, 2.88968608e-02, 1.95982475e-02,
            1.81658305e-02, 2.95782331e-02, 2.37750839e-02, 2.41608657e-02,
            2.36075968e-02, 1.97727382e-02],
           [8.26242287e-03, 1.12633966e-02, 7.77435023e-03, 5.15230978e-03,
            5.34707634e-03, 8.35722499e-03, 6.48146309e-03, 6.49051880e-03,
            6.36358187e-03, 4.85634245e-03],
           [2.22942140e-03, 3.04574636e-03, 2.04104371e-03, 1.34552340e-03,
            1.53527781e-03, 2.41069985e-03, 1.55047316e-03, 1.90655037e-03,
            1.82275265e-03, 1.09544152e-03],
           [5.50654950e-04, 7.18007912e-04, 4.89816477e-04, 3.36949015e-04,
            3.88171873e-04, 5.97794715e-04, 4.18648764e-04, 5.28033765e-04,
            4.75226465e-04, 1.45572543e-04],
           [1.36987539e-04, 1.20915436e-04, 8.46834519e-05, 6.95898125e-05,
            8.30818099e-05, 1.20386445e-04, 8.31685757e-05, 8.82869062e-05,
            7.36663133e-05, 1.64948033e-05],
           [6.13628217e-05, 4.09286113e-05, 2.97151109e-05, 2.36285578e-05,
            2.43750328e-05, 2.66684492e-05, 1.91748913e-05, 2.19847443e-05,
            1.58058992e-05, 7.93608706e-06]], dtype=float32)

  • point_counts
    (r, cond_bin)
    int64
    287662 7030 12841 ... 506873 121564

    array([[ 287662,    7030,   12841,   22144,   25301,   18857,   41750,
              84876,  112599,   53784],
           [ 669580,   15789,   29258,   47509,   53662,   38409,   88195,
             178450,  236495,  107170],
           [1739295,   38263,   68959,  107826,  121031,   87201,  203701,
             407798,  514908,  237014],
           [3973246,   96352,  156217,  246565,  268117,  216995,  472916,
             966913, 1236768,  565106],
           [4869626,  133980,  173700,  242150,  258574,  282663,  486460,
            1083889, 1323600,  641499],
           [4803819,  156891,  198026,  231604,  245282,  297658,  529726,
            1094385, 1418597,  567535],
           [4453661,  148372,  187064,  213789,  253502,  295299,  507057,
            1032266, 1342652,  489429],
           [4219456,  140874,  172438,  196033,  255568,  299087,  425895,
            1064670, 1350343,  387636],
           [3659303,  116606,  145301,  172368,  226881,  260412,  403778,
            1035340, 1236151,  180871],
           [3196351,   68949,   88204,  124995,  170504,  184137,  281648,
             607816,  672813,   71960],
           [5027282,   81946,  108673,  149018,  175642,  143224,  228000,
             531437,  506873,  121564]])

  • converged
    (r, cond_bin)
    bool
    True True True ... True True True

    array([[ True,  True,  True,  True,  True,  True,  True,  True,  True,
             True],
           [ True,  True,  True,  True,  True,  True,  True,  True,  True,
             True],
           [ True,  True,  True,  True,  True,  True,  True,  True,  True,
             True],
           [ True,  True,  True,  True,  True,  True,  True,  True,  True,
             True],
           [ True,  True,  True,  True,  True,  True,  True,  True,  True,
             True],
           [ True,  True,  True,  True,  True,  True,  True,  True,  True,
             True],
           [ True,  True,  True,  True,  True,  True,  True,  True,  True,
             True],
           [ True,  True,  True,  True,  True,  True,  True,  True,  True,
             True],
           [ True,  True,  True,  True,  True,  True,  True,  True,  True,
             True],
           [ True,  True,  True,  True,  True,  True,  True,  True,  True,
             True],
           [ True,  True,  True,  True,  True,  True,  True,  True,  True,
             True]])

  • r_bins
    (cond_bin, r_edge)
    float64
    1e+03 1.874e+03 ... 5.337e+05 1e+06

    array([[   1000.        ,    1873.81742286,    3511.19173422,
               6579.33224658,   12328.46739442,   23101.29700083,
              43287.61281083,   81113.08307897,  151991.10829529,
             284803.58684358,  533669.92312063, 1000000.        ],
           [   1000.        ,    1873.81742286,    3511.19173422,
               6579.33224658,   12328.46739442,   23101.29700083,
              43287.61281083,   81113.08307897,  151991.10829529,
             284803.58684358,  533669.92312063, 1000000.        ],
           [   1000.        ,    1873.81742286,    3511.19173422,
               6579.33224658,   12328.46739442,   23101.29700083,
              43287.61281083,   81113.08307897,  151991.10829529,
             284803.58684358,  533669.92312063, 1000000.        ],
           [   1000.        ,    1873.81742286,    3511.19173422,
               6579.33224658,   12328.46739442,   23101.29700083,
              43287.61281083,   81113.08307897,  151991.10829529,
             284803.58684358,  533669.92312063, 1000000.        ],
           [   1000.        ,    1873.81742286,    3511.19173422,
               6579.33224658,   12328.46739442,   23101.29700083,
              43287.61281083,   81113.08307897,  151991.10829529,
             284803.58684358,  533669.92312063, 1000000.        ],
           [   1000.        ,    1873.81742286,    3511.19173422,
               6579.33224658,   12328.46739442,   23101.29700083,
              43287.61281083,   81113.08307897,  151991.10829529,
             284803.58684358,  533669.92312063, 1000000.        ],
           [   1000.        ,    1873.81742286,    3511.19173422,
               6579.33224658,   12328.46739442,   23101.29700083,
              43287.61281083,   81113.08307897,  151991.10829529,
             284803.58684358,  533669.92312063, 1000000.        ],
           [   1000.        ,    1873.81742286,    3511.19173422,
               6579.33224658,   12328.46739442,   23101.29700083,
              43287.61281083,   81113.08307897,  151991.10829529,
             284803.58684358,  533669.92312063, 1000000.        ],
           [   1000.        ,    1873.81742286,    3511.19173422,
               6579.33224658,   12328.46739442,   23101.29700083,
              43287.61281083,   81113.08307897,  151991.10829529,
             284803.58684358,  533669.92312063, 1000000.        ],
           [   1000.        ,    1873.81742286,    3511.19173422,
               6579.33224658,   12328.46739442,   23101.29700083,
              43287.61281083,   81113.08307897,  151991.10829529,
             284803.58684358,  533669.92312063, 1000000.        ]])

  • theta_bins
    (cond_bin, theta_edge)
    float64
    -3.142 -2.967 ... 2.967 3.142

    array([[-3.14159265, -2.96705973, -2.7925268 , -2.61799388, -2.44346095,
            -2.26892803, -2.0943951 , -1.91986218, -1.74532925, -1.57079633,
            -1.3962634 , -1.22173048, -1.04719755, -0.87266463, -0.6981317 ,
            -0.52359878, -0.34906585, -0.17453293,  0.        ,  0.17453293,
             0.34906585,  0.52359878,  0.6981317 ,  0.87266463,  1.04719755,
             1.22173048,  1.3962634 ,  1.57079633,  1.74532925,  1.91986218,
             2.0943951 ,  2.26892803,  2.44346095,  2.61799388,  2.7925268 ,
             2.96705973,  3.14159265],
           [-3.14159265, -2.96705973, -2.7925268 , -2.61799388, -2.44346095,
            -2.26892803, -2.0943951 , -1.91986218, -1.74532925, -1.57079633,
            -1.3962634 , -1.22173048, -1.04719755, -0.87266463, -0.6981317 ,
            -0.52359878, -0.34906585, -0.17453293,  0.        ,  0.17453293,
             0.34906585,  0.52359878,  0.6981317 ,  0.87266463,  1.04719755,
             1.22173048,  1.3962634 ,  1.57079633,  1.74532925,  1.91986218,
             2.0943951 ,  2.26892803,  2.44346095,  2.61799388,  2.7925268 ,
             2.96705973,  3.14159265],
           [-3.14159265, -2.96705973, -2.7925268 , -2.61799388, -2.44346095,
            -2.26892803, -2.0943951 , -1.91986218, -1.74532925, -1.57079633,
            -1.3962634 , -1.22173048, -1.04719755, -0.87266463, -0.6981317 ,
            -0.52359878, -0.34906585, -0.17453293,  0.        ,  0.17453293,
    ...
             0.34906585,  0.52359878,  0.6981317 ,  0.87266463,  1.04719755,
             1.22173048,  1.3962634 ,  1.57079633,  1.74532925,  1.91986218,
             2.0943951 ,  2.26892803,  2.44346095,  2.61799388,  2.7925268 ,
             2.96705973,  3.14159265],
           [-3.14159265, -2.96705973, -2.7925268 , -2.61799388, -2.44346095,
            -2.26892803, -2.0943951 , -1.91986218, -1.74532925, -1.57079633,
            -1.3962634 , -1.22173048, -1.04719755, -0.87266463, -0.6981317 ,
            -0.52359878, -0.34906585, -0.17453293,  0.        ,  0.17453293,
             0.34906585,  0.52359878,  0.6981317 ,  0.87266463,  1.04719755,
             1.22173048,  1.3962634 ,  1.57079633,  1.74532925,  1.91986218,
             2.0943951 ,  2.26892803,  2.44346095,  2.61799388,  2.7925268 ,
             2.96705973,  3.14159265],
           [-3.14159265, -2.96705973, -2.7925268 , -2.61799388, -2.44346095,
            -2.26892803, -2.0943951 , -1.91986218, -1.74532925, -1.57079633,
            -1.3962634 , -1.22173048, -1.04719755, -0.87266463, -0.6981317 ,
            -0.52359878, -0.34906585, -0.17453293,  0.        ,  0.17453293,
             0.34906585,  0.52359878,  0.6981317 ,  0.87266463,  1.04719755,
             1.22173048,  1.3962634 ,  1.57079633,  1.74532925,  1.91986218,
             2.0943951 ,  2.26892803,  2.44346095,  2.61799388,  2.7925268 ,
             2.96705973,  3.14159265]])

• Attributes: (15)

  order :

  2

  function_type :

  default_vel

  window_size_theta :

  12

  window_size_r :

  3

  convergence_eps :

  0.001

  max_nbootstrap :

  5000

  initial_nbootstrap :

  5000

  bin_type :

  logarithmic

  variables :

  ['u', 'v']

  confidence_level :

  0.95

  bootstrappable_dimensions :

  y,x

  backend :

  loky

  conditioning_variable :

  sic

  conditioning_bin_edges :

  [np.float64(-0.01), np.float64(0.09200000000000001), np.float64(0.194), np.float64(0.29600000000000004), np.float64(0.398), np.float64(0.5), np.float64(0.6020000000000001), np.float64(0.7040000000000001), np.float64(0.806), np.float64(0.908), np.float64(1.01)]

  conditioning_bin_idx :

  0

Plot Second Order SF conditional (Sea-Ice)

# Get the data
r = iso_cond.r.values  # radial bins
cond_bins = iso_cond.cond_bin.values  # bin centers (SIC values)
sf_cond_data = iso_cond.sf.values  # shape: (n_r, n_cond_bin)

# ============================================================
# Plot 1: D(r) vs r - colored band style like your figure
# ============================================================
fig, ax = plt.subplots(figsize=(8, 6))

# Colormap - yellow to blue like your figure
cmap = plt.cm.plasma  # or try 'viridis', 'coolwarm', 'jet'
norm = mcolors.Normalize(vmin=cond_bins.min(), vmax=cond_bins.max())

# Plot each conditioning bin as a line
for i, sic_center in enumerate(cond_bins):
    color = cmap(norm(sic_center))
    ax.loglog(r, sf_cond_data[:, i], '-', color=color, linewidth=2, alpha=0.8)

# Colorbar
sm = cm.ScalarMappable(cmap=cmap, norm=norm)
sm.set_array([])
cbar = fig.colorbar(sm, ax=ax, pad=0.02)
cbar.set_label('Sea Ice Concentration', fontsize=12)

# Reference slope
r_ref = np.array([r.min(), r.max()])
# Adjust prefactor to fit your data
sf_ref = sf_cond_data[len(r)//2, :].mean()  # reference value
r_mid = r[len(r)//2]
prefactor = sf_ref / (r_mid**(2/3))
ax.loglog(r_ref, prefactor * r_ref**(2/3), 'k--', linewidth=2, label=r'$r^{2/3}$')

ax.set_xlabel(r'$r$ [m]', fontsize=12)
ax.set_ylabel(r'$D(r)$ [m²/s²]', fontsize=12)
ax.set_title('Structure Function conditioned on SIC', fontsize=14)
ax.legend(loc='lower right')
ax.grid(True, which='both', alpha=0.3, linestyle='-', linewidth=0.5)
plt.tight_layout()
plt.show()

# ============================================================
# Plot 2: D(SIC) at fixed r values
# ============================================================
fig, ax = plt.subplots(figsize=(8, 6))

# Select specific r values
r_values_km = [5, 10, 20, 50, 80]  # km
cmap_r = plt.cm.viridis
norm_r = mcolors.LogNorm(vmin=1, vmax=100)

for r_km in r_values_km:
    r_target = r_km * 1e3  # convert to m
    idx = np.argmin(np.abs(r - r_target))
    r_actual = r[idx]
    color = cmap_r(norm_r(r_km))
    ax.plot(cond_bins, sf_cond_data[idx,:], '-o', color=color,
            label=f'r = {r_actual/1e3:.1f} km', markersize=5, linewidth=2)

ax.set_xlabel('Sea Ice Concentration', fontsize=12)
ax.set_ylabel(r'$D(r; SIC)$ [m²/s²]', fontsize=12)
ax.set_yscale('log')
ax.set_xlim([0, 1])
ax.legend(fontsize=10)
ax.grid(True, alpha=0.3)
plt.tight_layout()
plt.show()

Plot Total Conditionned vs Total SF

plt.loglog(r,((iso_cond.sf*iso_cond.point_counts).sum('cond_bin'))/iso_cond.point_counts.sum('cond_bin'),label='Total Conditionnal 2nd order SF')
plt.loglog(r,iso_sf.sf,'k',label='Total 2nd order SF')
plt.xlabel('Separation Distance [m]')
plt.ylabel('[m²/s²]', fontsize=12)
plt.legend()
plt.tight_layout()
plt.show()

Physical Interpretation: Conditional Structure Functions in the Marginal Ice Zone

This analysis uses LLC4320 high-resolution ocean model data to examine how velocity structure functions vary with sea ice concentration (SIC):

• Conditional binning by SIC: The structure function is computed separately for different sea ice concentration ranges, revealing how turbulence properties change from open ocean (SIC ~ 0) to fully ice-covered regions (SIC ~ 1).

• Open ocean (low SIC): Mesoscale eddies dominate, showing typical geostrophic turbulence scaling with an inverse energy cascade.

• Marginal ice zone (intermediate SIC): Enhanced energy at small scales due to ice-ocean interactions, ice edge jets, and submesoscale instabilities along the ice edge.

• Ice-covered regions (high SIC): Suppressed surface velocity variance as sea ice dampens momentum transfer from the atmosphere and inhibits surface currents.

• Scale dependence: The relationship between SIC and turbulence may vary with scale — ice affects small-scale dynamics differently than large-scale geostrophic flows.

This conditional analysis is essential for understanding Arctic/Antarctic dynamics where sea ice modulates ocean turbulence and heat transport.