def get_delta_RK4(self, sc, time_step, model_time, pos, vel_field):
dt = timedelta(seconds=time_step)
dt_s = dt.seconds
t = model_time
v0 = vel_field.at(pos, t, extrapolate=self.extrapolate).data
d0 = FlatEarthProjection.meters_to_lonlat(v0 * dt_s / 2, pos)
v1 = vel_field.at(pos + d0, t + dt / 2,
extrapolate=self.extrapolate).data
d1 = FlatEarthProjection.meters_to_lonlat(v1 * dt_s / 2, pos)
v2 = vel_field.at(pos + d1, t + dt / 2,
extrapolate=self.extrapolate).data
d2 = FlatEarthProjection.meters_to_lonlat(v2 * dt_s, pos)
v3 = vel_field.at(pos + d2, t + dt, extrapolate=self.extrapolate).data
return dt_s / 6 * (v0 + 2 * v1 + 2 * v2 + v3)
def get_move(self, sc, time_step, model_time_datetime):
"""
Compute the move in (long,lat,z) space. It returns the delta move
for each element of the spill as a numpy array of size
(number_elements X 3) and dtype = gnome.basic_types.world_point_type
Base class returns an array of numpy.nan for delta to indicate the
get_move is not implemented yet.
Each class derived from Mover object must implement it's own get_move
:param sc: an instance of gnome.spill_container.SpillContainer class
:param time_step: time step in seconds
:param model_time_datetime: current model time as datetime object
All movers must implement get_move() since that's what the model calls
"""
status = sc['status_codes'] != oil_status.in_water
positions = sc['positions']
vels = self.grid.interpolated_velocities(model_time_datetime, positions[:, 0:2])
deltas = np.zeros_like(positions)
deltas[:] = 0.
deltas[:, 0:2] = vels * time_step
deltas = FlatEarthProjection.meters_to_lonlat(deltas, positions)
deltas[status] = (0, 0, 0)
pass
return deltas
def test_uncertainty():
sp = sample_sc_release(num_elements=1000, start_pos=(0.0, 0.0, 0.0))
u_sp = sample_sc_release(num_elements=1000,
start_pos=(0.0, 0.0, 0.0),
uncertain=True)
mover = simple_mover.SimpleMover(velocity=(10.0, 10.0, 0.0))
delta = mover.get_move(sp, time_step=100, model_time=None)
u_delta = mover.get_move(u_sp, time_step=100, model_time=None)
# expected = np.zeros_like(delta)
expected = proj.meters_to_lonlat((1000.0, 1000.0, 0.0), (0.0, 0.0, 0.0))
assert np.alltrue(delta == expected)
# but uncertain spills should be different:
assert not np.alltrue(u_delta == expected)
# the mean should be close:
# this is teh smallest tolerance that consitantly passed -- good enough?
assert np.allclose(np.mean(delta, 0), np.mean(u_delta, 0), rtol=1.7e-1)
def get_move(self, sc, time_step, model_time_datetime, num_method=None):
"""
Compute the move in (long,lat,z) space. It returns the delta move
for each element of the spill as a numpy array of size
(number_elements X 3) and dtype = gnome.basic_types.world_point_type
Base class returns an array of numpy.nan for delta to indicate the
get_move is not implemented yet.
Each class derived from Mover object must implement it's own get_move
:param sc: an instance of gnome.spill_container.SpillContainer class
:param time_step: time step in seconds
:param model_time_datetime: current model time as datetime object
All movers must implement get_move() since that's what the model calls
"""
positions = sc['positions']
if self.active and len(positions) > 0:
status = sc['status_codes'] != oil_status.in_water
pos = positions[:]
deltas = self.delta_method(num_method)(sc, time_step, model_time_datetime, pos, self.wind)
deltas[:, 0] *= sc['windages'] * self.wind_scale
deltas[:, 1] *= sc['windages'] * self.wind_scale
deltas = FlatEarthProjection.meters_to_lonlat(deltas, positions)
deltas[status] = (0, 0, 0)
else:
deltas = np.zeros_like(positions)
return deltas
def test_uncertainty():
sp = sample_sc_release(num_elements=1000, start_pos=(0.0, 0.0,
0.0))
u_sp = sample_sc_release(num_elements=1000, start_pos=(0.0, 0.0,
0.0), uncertain=True)
mover = simple_mover.SimpleMover(velocity=(10.0, 10.0, 0.0))
delta = mover.get_move(sp, time_step=100, model_time=None)
u_delta = mover.get_move(u_sp, time_step=100, model_time=None)
# expected = np.zeros_like(delta)
expected = proj.meters_to_lonlat((1000.0, 1000.0, 0.0), (0.0, 0.0,
0.0))
assert np.alltrue(delta == expected)
# but uncertain spills should be different:
assert not np.alltrue(u_delta == expected)
# the mean should be close:
# this is teh smallest tolerance that consitantly passed -- good enough?
assert np.allclose(np.mean(delta, 0), np.mean(u_delta, 0),
rtol=1.7e-1)
def get_move(self, sc, time_step, model_time_datetime, num_method = None):
"""
Compute the move in (long,lat,z) space. It returns the delta move
for each element of the spill as a numpy array of size
(number_elements X 3) and dtype = gnome.basic_types.world_point_type
Base class returns an array of numpy.nan for delta to indicate the
get_move is not implemented yet.
Each class derived from Mover object must implement it's own get_move
:param sc: an instance of gnome.spill_container.SpillContainer class
:param time_step: time step in seconds
:param model_time_datetime: current model time as datetime object
All movers must implement get_move() since that's what the model calls
"""
method = None
if num_method is None:
method = self.num_methods[self.default_num_method]
else:
method = self.num_method[num_method]
status = sc['status_codes'] != oil_status.in_water
positions = sc['positions']
deltas = np.zeros_like(positions)
pos = positions[:, 0:2]
deltas[:, 0:2] = method(sc, time_step, model_time_datetime, pos, self.wind)
deltas[:,0] *= sc['windages']
deltas[:,1] *= sc['windages']
deltas = FlatEarthProjection.meters_to_lonlat(deltas, positions)
deltas[status] = (0, 0, 0)
return deltas
def __init__(self,
image_size,
projection=None,
viewport=None,
preset_colors='BW',
background_color='transparent',
colordepth=8):
"""
create a new map image from scratch -- specifying the size
:param image_size: (width, height) tuple of the image size in pixels
Optional parameters
:param projection=None: gnome.utilities.projections object to use.
if None, it defaults to FlatEarthProjection()
:param viewport: viewport of map -- what gets drawn and on what
scale.
Default is full globe: (((-180, -90), (180, 90)))
:param preset_colors='BW': color set to preset. Options are:
'BW' - transparent, black, and white:
transparent background
'web' - the basic named colors for the web:
transparent background
'transparent' - transparent background,
no other colors set
None - no pre-allocated colors
-- the first one you allocate will
be the background color
:param background_color = 'transparent': color for the background
-- must be a color that exists
:param colordepth=8: only 8 bit color supported for now
maybe someday, 32 bit will be an option
"""
projection = (FlatEarthProjection()
if projection is None else projection)
self._image_size = image_size
if colordepth != 8:
raise NotImplementedError("only 8 bit color currently implemented")
self.background_color = background_color
self.create_images(preset_colors)
self.projection = projection
self._viewport = Viewport(((-180, -90), (180, 90)))
if viewport is not None:
self._viewport.BB = tuple(map(tuple, viewport))
self.projection.set_scale(self.viewport, self.image_size)
self.graticule = GridLines(self._viewport, self.projection)
def get_delta_Trapezoid(self, sc, time_step, model_time, pos, vel_field):
dt = timedelta(seconds=time_step)
dt_s = dt.seconds
t = model_time
v0 = vel_field.at(pos, t, extrapolate=self.extrapolate).data
d0 = FlatEarthProjection.meters_to_lonlat(v0 * dt_s, pos)
v1 = vel_field.at(pos + d0, t + dt, extrapolate=self.extrapolate).data
return dt_s / 2 * (v0 + v1)
def test_north():
sp = sample_sc_release(num_elements=10, start_pos=(20, 0.0, 0.0))
mover = simple_mover.SimpleMover(velocity=(0.0, 10, 0.0))
delta = mover.get_move(sp, time_step=100.0, model_time=None)
# expected = np.zeros_like(delta)
expected = proj.meters_to_lonlat((0.0, 1000.0, 0.0), (0.0, 0.0, 0.0))
assert np.alltrue(delta == expected)
def _expected_move(self):
"""
Put the expected move logic in separate (fixture) if it gets used
multiple times
"""
uv = r_theta_to_uv_wind(self.time_val['value'])
exp = np.zeros((self.sc.num_released, 3))
exp[:, 0] = self.sc['windages'] * uv[0, 0] * self.time_step
exp[:, 1] = self.sc['windages'] * uv[0, 1] * self.time_step
xform = FlatEarthProjection.meters_to_lonlat(exp, self.sc['positions'])
return xform
def test_north():
sp = sample_sc_release(num_elements=10, start_pos=(20, 0.0, 0.0))
mover = simple_mover.SimpleMover(velocity=(0.0, 10, 0.0))
delta = mover.get_move(sp, time_step=100.0, model_time=None)
# expected = np.zeros_like(delta)
expected = proj.meters_to_lonlat((0.0, 1000.0, 0.0), (0.0, 0.0,
0.0))
assert np.alltrue(delta == expected)
def _expected_move(self):
"""
Put the expected move logic in separate (fixture) if it gets used
multiple times
"""
uv = r_theta_to_uv_wind(self.time_val['value'])
exp = np.zeros((self.sc.num_released, 3))
exp[:, 0] = self.sc['windages'] * uv[0, 0] * self.time_step
exp[:, 1] = self.sc['windages'] * uv[0, 1] * self.time_step
xform = FlatEarthProjection.meters_to_lonlat(exp, self.sc['positions'])
return xform
def get_delta_RK2(self, sc, time_step, model_time, pos, vel_field):
dt = timedelta(seconds=time_step)
dt_s = dt.seconds
t = model_time
v0 = vel_field.at(pos, t)
d0 = FlatEarthProjection.meters_to_lonlat(v0 * dt_s, pos)
p1 = pos.copy()
p1 += d0
v1 = vel_field.at(p1, t + dt)
return dt_s / 2 * (v0 + v1)
def get_delta_RK2(self, sc, time_step, model_time, pos, vel_field):
dt = timedelta(seconds=time_step)
dt_s = dt.seconds
t = model_time
v0 = vel_field.at(pos, t)
d0 = FlatEarthProjection.meters_to_lonlat(v0 * dt_s, pos)
p1 = pos.copy()
p1 += d0
v1 = vel_field.at(p1, t + dt)
return dt_s / 2 * (v0 + v1)
def get_delta_RK4(self, sc, time_step, model_time, pos, vel_field):
dt = timedelta(seconds=time_step)
dt_s = dt.seconds
t = model_time
v0 = vel_field.at(pos, t)
d0 = FlatEarthProjection.meters_to_lonlat(v0 * dt_s / 2, pos)
p1 = pos.copy()
p1 += d0
v1 = vel_field.at(p1, t + dt / 2)
d1 = FlatEarthProjection.meters_to_lonlat(v1 * dt_s / 2, pos)
p2 = pos.copy()
p2 += d1
v2 = vel_field.at(p2, t + dt / 2)
d2 = FlatEarthProjection.meters_to_lonlat(v2 * dt_s, pos)
p3 = pos.copy()
p3 += d2
v3 = vel_field.at(p3, t + dt)
return dt_s / 6 * (v0 + 2 * v1 + 2 * v2 + v3)
def get_delta_RK4(self, sc, time_step, model_time, pos, vel_field):
dt = timedelta(seconds=time_step)
dt_s = dt.seconds
t = model_time
v0 = vel_field.at(pos, t)
d0 = FlatEarthProjection.meters_to_lonlat(v0 * dt_s / 2, pos)
p1 = pos.copy()
p1 += d0
v1 = vel_field.at(p1, t + dt / 2)
d1 = FlatEarthProjection.meters_to_lonlat(v1 * dt_s / 2, pos)
p2 = pos.copy()
p2 += d1
v2 = vel_field.at(p2, t + dt / 2)
d2 = FlatEarthProjection.meters_to_lonlat(v2 * dt_s, pos)
p3 = pos.copy()
p3 += d2
v3 = vel_field.at(p3, t + dt)
return dt_s / 6 * (v0 + 2 * v1 + 2 * v2 + v3)
def test_basic_move():
# initilizes to long, lat, z = 0.0, 0.0, 0.0
sp = sample_sc_release(num_elements=5)
mover = simple_mover.SimpleMover(velocity=(1.0, 10.0, 0.0))
delta = mover.get_move(sp, time_step=100.0, model_time=None)
print delta
# expected = np.zeros_like(delta)
expected = proj.meters_to_lonlat((100.0, 1000.0, 0.0), (0.0, 0.0, 0.0))
assert np.alltrue(delta == expected)
def test_basic_move():
# initilizes to long, lat, z = 0.0, 0.0, 0.0
sp = sample_sc_release(num_elements=5)
mover = simple_mover.SimpleMover(velocity=(1.0, 10.0, 0.0))
delta = mover.get_move(sp, time_step=100.0, model_time=None)
print delta
# expected = np.zeros_like(delta)
expected = proj.meters_to_lonlat((100.0, 1000.0, 0.0), (0.0, 0.0,
0.0))
assert np.alltrue(delta == expected)
def get_move(self, sc, time_step, model_time_datetime, num_method=None):
"""
Compute the move in (long,lat,z) space. It returns the delta move
for each element of the spill as a numpy array of size
(number_elements X 3) and dtype = gnome.basic_types.world_point_type
Base class returns an array of numpy.nan for delta to indicate the
get_move is not implemented yet.
Each class derived from Mover object must implement it's own get_move
:param sc: an instance of gnome.spill_container.SpillContainer class
:param time_step: time step in seconds
:param model_time_datetime: current model time as datetime object
All movers must implement get_move() since that's what the model calls
"""
positions = sc['positions']
if self.active and len(positions) > 0:
status = sc['status_codes'] != oil_status.in_water
pos = positions[:]
res = self.delta_method(num_method)(sc, time_step,
model_time_datetime, pos,
self.current)
if res.shape[1] == 2:
deltas = np.zeros_like(positions)
deltas[:, 0:2] = res
else:
deltas = res
deltas *= self.scale_value
deltas = FlatEarthProjection.meters_to_lonlat(deltas, positions)
deltas[status] = (0, 0, 0)
else:
deltas = np.zeros_like(positions)
return deltas
def test_variance1(start_loc, time_step):
"""
After a few timesteps the variance of the particle positions should be
similar to the computed value: var = Dt
"""
num_le = 1000
start_time = datetime.datetime(2012, 11, 10, 0)
sc = sample_sc_release(num_le, start_loc, start_time)
D = 100000
num_steps = 10
rand = RandomMover(diffusion_coef=D)
model_time = start_time
for i in range(num_steps):
model_time += datetime.timedelta(seconds=time_step)
sc.release_elements(time_step, model_time)
rand.prepare_for_model_step(sc, time_step, model_time)
delta = rand.get_move(sc, time_step, model_time)
# print "delta:", delta
sc['positions'] += delta
# print sc['positions']
# compute the variances:
# convert to meters
pos = FlatEarthProjection.lonlat_to_meters(sc['positions'],
start_loc)
var = np.var(pos, axis=0)
# D converted to meters^s/s
expected = 2.0 * (D * 1e-4) * num_steps * time_step
assert np.allclose(var, (expected, expected, 0.), rtol=0.1)
def test_variance1(start_loc, time_step):
"""
After a few timesteps the variance of the particle positions should be
similar to the computed value: var = Dt
"""
num_le = 1000
start_time = datetime.datetime(2012, 11, 10, 0)
sc = sample_sc_release(num_le, start_loc, start_time)
D = 100000
num_steps = 10
rand = RandomMover(diffusion_coef=D)
model_time = start_time
for i in range(num_steps):
model_time += datetime.timedelta(seconds=time_step)
sc.release_elements(time_step, model_time)
rand.prepare_for_model_step(sc, time_step, model_time)
delta = rand.get_move(sc, time_step, model_time)
# print "delta:", delta
sc['positions'] += delta
# print sc['positions']
# compute the variances:
# convert to meters
pos = FlatEarthProjection.lonlat_to_meters(sc['positions'], start_loc)
var = np.var(pos, axis=0)
# D converted to meters^s/s
expected = 2.0 * (D * 1e-4) * num_steps * time_step
assert np.allclose(var, (expected, expected, 0.), rtol=0.1)
def get_move(self, sc, time_step, model_time_datetime, num_method=None):
"""
Compute the move in (long,lat,z) space. It returns the delta move
for each element of the spill as a numpy array of size
(number_elements X 3) and dtype = gnome.basic_types.world_point_type
Base class returns an array of numpy.nan for delta to indicate the
get_move is not implemented yet.
Each class derived from Mover object must implement it's own get_move
:param sc: an instance of gnome.spill_container.SpillContainer class
:param time_step: time step in seconds
:param model_time_datetime: current model time as datetime object
All movers must implement get_move() since that's what the model calls
"""
method = None
if num_method is None:
method = self.num_methods[self.default_num_method]
else:
method = self.num_method[num_method]
status = sc["status_codes"] != oil_status.in_water
positions = sc["positions"]
pos = positions[:]
res = method(sc, time_step, model_time_datetime, pos, self.current)
if res.shape[1] == 2:
deltas = np.zeros_like(positions)
deltas[:, 0:2] = res
else:
deltas = res
deltas = FlatEarthProjection.meters_to_lonlat(deltas, positions)
deltas[status] = (0, 0, 0)
return deltas
class SimpleMover(Mover):
"""
simple_mover
a really simple mover -- moves all LEs a constant speed and direction
(not all that different than a constant wind mover, now that I
think about it)
"""
_schema = SimpleMoverSchema
def __init__(self, velocity, uncertainty_scale=0.5, **kwargs):
"""
simple_mover (velocity)
create a simple_mover instance
:param velocity: a (u, v, w) triple -- in meters per second
:param uncertainty_scale=0.5: the scale of the uncertainty of the velocity
Remaining kwargs are passed onto Mover's __init__ using super.
See Mover documentation for remaining valid kwargs.
"""
# use this, to be compatible with whatever we are using for location
self.velocity = np.asarray(velocity, dtype=mover_type).reshape((3, ))
self.uncertainty_scale = uncertainty_scale
super(SimpleMover, self).__init__(**kwargs)
def __repr__(self):
return "SimpleMover(velocity={!r}, uncertainty_scale={!r})".format(
self.velocity, self.uncertainty_scale)
def get_move(self, spill, time_step, model_time):
"""
moves the particles defined in the spill object
:param spill: spill is an instance of the gnome.spill.Spill class
:param time_step: time_step in seconds
:param model_time: current model time as a datetime object
In this case, it uses the:
positions
status_code
data arrays.
:returns delta: Nx3 numpy array of movement
-- in (long, lat, meters) units
"""
# Get the data:
try:
positions = spill['positions']
status_codes = spill['status_codes']
except KeyError, err:
raise ValueError("The spill doesn't have the required "
"data arrays\n{}".format(err))
# which ones should we move?
in_water_mask = status_codes == oil_status.in_water
# compute the move
delta = np.zeros_like(positions)
if self.active and self.on:
delta[in_water_mask] = self.velocity * time_step
# add some random stuff if uncertainty is on
if spill.uncertain:
num = sum(in_water_mask)
scale = self.uncertainty_scale * self.velocity \
* time_step
delta[in_water_mask,
0] += random.uniform(-scale[0], scale[0], num)
delta[in_water_mask,
1] += random.uniform(-scale[1], scale[1], num)
delta[in_water_mask,
2] += random.uniform(-scale[2], scale[2], num)
# scale for projection
# just the lat-lon...
delta = proj.meters_to_lonlat(delta, positions)
return delta
def set_newparticle_values(self, num_new_particles, current_time,
time_step, data_arrays):
"""
SpillContainer will release elements and initialize all data_arrays
to default initial value. The SpillContainer gets passed as input and
the data_arrays for 'position' get initialized correctly by the release
object: self.release.set_newparticle_positions()
If a Spill Amount is given, the Spill object also sets the 'mass' data
array; else 'mass' array remains '0'
:param int num_new_particles: number of new particles that were added.
Always greater than 0
:param current_time: current time
:type current_time: datetime.datetime
:param time_step: the time step, sometimes used to decide how many
should get released.
:type time_step: integer seconds
:param data_arrays: dict of data_arrays provided by the SpillContainer.
Look for 'positions' array in the dict and update positions for
latest num_new_particles that are released
:type data_arrays: dict containing numpy arrays for values
Also, the set_newparticle_values() method for all element_type gets
called so each element_type sets the values for its own data correctly
"""
mass_fluxes = [tam_drop.mass_flux for tam_drop in self.droplets]
delta_masses, proportions, total_mass = self._get_mass_distribution(mass_fluxes, time_step)
# set up LE distribution, the number of particles in each 'release point'
LE_distribution = [int(num_new_particles * p) for p in proportions]
diff = num_new_particles - sum(LE_distribution)
for i in range(0, diff):
LE_distribution[i % len(LE_distribution)] += 1
# compute release point location for each droplet
positions = [self.start_position + FlatEarthProjection.meters_to_lonlat(d.position, self.start_position) for d in self.droplets]
for p in positions:
p[0][2] -= self.start_position[2]
# for each release location, set the position and mass of the elements released at that location
total_rel = 0
for mass_dist, n_LEs, pos, droplet in zip(delta_masses, LE_distribution, positions, self.droplets):
start_idx = -num_new_particles + total_rel
if start_idx == 0:
break
end_idx = start_idx + n_LEs
if end_idx == 0:
end_idx = None
# print '{0} to {1}'.format(start_idx, end_idx)
if start_idx == end_idx:
continue
data_arrays['positions'][start_idx:end_idx] = pos
data_arrays['mass'][start_idx:end_idx] = mass_dist / n_LEs
data_arrays['init_mass'][start_idx:end_idx] = mass_dist / n_LEs
data_arrays['density'][start_idx:end_idx] = droplet.density
data_arrays['droplet_diameter'][start_idx:end_idx] = np.random.normal(droplet.radius * 2, droplet.radius * 0.15, (n_LEs))
v = data_arrays['rise_vel'][start_idx:end_idx]
rise_velocity_from_drop_size(v,
data_arrays['density'][start_idx:end_idx],
data_arrays['droplet_diameter'][start_idx:end_idx],
0.0000013, 1020)
data_arrays['rise_vel'][start_idx:end_idx] = v
total_rel += n_LEs
self.num_released += num_new_particles
self.amount_released += total_mass
class SimpleMover(Mover, serializable.Serializable):
"""
simple_mover
a really simple mover -- moves all LEs a constant speed and direction
(not all that different than a constant wind mover, now that I think about it)
"""
_state = copy.deepcopy(Mover._state)
_state.add(update=['uncertainty_scale', 'velocity'],
save=['uncertainty_scale', 'velocity'])
def __init__(self, velocity, uncertainty_scale=0.5, **kwargs):
"""
simple_mover (velocity)
create a simple_mover instance
:param velocity: a (u, v, w) triple -- in meters per second
Remaining kwargs are passed onto Mover's __init__ using super.
See Mover documentation for remaining valid kwargs.
"""
self.velocity = np.asarray(
velocity, dtype=basic_types.mover_type
).reshape(
(3, )
) # use this, to be compatible with whatever we are using for location
self.uncertainty_scale = uncertainty_scale
super(SimpleMover, self).__init__(**kwargs)
def __repr__(self):
return 'SimpleMover(<%s>)' % self.id
def velocity_to_dict(self):
"""
convert velocity back into a tuple
"""
return tuple(self.velocity.tolist())
def get_move(
self,
spill,
time_step,
model_time,
):
"""
moves the particles defined in the spill object
:param spill: spill is an instance of the gnome.spill.Spill class
:param time_step: time_step in seconds
:param model_time: current model time as a datetime object
In this case, it uses the:
positions
status_code
data arrays.
:returns delta: Nx3 numpy array of movement -- in (long, lat, meters) units
"""
# Get the data:
try:
positions = spill['positions']
status_codes = spill['status_codes']
except KeyError, err:
raise ValueError(
'The spill does not have the required data arrays\n' +
err.message)
# which ones should we move?
in_water_mask = status_codes == basic_types.oil_status.in_water
# compute the move
delta = np.zeros_like(positions)
if self.active and self.on:
delta[in_water_mask] = self.velocity * time_step
# add some random stuff if uncertainty is on
if spill.uncertain:
num = sum(in_water_mask)
scale = self.uncertainty_scale * self.velocity \
* time_step
delta[in_water_mask,
0] += random.uniform(-scale[0], scale[0], num)
delta[in_water_mask,
1] += random.uniform(-scale[1], scale[1], num)
delta[in_water_mask,
2] += random.uniform(-scale[2], scale[2], num)
# scale for projection
delta = proj.meters_to_lonlat(delta,
positions) # just the lat-lon...
return delta