├── ftldrive ├── tests │ ├── __init__.py │ └── test_numba_ensrf.py ├── __init__.py ├── spatial.py └── core.py ├── README.md ├── ci ├── requirements-py35.yml └── requirements-py36.yml ├── .travis.yml ├── .gitignore ├── setup.py └── LICENSE /ftldrive/tests/__init__.py: -------------------------------------------------------------------------------- 1 | -------------------------------------------------------------------------------- /ftldrive/__init__.py: -------------------------------------------------------------------------------- 1 | from ftldrive.core import serial_ensrf 2 | -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | # ftldrive 2 | 3 | [![Travis-CI Build Status](https://travis-ci.org/brews/ftldrive.svg?branch=master)](https://travis-ci.org/brews/ftldrive) 4 | 5 | Very experimental ensemble kalman filter engine. 6 | -------------------------------------------------------------------------------- /ci/requirements-py35.yml: -------------------------------------------------------------------------------- 1 | name: test_env 2 | channels: 3 | - conda-forge 4 | - defaults 5 | dependencies: 6 | - coverage 7 | - docutils 8 | - numba 9 | - numpy 10 | - pytest 11 | - python=3.5 12 | - scipy 13 | - tox 14 | -------------------------------------------------------------------------------- /ci/requirements-py36.yml: -------------------------------------------------------------------------------- 1 | name: test_env 2 | channels: 3 | - conda-forge 4 | - defaults 5 | dependencies: 6 | - coverage 7 | - docutils 8 | - numba 9 | - numpy 10 | - pytest 11 | - python=3.5 12 | - scipy 13 | - tox 14 | -------------------------------------------------------------------------------- /.travis.yml: -------------------------------------------------------------------------------- 1 | language: python 2 | notifications: 3 | email: false 4 | 5 | matrix: 6 | fast_finish: true 7 | include: 8 | - python: 3.5 9 | env: 10 | - CONDA_ENV=py35 11 | - JOB_OS=Linux 12 | - python: 3.6 13 | env: 14 | - CONDA_ENV=py36 15 | - JOB_OS=Linux 16 | 17 | before_install: 18 | - if [[ "$TRAVIS_PYTHON_VERSION" == "2.7" ]]; then 19 | wget https://repo.continuum.io/miniconda/Miniconda2-latest-$JOB_OS-x86_64.sh -O miniconda.sh; 20 | else 21 | wget https://repo.continuum.io/miniconda/Miniconda3-latest-$JOB_OS-x86_64.sh -O miniconda.sh; 22 | fi 23 | - bash miniconda.sh -b -p $HOME/miniconda 24 | - export PATH="$HOME/miniconda/bin:$PATH" 25 | - hash -r 26 | - conda config --set always_yes yes --set changeps1 no 27 | - conda update -q conda 28 | # Useful for debugging any issues with conda 29 | - conda info -a 30 | 31 | install: 32 | - conda env create -q -f ci/requirements-$CONDA_ENV.yml 33 | - source activate test_env 34 | - pip install . 35 | 36 | script: 37 | - pytest --pyargs ftldrive 38 | -------------------------------------------------------------------------------- /.gitignore: -------------------------------------------------------------------------------- 1 | # Byte-compiled / optimized / DLL files 2 | __pycache__/ 3 | *.py[cod] 4 | *$py.class 5 | 6 | # C extensions 7 | *.so 8 | 9 | # Distribution / packaging 10 | .Python 11 | env/ 12 | build/ 13 | develop-eggs/ 14 | dist/ 15 | downloads/ 16 | eggs/ 17 | .eggs/ 18 | lib/ 19 | lib64/ 20 | parts/ 21 | sdist/ 22 | var/ 23 | wheels/ 24 | *.egg-info/ 25 | .installed.cfg 26 | *.egg 27 | 28 | # PyInstaller 29 | # Usually these files are written by a python script from a template 30 | # before PyInstaller builds the exe, so as to inject date/other infos into it. 31 | *.manifest 32 | *.spec 33 | 34 | # Installer logs 35 | pip-log.txt 36 | pip-delete-this-directory.txt 37 | 38 | # Unit test / coverage reports 39 | htmlcov/ 40 | .tox/ 41 | .coverage 42 | .coverage.* 43 | .cache 44 | nosetests.xml 45 | coverage.xml 46 | *.cover 47 | .hypothesis/ 48 | 49 | # Translations 50 | *.mo 51 | *.pot 52 | 53 | # Django stuff: 54 | *.log 55 | local_settings.py 56 | 57 | # Flask stuff: 58 | instance/ 59 | .webassets-cache 60 | 61 | # Scrapy stuff: 62 | .scrapy 63 | 64 | # Sphinx documentation 65 | docs/_build/ 66 | 67 | # PyBuilder 68 | target/ 69 | 70 | # Jupyter Notebook 71 | .ipynb_checkpoints 72 | 73 | # pyenv 74 | .python-version 75 | 76 | # celery beat schedule file 77 | celerybeat-schedule 78 | 79 | # SageMath parsed files 80 | *.sage.py 81 | 82 | # dotenv 83 | .env 84 | 85 | # virtualenv 86 | .venv 87 | venv/ 88 | ENV/ 89 | 90 | # Spyder project settings 91 | .spyderproject 92 | .spyproject 93 | 94 | # Rope project settings 95 | .ropeproject 96 | 97 | # mkdocs documentation 98 | /site 99 | 100 | # mypy 101 | .mypy_cache/ 102 | 103 | # pycharm 104 | .idea/ 105 | -------------------------------------------------------------------------------- /setup.py: -------------------------------------------------------------------------------- 1 | import os 2 | from setuptools import setup, find_packages 3 | 4 | MAJOR = 0 5 | MINOR = 0 6 | MICRO = '1a1' 7 | VERSION = '{}.{}.{}'.format(MAJOR, MINOR, MICRO) 8 | FULLVERSION = VERSION 9 | 10 | 11 | def write_version_py(filename=None): 12 | cnt = """\ 13 | version = '%s' 14 | short_version = '%s' 15 | """ 16 | if not filename: 17 | filename = os.path.join( 18 | os.path.dirname(__file__), 'ftldrive', 'version.py') 19 | 20 | a = open(filename, 'w') 21 | try: 22 | a.write(cnt % (FULLVERSION, VERSION)) 23 | finally: 24 | a.close() 25 | 26 | write_version_py() 27 | 28 | 29 | 30 | setup_kwargs = dict(name='ftldrive', 31 | version=FULLVERSION, 32 | description='ftldrive', 33 | url='https://github.com/brews/ftldrive', 34 | author='S. Brewster Malevich', 35 | author_email='malevich@email.arizona.edu', 36 | license='GPLv3', 37 | classifiers=[ 38 | 'Development Status :: 1 - Planning', 39 | 40 | 'Operating System :: POSIX', 41 | 42 | 'Intended Audience :: Developers', 43 | 'Intended Audience :: Science/Research', 44 | 45 | 'Topic :: Scientific/Engineering', 46 | 'Topic :: Software Development', 47 | 48 | 'License :: OSI Approved :: GNU General Public License v3 or later (GPLv3+)', 49 | 50 | 'Programming Language :: Python :: 3'], 51 | keywords='assimilation kalman', 52 | install_requires=['numpy', 'numba', 'scipy'], 53 | packages=find_packages(), 54 | ) 55 | 56 | setup(**setup_kwargs) 57 | -------------------------------------------------------------------------------- /ftldrive/spatial.py: -------------------------------------------------------------------------------- 1 | import numpy as np 2 | 3 | 4 | def haversine_distance(latlon1, latlon2, sphere_radius=6378.137): 5 | """haversine distance between two sequences of (lat, lon) points 6 | 7 | Parameters 8 | ---------- 9 | latlon1 : sequence of tuples 10 | (latitude, longitude) for one set of points. 11 | latlon2 : sequence of tuples 12 | A sequence of (latitude, longitude) for another set of points. 13 | sphere_radius: float 14 | Radius of sphere we are calculating distances on. Default is 6378.137, 15 | Earth radius in km. 16 | 17 | Returns 18 | ------- 19 | dists : 2d array 20 | An mxn array of Earth haversine distances [1]_ between points in 21 | latlon1 and latlon2. Units is based on `sphere_radius` used. 22 | 23 | References 24 | ---------- 25 | .. [1] https://en.wikipedia.org/wiki/Haversine_formula 26 | 27 | """ 28 | latlon1 = np.atleast_2d(latlon1) 29 | latlon2 = np.atleast_2d(latlon2) 30 | 31 | n = latlon1.shape[0] 32 | m = latlon2.shape[0] 33 | 34 | paired = np.hstack((np.kron(latlon1, np.ones((m, 1))), 35 | np.kron(np.ones((n, 1)), latlon2))) 36 | latdif = np.deg2rad(paired[:, 0] - paired[:, 2]) 37 | londif = np.deg2rad(paired[:, 1] - paired[:, 3]) 38 | 39 | a = (np.sin(latdif / 2)**2 + np.cos(np.deg2rad(paired[:, 0])) 40 | * np.cos(np.deg2rad(paired[:, 2])) * np.sin(londif / 2)**2) 41 | c = 2 * np.arctan2(np.sqrt(a), np.sqrt(1 - a)) 42 | 43 | return sphere_radius * c 44 | 45 | 46 | def chordal_distance(latlon1, latlon2, sphere_radius=6378.137): 47 | """Chordal distance between two sequences of (lat, lon) points 48 | 49 | Parameters 50 | ---------- 51 | latlon1 : sequence of tuples 52 | (latitude, longitude) for one set of points. 53 | latlon2 : sequence of tuples 54 | A sequence of (latitude, longitude) for another set of points. 55 | sphere_radius: float 56 | Radius of sphere we are calculating distances on. Default is 6378.137, 57 | Earth radius in km. 58 | 59 | Returns 60 | ------- 61 | dists : 2d array 62 | An mxn array of Earth chordal distances [1]_ (km) between points in 63 | latlon1 and latlon2. 64 | 65 | References 66 | ---------- 67 | .. [1] https://en.wikipedia.org/wiki/Chord_(geometry) 68 | 69 | """ 70 | sphere_radius = 6378.137 # in km 71 | 72 | latlon1 = np.atleast_2d(latlon1) 73 | latlon2 = np.atleast_2d(latlon2) 74 | 75 | n = latlon1.shape[0] 76 | m = latlon2.shape[0] 77 | 78 | paired = np.hstack((np.kron(latlon1, np.ones((m, 1))), 79 | np.kron(np.ones((n, 1)), latlon2))) 80 | 81 | latdif = np.deg2rad(paired[:, 0] - paired[:, 2]) 82 | londif = np.deg2rad(paired[:, 1] - paired[:, 3]) 83 | 84 | a = np.sin(latdif / 2) ** 2 85 | b = np.cos(np.deg2rad(paired[:, 0])) 86 | c = np.cos(np.deg2rad(paired[:, 2])) 87 | d = np.sin(np.abs(londif) / 2) ** 2 88 | 89 | half_angles = np.arcsin(np.sqrt(a + b * c * d)) 90 | 91 | dists = 2 * sphere_radius * np.sin(half_angles) 92 | 93 | return dists.reshape(m, n) 94 | 95 | 96 | def gasparicohn_localization(dists, local_radius): 97 | """Gaspari-Cohn distance-weights 98 | 99 | Parameters 100 | ---------- 101 | dists : ndarray 102 | Distances from a given point. 103 | local_radius : float 104 | Distance at which correlation drops to zero. 105 | 106 | Returns 107 | ------- 108 | weights : ndarray 109 | Correlations weighted on distance. 110 | 111 | 112 | References 113 | ---------- 114 | Hamill, T. M., Whitaker, J. S., & Snyder, C. (2001). Distance-Dependent Filtering of Background Error Covariance Estimates in an Ensemble Kalman Filter. Monthly Weather Review, 129(11), 2776–2790. https://doi.org/10.1175/1520-0493(2001)129<2776:DDFOBE>2.0.CO;2 115 | 116 | Gaspari Gregory, & Cohn Stephen E. (2006). Construction of correlation functions in two and three dimensions. Quarterly Journal of the Royal Meteorological Society, 125(554), 723–757. https://doi.org/10.1002/qj.49712555417 117 | 118 | """ 119 | weights = np.ones(shape=dists.shape, dtype=np.float64) 120 | hlr = 0.5 * local_radius # work with half the localization radius 121 | r = dists / hlr 122 | 123 | weights = np.piecewise(r, condlist=[dists <= hlr, dists > hlr, dists > 2 * hlr], 124 | funclist=[_gasparicohn_near, _gasparicohn_far, lambda x: 0]) 125 | 126 | # prevent negative values: calc. above may produce tiny negative 127 | weights[weights < 0] = 0 128 | 129 | return weights 130 | 131 | 132 | def _gasparicohn_near(x): 133 | """Support funciton for `gasparicohn_localization` 134 | """ 135 | return (((-0.25 * x + 0.5) * x + 0.625) * x - (5 / 3)) * (x**2) + 1 136 | 137 | 138 | def _gasparicohn_far(x): 139 | """Support function for `gasparicohn_localization` 140 | """ 141 | return ((((x / 12 - 0.5) * x + 0.625) * x + 5 / 3) * x - 5) * x + 4 - 2 / (3 * x) 142 | -------------------------------------------------------------------------------- /ftldrive/tests/test_numba_ensrf.py: -------------------------------------------------------------------------------- 1 | import pytest 2 | import numpy as np 3 | from ftldrive.core import (ensemble_mean, kalman_gain, 4 | modified_kalman_gain, analysis_deviation, analysis_mean, update, 5 | obs_assimilation_loop, serial_ensrf, inflate_state_variance) 6 | 7 | 8 | def test_ensemble_mean(): 9 | victim = np.arange(15).reshape(5, 3) 10 | goal = np.array([[1.0, 4.0, 7.0, 10.0, 13.0]]).T 11 | actual = ensemble_mean(victim) 12 | np.testing.assert_allclose(actual, goal, atol=1e-15, rtol=0) 13 | 14 | 15 | def test_inflate_state_variance(): 16 | state = np.arange(15).reshape(5, 3) 17 | inflation = np.ones((state.shape[0], 1)) * 2.0 18 | goal = np.array([[-1., 1., 3.], 19 | [2., 4., 6.], 20 | [5., 7., 9.], 21 | [8., 10., 12.], 22 | [11., 13., 15.]]) 23 | actual = inflate_state_variance(x=state, infl=inflation) 24 | np.testing.assert_allclose(actual, goal, atol=1e-15, rtol=0) 25 | 26 | 27 | def test_kalman_gain(): 28 | xb_yb_cov = np.array([0.5, 0.5, 0.5, 0.5, 0.5]) 29 | yb_var = 1.0 30 | r = 0.25 31 | actual = kalman_gain(xbye_cov=xb_yb_cov, yb_prime_var=yb_var, r=r) 32 | goal = np.array([[0.4, 0.4, 0.4, 0.4, 0.4]]).T 33 | np.testing.assert_allclose(actual, goal, atol=1e-15, rtol=0) 34 | 35 | 36 | def test_modified_kalman_gain(): 37 | yb_var = 1.0 38 | r = 0.25 39 | k = np.array([[0.4, 0.4, 0.4, 0.4, 0.4]]).T 40 | actual = modified_kalman_gain(yb_prime_var=yb_var, r=r, k=k) 41 | goal = np.array([[0.2763932, 0.2763932, 0.2763932, 0.2763932, 0.2763932]]).T 42 | np.testing.assert_allclose(actual, goal, atol=1e-6, rtol=0) 43 | 44 | 45 | def test_analysis_deviation(): 46 | y_prime = np.array([-1, 0, 1]) 47 | x = np.arange(15).reshape((5, 3)) 48 | x_prime = x - x.mean(axis=1)[:, None] 49 | k_tilde = np.array([[0.2763932, 0.2763932, 0.2763932, 0.2763932, 0.2763932]]).T 50 | actual = analysis_deviation(xb_prime=x_prime, k_tilde=k_tilde, 51 | yb_prime=y_prime) 52 | goal = np.array([[-0.7236068, 0, 0.7236068], 53 | [-0.7236068, 0, 0.7236068], 54 | [-0.7236068, 0, 0.7236068], 55 | [-0.7236068, 0, 0.7236068], 56 | [-0.7236068, 0, 0.7236068]]) 57 | np.testing.assert_allclose(actual, goal, atol=1e-7, rtol=0) 58 | 59 | 60 | def test_analysis_mean(): 61 | x = np.arange(15).reshape((5, 3)) 62 | xb_bar = ensemble_mean(x) 63 | y0 = 6.5 64 | yb_bar = 7.0 65 | k = np.array([[0.4, 0.4, 0.4, 0.4, 0.4]]).T 66 | actual = analysis_mean(xb_bar=xb_bar, k=k, y0=y0, yb_bar=yb_bar) 67 | goal = np.array([[0.8, 3.8, 6.8, 9.8, 12.8]]).T 68 | np.testing.assert_allclose(actual, goal, atol=1e-7, rtol=0) 69 | 70 | 71 | def test_update(): 72 | xb = np.arange(15).reshape(5, 3) 73 | yb = xb[2, :] 74 | y0 = 6.5 75 | r = 0.25 76 | loc = 1 77 | actual = update(xb=xb, yb=yb, y0=y0, r=r, loc=loc) 78 | goal = np.array([[ 0.1527864, 0.6, 1.0472136], 79 | [ 3.1527864, 3.6, 4.0472136], 80 | [ 6.1527864, 6.6, 7.0472136], 81 | [ 9.1527864, 9.6, 10.0472136], 82 | [ 12.1527864, 12.6, 13.0472136]]) 83 | np.testing.assert_allclose(actual, goal, atol=1e-7, rtol=0) 84 | 85 | 86 | def test_obs_assimilation_loop_1obs(): 87 | state = np.arange(15).reshape(5, 3) 88 | obs = np.array([6.5]) 89 | obs_error = np.array([0.25]) 90 | obs_idx = np.array([2, ]) 91 | inflation = np.ones((state.shape[0], 1)) 92 | localization = np.ones((len(obs), state.shape[0])) 93 | actual = obs_assimilation_loop(state=state, obs=obs, obs_error=obs_error, 94 | obs_idx=obs_idx, inflation=inflation, 95 | localization=localization) 96 | goal = np.array([[ 0.1527864, 0.6, 1.0472136], 97 | [ 3.1527864, 3.6, 4.0472136], 98 | [ 6.1527864, 6.6, 7.0472136], 99 | [ 9.1527864, 9.6, 10.0472136], 100 | [ 12.1527864, 12.6, 13.0472136]]) 101 | np.testing.assert_allclose(actual, goal, atol=1e-7, rtol=0) 102 | 103 | 104 | def test_obs_assimilation_loop_1obs2(): 105 | state = np.array([[ 0.1527864, 0.6, 1.0472136], 106 | [ 3.1527864, 3.6, 4.0472136], 107 | [ 6.1527864, 6.6, 7.0472136], 108 | [ 9.1527864, 9.6, 10.0472136], 109 | [ 12.1527864, 12.6, 13.0472136]]) 110 | obs = np.array([0.4]) 111 | obs_error = np.array([0.35]) 112 | obs_idx = np.array([0, ]) 113 | inflation = np.ones((state.shape[0], 1)) 114 | localization = np.ones((len(obs), state.shape[0])) 115 | actual = obs_assimilation_loop(state=state, obs=obs, obs_error=obs_error, 116 | obs_idx=obs_idx, inflation=inflation, 117 | localization=localization) 118 | goal = np.array([[ 0.17051969, 0.52727273, 0.88402576], 119 | [ 3.17051969, 3.52727273, 3.88402576], 120 | [ 6.17051969, 6.52727273, 6.88402576], 121 | [ 9.17051969, 9.52727273, 9.88402576], 122 | [ 12.17051969, 12.52727273, 12.88402576]]) 123 | np.testing.assert_allclose(actual, goal, atol=1e-7, rtol=0) 124 | 125 | 126 | def test_obs_assimilation_loop_2obs(): 127 | state = np.arange(15).reshape(5, 3) 128 | obs = np.array([6.5, 0.4]) 129 | obs_error = np.array([0.25, 0.35]) 130 | obs_idx = np.array([2, 0]) 131 | inflation = np.ones((state.shape[0], 1)) 132 | localization = np.ones((len(obs))) 133 | actual = obs_assimilation_loop(state=state, obs=obs, obs_error=obs_error, 134 | obs_idx=obs_idx, inflation=inflation, 135 | localization=localization) 136 | goal = np.array([[ 0.17051969, 0.52727273, 0.88402576], 137 | [ 3.17051969, 3.52727273, 3.88402576], 138 | [ 6.17051969, 6.52727273, 6.88402576], 139 | [ 9.17051969, 9.52727273, 9.88402576], 140 | [ 12.17051969, 12.52727273, 12.88402576]]) 141 | np.testing.assert_allclose(actual, goal, atol=1e-7, rtol=0) 142 | 143 | 144 | def test_serial_ensrf(): 145 | state = np.arange(15).reshape(5, 3) 146 | obs = np.array([6.5, 0.4]) 147 | obs_error = np.array([0.25, 0.35]) 148 | obs_idx = np.array([2, 0]) 149 | goal = np.array([[ 0.17051969, 0.52727273, 0.88402576], 150 | [ 3.17051969, 3.52727273, 3.88402576], 151 | [ 6.17051969, 6.52727273, 6.88402576], 152 | [ 9.17051969, 9.52727273, 9.88402576], 153 | [ 12.17051969, 12.52727273, 12.88402576]]) 154 | actual = serial_ensrf(state=state, obs_value=obs, obs_error=obs_error, 155 | obs_idx=obs_idx, inflation=None, localization=None) 156 | np.testing.assert_allclose(actual, goal, atol=1e-7, rtol=0) 157 | 158 | 159 | def test_serial_ensrf_inflation_1obs(): 160 | state = np.arange(15).reshape(5, 3) 161 | m = state.shape[0] 162 | obs = np.array([0.4]) 163 | obs_error = np.array([0.35]) 164 | obs_idx = np.array([0]) 165 | inflation = np.ones((m, 1)) * 2.0 166 | goal = np.array([[ -0.11903277, 0.44827586, 1.01558449], 167 | [ 2.88096723, 3.44827586, 4.01558449], 168 | [ 5.88096723, 6.44827586, 7.01558449], 169 | [ 8.88096723, 9.44827586, 10.01558449], 170 | [ 11.88096723, 12.44827586, 13.01558449]]) 171 | actual = serial_ensrf(state=state, obs_value=obs, obs_error=obs_error, 172 | obs_idx=obs_idx, inflation=inflation, localization=None) 173 | np.testing.assert_allclose(actual, goal, atol=1e-7, rtol=0) 174 | 175 | 176 | def test_serial_ensrf_inflation_2obs(): 177 | state = np.arange(15).reshape(5, 3) 178 | m = state.shape[0] 179 | obs = np.array([6.5, 0.4]) 180 | obs_error = np.array([0.25, 0.35]) 181 | obs_idx = np.array([2, 0]) 182 | inflation = np.ones((m, 1)) * 2.0 183 | goal = np.array([[ 0.10228226, 0.47738693, 0.85249161], 184 | [ 3.10228226, 3.47738693, 3.85249161], 185 | [ 6.10228226, 6.47738693, 6.85249161], 186 | [ 9.10228226, 9.47738693, 9.85249161], 187 | [ 12.10228226, 12.47738693, 12.85249161]]) 188 | actual = serial_ensrf(state=state, obs_value=obs, obs_error=obs_error, 189 | obs_idx=obs_idx, inflation=inflation, localization=None) 190 | np.testing.assert_allclose(actual, goal, atol=1e-7, rtol=0) 191 | 192 | 193 | def test_serial_ensrf_localization(): 194 | state = np.arange(15).reshape(5, 3) 195 | obs = np.array([6.5, 0.4]) 196 | obs_error = np.array([0.25, 0.35]) 197 | obs_idx = np.array([2, 0]) 198 | localization = np.array([[0, 0.5, 1.0, 0.5, 0], 199 | [1.0, 0.5, 0, 0, 0]]) 200 | goal = np.array([[ 0.04638048, 0.55555556, 1.06473063], 201 | [ 3.09317382, 3.63919849, 4.18522316], 202 | [ 6.1527864 , 6.6 , 7.0472136 ], 203 | [ 9.0763932 , 9.8 , 10.5236068 ], 204 | [ 12. , 13. , 14. ]]) 205 | actual = serial_ensrf(state=state, obs_value=obs, obs_error=obs_error, 206 | obs_idx=obs_idx, inflation=None, 207 | localization=localization) 208 | np.testing.assert_allclose(actual, goal, atol=1e-7, rtol=0) 209 | 210 | 211 | def test_serial_ensrf_localization_inflation(): 212 | state = np.arange(15).reshape(5, 3) 213 | m = state.shape[0] 214 | obs = np.array([6.5, 0.4]) 215 | obs_error = np.array([0.25, 0.35]) 216 | obs_idx = np.array([2, 0]) 217 | inflation = np.ones((m, 1)) * 2.0 218 | localization = np.array([[0, 0.5, 1.0, 0.5, 0], 219 | [1.0, 0.5, 0, 0, 0]]) 220 | goal = np.array([[ -0.11903277, 0.44827586, 1.01558449], 221 | [ 2.79582855, 3.59332166, 4.39081477], 222 | [ 6.04434051, 6.52941176, 7.01448301], 223 | [ 8.52217026, 9.76470588, 11.00724151], 224 | [ 11. , 13. , 15. ]]) 225 | actual = serial_ensrf(state=state, obs_value=obs, obs_error=obs_error, 226 | obs_idx=obs_idx, inflation=inflation, 227 | localization=localization) 228 | np.testing.assert_allclose(actual, goal, atol=1e-7, rtol=0) 229 | -------------------------------------------------------------------------------- /ftldrive/core.py: -------------------------------------------------------------------------------- 1 | import numpy as np 2 | from numba import njit, prange 3 | 4 | 5 | def serial_ensrf(state, obs_value, obs_error, obs_idx, inflation=None, 6 | localization=None): 7 | """Serial ensemble square root filter. 8 | 9 | Parameters 10 | ---------- 11 | state : array_like 12 | 2D (m x n) state ensemble where m is the state vector size and n 13 | is ensemble size. 14 | obs_value : array_like 15 | 1D (p) array of observations to be assimilated. 16 | obs_error : array_like 17 | 1D (p) array of error variances for each observation. 18 | obs_idx : array_like 19 | 1D (p) array containing indexes indicating which state vector element 20 | corresponds to a given observation. 21 | inflation : array_like 22 | 1D (m) inflation factor applied to background state deviations before 23 | the background error covariance is calculated and before any 24 | observations are assimilated. A value of 1 leaves the covariance 25 | unchanged. 26 | localization : array_like 27 | 2D (p x m) or 1D (p) array of values multiplied against the background error 28 | covariance matrix to limit the influence of each observation on the 29 | state. Values of 1 will leave the covariance unchanged. 30 | 31 | Returns 32 | ------- 33 | updated_state : ndarray 34 | 2D (m x n) updated state vector ensemble. 35 | 36 | References 37 | ---------- 38 | Compo, G. P., Whitaker, J. S., & Sardeshmukh, P. D. (2006). Feasibility of 39 | a 100-Year Reanalysis Using Only Surface Pressure Data. Bulletin of the 40 | American Meteorological Society, 87(2), 175–190. 41 | https://doi.org/10.1175/BAMS-87-2-175 42 | Whitaker, J. S., Compo, G. P., Wei, X., & Hamill, T. M. (2004). Reanalysis 43 | without Radiosondes Using Ensemble Data Assimilation. Monthly Weather 44 | Review, 132(5), 1190–1200. 45 | https://doi.org/10.1175/1520-0493(2004)132<1190:RWRUED>2.0.CO;2 46 | """ 47 | state = np.atleast_2d(state) 48 | obs_value = np.atleast_1d(obs_value) 49 | obs_error = np.atleast_1d(obs_error) 50 | obs_idx = np.atleast_1d(obs_idx) 51 | 52 | obs, error, idx = np.broadcast_arrays(obs_value, obs_error, obs_idx) 53 | # Below line guarantee contiguous in memory. Might be too bold if we're expecting our users to handle this. 54 | # obs, error, idx = [np.array(a) for a in np.broadcast_arrays(obs_value, obs_error, obs_idx)] 55 | 56 | # Ghetto check that all our sizes are good. 57 | m, n = state.shape 58 | p = len(obs_value) 59 | 60 | if inflation is not None: 61 | inflation = np.atleast_2d(inflation) 62 | else: 63 | inflation = np.ones((m, 1)) 64 | 65 | if localization is not None: 66 | localization = np.atleast_2d(localization) 67 | else: 68 | localization = np.ones(p) 69 | 70 | assert p == len(error), 'obs_value and obs_error need to have the same first dim size' 71 | assert p == len(idx), 'obs_value and obs_idx need to have the same first dim size' 72 | assert inflation.shape[0] == m, 'state and inflation need to have the same first dim size' 73 | assert localization.shape == (p, m) or localization.shape == (p,), 'state and localization need to have the same first dim size' 74 | 75 | updated_state = obs_assimilation_loop(state, obs, error, idx, inflation, localization) 76 | 77 | return updated_state 78 | 79 | 80 | @njit 81 | def obs_assimilation_loop(state, obs, obs_error, obs_idx, inflation, 82 | localization): 83 | """Serial ensemble square root filter observation assimilation loop. 84 | 85 | This function is intended to run fast, with little or no error checking. 86 | 87 | Parameters 88 | ---------- 89 | state : array_like 90 | 2D (m x n) state ensemble where m is the state vector size and n 91 | is ensemble size. 92 | obs : array_like 93 | 1D (p) array of observations to be assimilated. 94 | obs_error : array_like 95 | 1D (p) array of error variances for each observation. 96 | obs_idx : array_like 97 | 1D (p) array containing indexes indicating which state vector element 98 | corresponds to a given observation. 99 | inflation : array_like 100 | 1D (m) inflation factor applied to background state deviations before 101 | the background error covariance is calculated and before any 102 | observations are assimilated. A value of 1 leaves the covariance 103 | unchanged. 104 | localization : array_like 105 | 2D (p x m) or 1D (p) array of values multiplied against the background 106 | error covariance matrix to limit the influence of each observation on 107 | the state. Values of 1 will leave the covariance unchanged. 108 | 109 | Returns 110 | ------- 111 | updated_state : ndarray 112 | 2D (m x n) updated state vector ensemble. 113 | """ 114 | p = len(obs) 115 | 116 | updated_state = inflate_state_variance(x=state, infl=inflation) 117 | updated_state[:] = update(xb=updated_state, yb=updated_state[obs_idx[0]], y0=obs[0], r=obs_error[0], 118 | loc=localization[0]) 119 | 120 | if p == 1: 121 | return updated_state 122 | 123 | for i in range(1, p): 124 | updated_state[:] = update(xb=updated_state, yb=updated_state[obs_idx[i]], y0=obs[i], r=obs_error[i], 125 | loc=localization[i]) 126 | return updated_state 127 | 128 | 129 | @njit 130 | def update(xb, yb, y0, r, loc=1): 131 | """Serial ensemble square root filter update step. 132 | 133 | Parameters 134 | ---------- 135 | xb : array_like 136 | 2D (m x n) background state vector ensemble where m is the state vector 137 | size and n is ensemble size. 138 | yb : array_like 139 | n-length ensemble of estimates for the observation. 140 | y0 : scalar 141 | Observation. 142 | r : scalar 143 | Observation error (variance). 144 | loc : array_like 145 | 1D (m) covariance localization weights for background error covariance. 146 | A value of 1 leaves the covariance unchanged. 147 | 148 | Returns 149 | ------- 150 | xa : ndarray 151 | 2D (m x n) updated state vector ensemble. 152 | 153 | References 154 | ---------- 155 | Compo, G. P., Whitaker, J. S., & Sardeshmukh, P. D. (2006). Feasibility of 156 | a 100-Year Reanalysis Using Only Surface Pressure Data. Bulletin of the 157 | American Meteorological Society, 87(2), 175–190. 158 | https://doi.org/10.1175/BAMS-87-2-175 159 | Whitaker, J. S., Compo, G. P., Wei, X., & Hamill, T. M. (2004). Reanalysis 160 | without Radiosondes Using Ensemble Data Assimilation. Monthly Weather 161 | Review, 132(5), 1190–1200. 162 | https://doi.org/10.1175/1520-0493(2004)132<1190:RWRUED>2.0.CO;2 163 | """ 164 | # Background state mean and deviation. 165 | m, n = xb.shape 166 | xb_bar = ensemble_mean(xb) # (m x 1) 167 | xb_prime = xb - xb_bar # (m x n) 168 | 169 | # Obs estimate mean and deviation. 170 | yb_bar = np.mean(yb) # (scalar) 171 | yb_prime = yb - yb_bar # (m) 172 | 173 | # Obs estimate deviation's sample variance. 174 | yb_prime_var = np.dot(yb_prime, yb_prime) / (n - 1) # (scalar) 175 | # Obs estimate sample covariance with background state 176 | # (i.e. background error covariance) 177 | xb_prime_yb_prime_cov = np.dot(xb_prime, yb_prime) / (n - 1) # (m) 178 | 179 | # Apply covariance localization weights. 180 | xb_prime_yb_prime_cov *= loc 181 | 182 | # Assemble kalman gains 183 | k = kalman_gain(xbye_cov=xb_prime_yb_prime_cov, yb_prime_var=yb_prime_var, r=r) 184 | k_tilde = modified_kalman_gain(yb_prime_var=yb_prime_var, r=r, k=k) 185 | 186 | # Analysis state mean and deviations. 187 | xa_prime = analysis_deviation(xb_prime, k_tilde, yb_prime) 188 | xa_bar = analysis_mean(xb_bar, k, y0, yb_bar) 189 | 190 | return xa_bar + xa_prime 191 | 192 | 193 | @njit(parallel=True) 194 | def ensemble_mean(x): 195 | """Calculate the mean across ensembles for a state vector. 196 | 197 | This function is needed so that we can release the GIL while calculating an 198 | axis-targetting mean. 199 | 200 | Parameters 201 | ---------- 202 | x : ndarray 203 | 2D (m x n) state vector ensemble, where m the state vector size and n is 204 | ensemble size. 205 | 206 | Returns 207 | ------- 208 | x_bar : ndarray 209 | 2D (m x 1) state vector mean. 210 | """ 211 | m = x.shape[0] 212 | x_bar = np.zeros((m, 1)) 213 | n = x.shape[1] 214 | 215 | for i in prange(m): 216 | x_bar[i, 0] = np.nanmean(x[i, :]) 217 | 218 | return x_bar 219 | 220 | 221 | @njit 222 | def inflate_state_variance(x, infl): 223 | """Inflate state vector ensemble deviations by a factor. 224 | 225 | Parameters 226 | ---------- 227 | x : array_like 228 | 2D (m x n) State vector with `m` elements and `n` ensemble members. 229 | infl : array_like 230 | 2D (m x 1) Inflation factor to apply to state vector deviations. 231 | 232 | Returns 233 | ------- 234 | out : array_like 235 | 2D (m x n) inflated state vector ensemble. 236 | """ 237 | x_bar = ensemble_mean(x) # (m x 1) 238 | return x_bar + (x - x_bar) * infl 239 | 240 | 241 | @njit 242 | def kalman_gain(xbye_cov, yb_prime_var, r): 243 | """Kalman gain (K) for sequential ensemble square root filter. 244 | 245 | Parameters 246 | ---------- 247 | xbye_cov : array_like 248 | 1D (m) array sample covariance between the background state deviations 249 | (xb_prime) and observation estimate deviations(ye_prime). Often noted as PbHt. 250 | yb_prime_var : scalar 251 | Sample variance of observation estimate deviations (yb_prime). Often noted as HPbHt. 252 | r : scalar 253 | Observation error variance. 254 | 255 | Returns 256 | ------- 257 | k : ndarray 258 | 2D (m x 1) array kalman gain. 259 | 260 | References 261 | ---------- 262 | Compo, G. P., Whitaker, J. S., & Sardeshmukh, P. D. (2006). Feasibility of 263 | a 100-Year Reanalysis Using Only Surface Pressure Data. Bulletin of the 264 | American Meteorological Society, 87(2), 175–190. 265 | https://doi.org/10.1175/BAMS-87-2-175 266 | Whitaker, J. S., Compo, G. P., Wei, X., & Hamill, T. M. (2004). Reanalysis 267 | without Radiosondes Using Ensemble Data Assimilation. Monthly Weather 268 | Review, 132(5), 1190–1200. 269 | https://doi.org/10.1175/1520-0493(2004)132<1190:RWRUED>2.0.CO;2 270 | """ 271 | m = len(xbye_cov) 272 | k = np.ones((m, 1)) 273 | k[:, 0] = np.divide(xbye_cov, yb_prime_var + r) 274 | return k 275 | 276 | 277 | @njit 278 | def modified_kalman_gain(yb_prime_var, r, k): 279 | """Modified kalman gain (~K) for sequential ensemble square root filter. 280 | 281 | Parameters 282 | ---------- 283 | yb_prime_var: scalar 284 | Sample variance of observation estimate deviations (yb_prime). Often noted as HPbHt. 285 | r: scalar 286 | Observation error variance. 287 | k: array_like 288 | 2D (m x 1) Kalman gain (K). 289 | 290 | Returns 291 | ------- 292 | k_tilde : ndarray 293 | 2D (m x 1) array modified kalman gain. 294 | 295 | References 296 | ---------- 297 | Compo, G. P., Whitaker, J. S., & Sardeshmukh, P. D. (2006). Feasibility of 298 | a 100-Year Reanalysis Using Only Surface Pressure Data. Bulletin of the 299 | American Meteorological Society, 87(2), 175–190. 300 | https://doi.org/10.1175/BAMS-87-2-175 301 | Whitaker, J. S., Compo, G. P., Wei, X., & Hamill, T. M. (2004). Reanalysis 302 | without Radiosondes Using Ensemble Data Assimilation. Monthly Weather 303 | Review, 132(5), 1190–1200. 304 | https://doi.org/10.1175/1520-0493(2004)132<1190:RWRUED>2.0.CO;2 305 | """ 306 | return np.divide(1, 1 + np.sqrt(np.divide(r, yb_prime_var + r))) * k 307 | 308 | 309 | @njit 310 | def analysis_mean(xb_bar, k, y0, yb_bar): 311 | """Update step for analysis mean (xabar). 312 | 313 | Parameters 314 | ---------- 315 | xb_bar : array_like 316 | 1D (m) background state ensemble mean. 317 | k : array_like 318 | 2D (m x 1) kalman gain (K). 319 | y0 : scalar 320 | Observation to be assimilated. 321 | yb_bar : scalar 322 | Ensemble mean estimate for observation. 323 | 324 | Returns 325 | ------- 326 | xa_bar : ndarray 327 | 1D (m) ensemble mean for the updated state. 328 | 329 | References 330 | ---------- 331 | Compo, G. P., Whitaker, J. S., & Sardeshmukh, P. D. (2006). Feasibility of 332 | a 100-Year Reanalysis Using Only Surface Pressure Data. Bulletin of the 333 | American Meteorological Society, 87(2), 175–190. 334 | https://doi.org/10.1175/BAMS-87-2-175 335 | Whitaker, J. S., Compo, G. P., Wei, X., & Hamill, T. M. (2004). Reanalysis 336 | without Radiosondes Using Ensemble Data Assimilation. Monthly Weather 337 | Review, 132(5), 1190–1200. 338 | https://doi.org/10.1175/1520-0493(2004)132<1190:RWRUED>2.0.CO;2 339 | """ 340 | return xb_bar + k * (y0 - yb_bar) 341 | 342 | 343 | @njit 344 | def analysis_deviation(xb_prime, k_tilde, yb_prime): 345 | """Update step for analysis deviation (xa_prime). 346 | 347 | Parameters 348 | ---------- 349 | xb_prime : array_like 350 | 1D (m) background state ensemble deviation. 351 | k_tilde: array_like 352 | 2D (m x 1) modified kalman gain. 353 | yb_prime : array_like 354 | 1D (n) Ensemble deviation for observation estimate. 355 | 356 | Returns 357 | ------- 358 | xa_prime : ndarray 359 | 2D (m x n) ensemble deviation for the updated state. 360 | 361 | References 362 | ---------- 363 | Compo, G. P., Whitaker, J. S., & Sardeshmukh, P. D. (2006). Feasibility of 364 | a 100-Year Reanalysis Using Only Surface Pressure Data. Bulletin of the 365 | American Meteorological Society, 87(2), 175–190. 366 | https://doi.org/10.1175/BAMS-87-2-175 367 | Whitaker, J. S., Compo, G. P., Wei, X., & Hamill, T. M. (2004). Reanalysis 368 | without Radiosondes Using Ensemble Data Assimilation. Monthly Weather 369 | Review, 132(5), 1190–1200. 370 | https://doi.org/10.1175/1520-0493(2004)132<1190:RWRUED>2.0.CO;2 371 | """ 372 | return xb_prime - k_tilde * yb_prime 373 | -------------------------------------------------------------------------------- /LICENSE: -------------------------------------------------------------------------------- 1 | GNU GENERAL PUBLIC LICENSE 2 | Version 3, 29 June 2007 3 | 4 | Copyright (C) 2007 Free Software Foundation, Inc. 5 | Everyone is permitted to copy and distribute verbatim copies 6 | of this license document, but changing it is not allowed. 7 | 8 | Preamble 9 | 10 | The GNU General Public License is a free, copyleft license for 11 | software and other kinds of works. 12 | 13 | The licenses for most software and other practical works are designed 14 | to take away your freedom to share and change the works. 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No Surrender of Others' Freedom. 541 | 542 | If conditions are imposed on you (whether by court order, agreement or 543 | otherwise) that contradict the conditions of this License, they do not 544 | excuse you from the conditions of this License. If you cannot convey a 545 | covered work so as to satisfy simultaneously your obligations under this 546 | License and any other pertinent obligations, then as a consequence you may 547 | not convey it at all. For example, if you agree to terms that obligate you 548 | to collect a royalty for further conveying from those to whom you convey 549 | the Program, the only way you could satisfy both those terms and this 550 | License would be to refrain entirely from conveying the Program. 551 | 552 | 13. Use with the GNU Affero General Public License. 553 | 554 | Notwithstanding any other provision of this License, you have 555 | permission to link or combine any covered work with a work licensed 556 | under version 3 of the GNU Affero General Public License into a single 557 | combined work, and to convey the resulting work. The terms of this 558 | License will continue to apply to the part which is the covered work, 559 | but the special requirements of the GNU Affero General Public License, 560 | section 13, concerning interaction through a network will apply to the 561 | combination as such. 562 | 563 | 14. Revised Versions of this License. 564 | 565 | The Free Software Foundation may publish revised and/or new versions of 566 | the GNU General Public License from time to time. Such new versions will 567 | be similar in spirit to the present version, but may differ in detail to 568 | address new problems or concerns. 569 | 570 | Each version is given a distinguishing version number. If the 571 | Program specifies that a certain numbered version of the GNU General 572 | Public License "or any later version" applies to it, you have the 573 | option of following the terms and conditions either of that numbered 574 | version or of any later version published by the Free Software 575 | Foundation. If the Program does not specify a version number of the 576 | GNU General Public License, you may choose any version ever published 577 | by the Free Software Foundation. 578 | 579 | If the Program specifies that a proxy can decide which future 580 | versions of the GNU General Public License can be used, that proxy's 581 | public statement of acceptance of a version permanently authorizes you 582 | to choose that version for the Program. 583 | 584 | Later license versions may give you additional or different 585 | permissions. However, no additional obligations are imposed on any 586 | author or copyright holder as a result of your choosing to follow a 587 | later version. 588 | 589 | 15. Disclaimer of Warranty. 590 | 591 | THERE IS NO WARRANTY FOR THE PROGRAM, TO THE EXTENT PERMITTED BY 592 | APPLICABLE LAW. EXCEPT WHEN OTHERWISE STATED IN WRITING THE COPYRIGHT 593 | HOLDERS AND/OR OTHER PARTIES PROVIDE THE PROGRAM "AS IS" WITHOUT WARRANTY 594 | OF ANY KIND, EITHER EXPRESSED OR IMPLIED, INCLUDING, BUT NOT LIMITED TO, 595 | THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR 596 | PURPOSE. THE ENTIRE RISK AS TO THE QUALITY AND PERFORMANCE OF THE PROGRAM 597 | IS WITH YOU. SHOULD THE PROGRAM PROVE DEFECTIVE, YOU ASSUME THE COST OF 598 | ALL NECESSARY SERVICING, REPAIR OR CORRECTION. 599 | 600 | 16. Limitation of Liability. 601 | 602 | IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN WRITING 603 | WILL ANY COPYRIGHT HOLDER, OR ANY OTHER PARTY WHO MODIFIES AND/OR CONVEYS 604 | THE PROGRAM AS PERMITTED ABOVE, BE LIABLE TO YOU FOR DAMAGES, INCLUDING ANY 605 | GENERAL, SPECIAL, INCIDENTAL OR CONSEQUENTIAL DAMAGES ARISING OUT OF THE 606 | USE OR INABILITY TO USE THE PROGRAM (INCLUDING BUT NOT LIMITED TO LOSS OF 607 | DATA OR DATA BEING RENDERED INACCURATE OR LOSSES SUSTAINED BY YOU OR THIRD 608 | PARTIES OR A FAILURE OF THE PROGRAM TO OPERATE WITH ANY OTHER PROGRAMS), 609 | EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF THE POSSIBILITY OF 610 | SUCH DAMAGES. 611 | 612 | 17. Interpretation of Sections 15 and 16. 613 | 614 | If the disclaimer of warranty and limitation of liability provided 615 | above cannot be given local legal effect according to their terms, 616 | reviewing courts shall apply local law that most closely approximates 617 | an absolute waiver of all civil liability in connection with the 618 | Program, unless a warranty or assumption of liability accompanies a 619 | copy of the Program in return for a fee. 620 | 621 | END OF TERMS AND CONDITIONS 622 | 623 | How to Apply These Terms to Your New Programs 624 | 625 | If you develop a new program, and you want it to be of the greatest 626 | possible use to the public, the best way to achieve this is to make it 627 | free software which everyone can redistribute and change under these terms. 628 | 629 | To do so, attach the following notices to the program. It is safest 630 | to attach them to the start of each source file to most effectively 631 | state the exclusion of warranty; and each file should have at least 632 | the "copyright" line and a pointer to where the full notice is found. 633 | 634 | 635 | Copyright (C) 636 | 637 | This program is free software: you can redistribute it and/or modify 638 | it under the terms of the GNU General Public License as published by 639 | the Free Software Foundation, either version 3 of the License, or 640 | (at your option) any later version. 641 | 642 | This program is distributed in the hope that it will be useful, 643 | but WITHOUT ANY WARRANTY; without even the implied warranty of 644 | MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 645 | GNU General Public License for more details. 646 | 647 | You should have received a copy of the GNU General Public License 648 | along with this program. If not, see . 649 | 650 | Also add information on how to contact you by electronic and paper mail. 651 | 652 | If the program does terminal interaction, make it output a short 653 | notice like this when it starts in an interactive mode: 654 | 655 | Copyright (C) 656 | This program comes with ABSOLUTELY NO WARRANTY; for details type `show w'. 657 | This is free software, and you are welcome to redistribute it 658 | under certain conditions; type `show c' for details. 659 | 660 | The hypothetical commands `show w' and `show c' should show the appropriate 661 | parts of the General Public License. Of course, your program's commands 662 | might be different; for a GUI interface, you would use an "about box". 663 | 664 | You should also get your employer (if you work as a programmer) or school, 665 | if any, to sign a "copyright disclaimer" for the program, if necessary. 666 | For more information on this, and how to apply and follow the GNU GPL, see 667 | . 668 | 669 | The GNU General Public License does not permit incorporating your program 670 | into proprietary programs. If your program is a subroutine library, you 671 | may consider it more useful to permit linking proprietary applications with 672 | the library. If this is what you want to do, use the GNU Lesser General 673 | Public License instead of this License. But first, please read 674 | . 675 | --------------------------------------------------------------------------------