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model.py

Models 


              flowchart TD
              erosionfront.geomorphic.model.Models[Models]

              

              click erosionfront.geomorphic.model.Models href "" "erosionfront.geomorphic.model.Models"

Model type enumeration.

Attributes

str = "isotropic"

Isotropic eikonal model.

str = "step"

Simple error-function smooth-step model.

str = "exponential-activation"

Smooth-step model based on exp(-β_t/β) shape.

BaseStepModel dataclass 

BaseStepModel(
    type: str = "",
    label: str = "",
    ξ_min: float = 0.2,
    ξ_max: float = 1,
    ξ_trend: float = 0.05,
    β_t: float = pi / 4,
    σ_ξ: float = 0.1,
    n_ξ: float = 2,
    Δt_slabfailure: float = 0.05,
    do_slabfailure: bool = True,
)


              flowchart TD
              erosionfront.geomorphic.model.BaseStepModel[BaseStepModel]

              

              click erosionfront.geomorphic.model.BaseStepModel href "" "erosionfront.geomorphic.model.BaseStepModel"

Base class for step-type model erosion speed functions.

Subclassed models are all explicitly defined, rather than derived from Sympy theoretical development.

Parameters

ξ_min: float = 0.2 Asymptotic minimum ξ for β → 0. ξ_max: float = 1 Asymptotic minimum ξ for β → ∞. ξ_trend: float = 0.05 Overall "tilt" to ξ model as β rises. β_t: float = np.pi/4 Transition slope angle β where ξ switches from low to high. σ_ξ: float = 0.1 Smoothness (in Gaussian std dev sense w β) of ξ transition. n_ξ: float = 2 Order of transition smoothness (power exponent; alt to σ_ξ). Δt_slabfailure: float = 0.05 Time between periodic removal of overhangs. do_slabfailure: bool = True Do periodic removal of overhangs.

Attributes

type: str = "" Model name (long form). label: str = "" Model name (short form). ξ_min: float = 0.2 Asymptotic minimum ξ for β → 0. ξ_max: float = 1 Asymptotic minimum ξ for β → ∞. ξ_trend: float = 0.05 Overall "tilt" to ξ model as β rises. β_t: float = np.pi/4 Transition slope angle β where ξ switches from low to high. σ_ξ: float = 0.1 Smoothness (in Gaussian std dev sense w β) of ξ transition. n_ξ: float = 2 Order of transition smoothness (power exponent; alt to σ_ξ). Δt_slabfailure: float = 0.05 Time between periodic removal of overhangs. do_slabfailure: bool = True Do periodic removal of overhangs.

Methods:

  • __post_init__ –

    Select the model type and assign its short-form label.

  • dHlsdφ_fn_Δφx –

    Derivative of ls H_{∂φ/∂x} wrt ∂φ/∂x that must be set by model class.

  • dHlsdφ_fn_Δφz –

    Derivative of ls H_{∂φ/∂z} wrt ∂φ/∂z that must be set by model class.

  • ξ_fn_β –

    Erosion speed ξ(β) method that must be implemented by each model class.

__post_init__ 

__post_init__() -> None

Select the model type and assign its short-form label.

Attributes

label: str Short-form model type string.

Source code in erosionfront/geomorphic/model.py 
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def __post_init__(self) -> None:
    """
    Select the model type and assign its short-form label.

    Attributes
    ----------
    label: str
        Short-form model type string.
    """
    if self.type not in Models._value2member_map_.keys():
        raise ValueError("Invalid model type")
    model_labels: dict = {
        Models.ISO: "ISO",
        Models.STEP: "STP",
        Models.EA: "EA"
    }
    self.label = model_labels[self.type]

dHlsdφ_fn_Δφx abstractmethod 

dHlsdφ_fn_Δφx(
    Δφ_x: float | NDArray | MaskedArray, Δφ_z: float | NDArray | MaskedArray
) -> float | NDArray | MaskedArray

Derivative of ls H_{∂φ/∂x} wrt ∂φ/∂x that must be set by model class.

Parameters

Δφ_x: float | NDArray | MaskedArray Finite difference approximation of ∂φ/∂x. Δφ_z: float | NDArray | MaskedArray Finite difference approximation of ∂φ/∂z.

Returns

float | NDArray | MaskedArray: Derivative of ls H_{∂φ/∂x} wrt ∂φ/∂x.

Source code in erosionfront/geomorphic/model.py 
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@abstractmethod
def dHlsdφ_fn_Δφx(
        self, 
        Δφ_x: float | NDArray | MaskedArray,
        Δφ_z: float | NDArray | MaskedArray,
    ) -> float | NDArray | MaskedArray:
    """
    Derivative of ls H_{∂φ/∂x} wrt ∂φ/∂x that must be set by model class.

    Parameters
    ----------
    Δφ_x: float | NDArray | MaskedArray
        Finite difference approximation of ∂φ/∂x.
    Δφ_z: float | NDArray | MaskedArray
        Finite difference approximation of ∂φ/∂z.

    Returns
    -------
    float | NDArray | MaskedArray:
        Derivative of ls H_{∂φ/∂x} wrt ∂φ/∂x.
    """
    pass

dHlsdφ_fn_Δφz abstractmethod 

dHlsdφ_fn_Δφz(
    Δφ_x: float | NDArray | MaskedArray, Δφ_z: float | NDArray | MaskedArray
) -> float | NDArray | MaskedArray

Derivative of ls H_{∂φ/∂z} wrt ∂φ/∂z that must be set by model class.

Parameters

Δφ_x: float | NDArray | MaskedArray Finite difference approximation of ∂φ/∂x. Δφ_z: float | NDArray | MaskedArray Finite difference approximation of ∂φ/∂z.

Returns

float | NDArray | MaskedArray: Derivative of ls H_{∂φ/∂z} wrt ∂φ/∂z.

Source code in erosionfront/geomorphic/model.py 
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@abstractmethod
def dHlsdφ_fn_Δφz(
        self, 
        Δφ_x: float | NDArray | MaskedArray,
        Δφ_z: float | NDArray | MaskedArray,
    ) -> float | NDArray | MaskedArray:
    """
    Derivative of ls H_{∂φ/∂z} wrt ∂φ/∂z that must be set by model class.

    Parameters
    ----------
    Δφ_x: float | NDArray | MaskedArray
        Finite difference approximation of ∂φ/∂x.
    Δφ_z: float | NDArray | MaskedArray
        Finite difference approximation of ∂φ/∂z.

    Returns
    -------
    float | NDArray | MaskedArray:
        Derivative of ls H_{∂φ/∂z} wrt ∂φ/∂z.
    """
    pass

ξ_fn_β abstractmethod 

ξ_fn_β(β: float | NDArray | MaskedArray) -> float | NDArray | MaskedArray

Erosion speed ξ(β) method that must be implemented by each model class.

Parameters

β: float | NDArray | MaskedArray Surface tilt angle(s).

Returns

float | NDArray | MaskedArray: Surface-normal erosion speed(s) ξ(β).

Source code in erosionfront/geomorphic/model.py 
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@abstractmethod
def ξ_fn_β(
        self, 
        β: float | NDArray | MaskedArray,
    ) -> float | NDArray | MaskedArray:
    """
    Erosion speed ξ(β) method that must be implemented by each model class.

    Parameters
    ----------
    β: float | NDArray | MaskedArray
        Surface tilt angle(s).

    Returns
    -------
    float | NDArray | MaskedArray:
        Surface-normal erosion speed(s) ξ(β).
    """
    pass

Isotropic dataclass 

Isotropic(
    type: str = "",
    label: str = "",
    ξ_min: float = 0.2,
    ξ_max: float = 1,
    ξ_trend: float = 0.05,
    β_t: float = pi / 4,
    σ_ξ: float = 0.1,
    n_ξ: float = 2,
    Δt_slabfailure: float = 0.05,
    do_slabfailure: bool = True,
)


              flowchart TD
              erosionfront.geomorphic.model.Isotropic[Isotropic]
              erosionfront.geomorphic.model.BaseStepModel[BaseStepModel]

                              erosionfront.geomorphic.model.BaseStepModel --> erosionfront.geomorphic.model.Isotropic
                


              click erosionfront.geomorphic.model.Isotropic href "" "erosionfront.geomorphic.model.Isotropic"
              click erosionfront.geomorphic.model.BaseStepModel href "" "erosionfront.geomorphic.model.BaseStepModel"

Isotropic erosion speed model, i.e., simple eikonal model.

Parameters

See base class.

Attributes

See base class.

Returned by: Used by:

Methods:

  • __post_init__ –

    Select the model type and assign its short-form label.

  • dHlsdφ_fn_Δφx –

    Derivative of ls H_{∂φ/∂x} wrt ∂φ/∂x for isotropic erosion speed model.

  • dHlsdφ_fn_Δφz –

    Derivative of ls H_{∂φ/∂z} wrt ∂φ/∂z for isotropic erosion speed model.

  • ξ_fn_β –

    Surface-normal erosion speed ξ(β) for isotropic erosion speed model.

__post_init__ 

__post_init__() -> None

Select the model type and assign its short-form label.

Attributes

label: str Short-form model type string.

Source code in erosionfront/geomorphic/model.py 
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def __post_init__(self) -> None:
    """
    Select the model type and assign its short-form label.

    Attributes
    ----------
    label: str
        Short-form model type string.
    """
    if self.type not in Models._value2member_map_.keys():
        raise ValueError("Invalid model type")
    model_labels: dict = {
        Models.ISO: "ISO",
        Models.STEP: "STP",
        Models.EA: "EA"
    }
    self.label = model_labels[self.type]

dHlsdφ_fn_Δφx 

dHlsdφ_fn_Δφx(
    Δφ_x: float | NDArray | MaskedArray, Δφ_z: float | NDArray | MaskedArray
) -> float | NDArray | MaskedArray

Derivative of ls H_{∂φ/∂x} wrt ∂φ/∂x for isotropic erosion speed model.

Parameters

Δφ_x: float | NDArray | MaskedArray Finite difference approximation of ∂φ/∂x. Δφ_z: float | NDArray | MaskedArray Finite difference approximation of ∂φ/∂z.

Returns

float | NDArray | MaskedArray: Derivative of ls H_{∂φ/∂x} wrt ∂φ/∂x.

Source code in erosionfront/geomorphic/model.py 
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def dHlsdφ_fn_Δφx(
        self, 
        Δφ_x: float | NDArray | MaskedArray,
        Δφ_z: float | NDArray | MaskedArray,
    ) -> float | NDArray | MaskedArray:
    """
    Derivative of ls H_{∂φ/∂x} wrt ∂φ/∂x for isotropic erosion speed model.

    Parameters
    ----------
    Δφ_x: float | NDArray | MaskedArray
        Finite difference approximation of ∂φ/∂x.
    Δφ_z: float | NDArray | MaskedArray
        Finite difference approximation of ∂φ/∂z.

    Returns
    -------
    float | NDArray | MaskedArray:
        Derivative of ls H_{∂φ/∂x} wrt ∂φ/∂x.
    """
    if type(Δφ_x) is np.ndarray or type(Δφ_x) is np.ma.masked_array:
        return np.zeros_like(Δφ_x)
    else:
        return float(0)

dHlsdφ_fn_Δφz 

dHlsdφ_fn_Δφz(
    Δφ_x: float | NDArray | MaskedArray, Δφ_z: float | NDArray | MaskedArray
) -> float | NDArray | MaskedArray

Derivative of ls H_{∂φ/∂z} wrt ∂φ/∂z for isotropic erosion speed model.

Parameters

Δφ_x: float | NDArray | MaskedArray Finite difference approximation of ∂φ/∂x. Δφ_z: float | NDArray | MaskedArray Finite difference approximation of ∂φ/∂z.

Returns

float | NDArray | MaskedArray: Derivative of ls H_{∂φ/∂z} wrt ∂φ/∂z.

Source code in erosionfront/geomorphic/model.py 
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def dHlsdφ_fn_Δφz(
        self, 
        Δφ_x: float | NDArray | MaskedArray,
        Δφ_z: float | NDArray | MaskedArray,
    ) -> float | NDArray | MaskedArray:
    """
    Derivative of ls H_{∂φ/∂z} wrt ∂φ/∂z for isotropic erosion speed model.

    Parameters
    ----------
    Δφ_x: float | NDArray | MaskedArray
        Finite difference approximation of ∂φ/∂x.
    Δφ_z: float | NDArray | MaskedArray
        Finite difference approximation of ∂φ/∂z.

    Returns
    -------
    float | NDArray | MaskedArray:
        Derivative of ls H_{∂φ/∂z} wrt ∂φ/∂z.
    """
    if type(Δφ_z) is np.ndarray or type(Δφ_z) is np.ma.masked_array:
        return np.zeros_like(Δφ_z)
    else:
        return float(0)

ξ_fn_β 

ξ_fn_β(β: float | NDArray | MaskedArray) -> float | NDArray | MaskedArray

Surface-normal erosion speed ξ(β) for isotropic erosion speed model.

Parameters

β: float | NDArray | MaskedArray Surface tilt angle(s).

Attributes

References ξ_min attribute.

Returns

float | NDArray | MaskedArray: Surface-normal erosion speed(s) ξ(β) for isotropic model.

Source code in erosionfront/geomorphic/model.py 
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def ξ_fn_β(
        self, 
        β: float | NDArray | MaskedArray,
    ) -> float | NDArray | MaskedArray:
    r"""
    Surface-normal erosion speed ξ(β) for isotropic erosion speed model.

    Parameters
    ----------
    β: float | NDArray | MaskedArray
        Surface tilt angle(s).

    Attributes
    ----------
    References ξ_min attribute.

    Returns
    -------
    float | NDArray | MaskedArray:
        Surface-normal erosion speed(s) ξ(β) for isotropic model.
    """
    if type(β) is np.ndarray or type(β) is np.ma.masked_array:
        return np.ones_like(β)*self.ξ_max
    else:
        return float(self.ξ_max)

Step dataclass 

Step(
    type: str = "",
    label: str = "",
    ξ_min: float = 0.2,
    ξ_max: float = 1,
    ξ_trend: float = 0.05,
    β_t: float = pi / 4,
    σ_ξ: float = 0.1,
    n_ξ: float = 2,
    Δt_slabfailure: float = 0.05,
    do_slabfailure: bool = True,
)


              flowchart TD
              erosionfront.geomorphic.model.Step[Step]
              erosionfront.geomorphic.model.BaseStepModel[BaseStepModel]

                              erosionfront.geomorphic.model.BaseStepModel --> erosionfront.geomorphic.model.Step
                


              click erosionfront.geomorphic.model.Step href "" "erosionfront.geomorphic.model.Step"
              click erosionfront.geomorphic.model.BaseStepModel href "" "erosionfront.geomorphic.model.BaseStepModel"

Smoothed-step erosion speed model with switch at threshold β.

Parameters

See base class.

Attributes

See base class.

Returned by: Used by:

Methods:

  • __post_init__ –

    Select the model type and assign its short-form label.

  • dHlsdφ_fn_Δφx –

    Derivative of ls H_{∂φ/∂x} wrt ∂φ/∂x for smoothed-step erosion model.

  • dHlsdφ_fn_Δφz –

    Derivative of ls H_{∂φ/∂z} wrt ∂φ/∂z for smoothed-step erosion model.

  • ξ_fn_β –

    Compute smoothed-step erosion speed with switch at threshold β.

__post_init__ 

__post_init__() -> None

Select the model type and assign its short-form label.

Attributes

label: str Short-form model type string.

Source code in erosionfront/geomorphic/model.py 
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def __post_init__(self) -> None:
    """
    Select the model type and assign its short-form label.

    Attributes
    ----------
    label: str
        Short-form model type string.
    """
    if self.type not in Models._value2member_map_.keys():
        raise ValueError("Invalid model type")
    model_labels: dict = {
        Models.ISO: "ISO",
        Models.STEP: "STP",
        Models.EA: "EA"
    }
    self.label = model_labels[self.type]

dHlsdφ_fn_Δφx 

dHlsdφ_fn_Δφx(
    Δφ_x: float | NDArray | MaskedArray, Δφ_z: float | NDArray | MaskedArray
) -> float | NDArray | MaskedArray

Derivative of ls H_{∂φ/∂x} wrt ∂φ/∂x for smoothed-step erosion model.

Parameters

Δφ_x: float | NDArray | MaskedArray Finite difference approximation of ∂φ/∂x. Δφ_z: float | NDArray | MaskedArray Finite difference approximation of ∂φ/∂z.

Returns

float | NDArray | MaskedArray: Derivative of ls H_{∂φ/∂x} wrt ∂φ/∂x.

Source code in erosionfront/geomorphic/model.py 
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def dHlsdφ_fn_Δφx(
        self, 
        Δφ_x: float | NDArray | MaskedArray,
        Δφ_z: float | NDArray | MaskedArray,
    ) -> float | NDArray | MaskedArray:
    """
    Derivative of ls H_{∂φ/∂x} wrt ∂φ/∂x for smoothed-step erosion model.

    Parameters
    ----------
    Δφ_x: float | NDArray | MaskedArray
        Finite difference approximation of ∂φ/∂x.
    Δφ_z: float | NDArray | MaskedArray
        Finite difference approximation of ∂φ/∂z.

    Returns
    -------
    float | NDArray | MaskedArray:
        Derivative of ls H_{∂φ/∂x} wrt ∂φ/∂x.
    """
    return Δφ_x*0

dHlsdφ_fn_Δφz 

dHlsdφ_fn_Δφz(
    Δφ_x: float | NDArray | MaskedArray, Δφ_z: float | NDArray | MaskedArray
) -> float | NDArray | MaskedArray

Derivative of ls H_{∂φ/∂z} wrt ∂φ/∂z for smoothed-step erosion model.

Parameters

Δφ_x: float | NDArray | MaskedArray Finite difference approximation of ∂φ/∂x. Δφ_z: float | NDArray | MaskedArray Finite difference approximation of ∂φ/∂z.

Returns

float | NDArray | MaskedArray: Derivative of ls H_{∂φ/∂z} wrt ∂φ/∂z.

Source code in erosionfront/geomorphic/model.py 
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def dHlsdφ_fn_Δφz(
        self, 
        Δφ_x: float | NDArray | MaskedArray,
        Δφ_z: float | NDArray | MaskedArray,
    ) -> float | NDArray | MaskedArray:
    """
    Derivative of ls H_{∂φ/∂z} wrt ∂φ/∂z for smoothed-step erosion model.

    Parameters
    ----------
    Δφ_x: float | NDArray | MaskedArray
        Finite difference approximation of ∂φ/∂x.
    Δφ_z: float | NDArray | MaskedArray
        Finite difference approximation of ∂φ/∂z.

    Returns
    -------
    float | NDArray | MaskedArray:
        Derivative of ls H_{∂φ/∂z} wrt ∂φ/∂z.
    """
    return Δφ_z*0

ξ_fn_β 

ξ_fn_β(β: float | NDArray | MaskedArray) -> float | NDArray | MaskedArray

Compute smoothed-step erosion speed with switch at threshold β.

Parameters

β: float | NDArray | MaskedArray Surface tilt angle(s).

Attributes

References β_t, σ_ξ, ξ_min, ξ_max. Also now: ξ_trend to provide tilt.

Returns

float | NDArray | MaskedArray: Surface-normal erosion speed(s) ξ(β) for smoothed-step model.

Source code in erosionfront/geomorphic/model.py 
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def ξ_fn_β(
        self, 
        β: float | NDArray | MaskedArray,
    ) -> float | NDArray | MaskedArray:
    """
    Compute smoothed-step erosion speed with switch at threshold β.

    Parameters
    ----------
    β: float | NDArray | MaskedArray
        Surface tilt angle(s).

    Attributes
    ----------
    References β_t, σ_ξ, ξ_min, ξ_max.
    Also now: ξ_trend to provide tilt.

    Returns
    -------
    float | NDArray | MaskedArray:
        Surface-normal erosion speed(s) ξ(β) for smoothed-step model.
    """
    switch: float | NDArray | MaskedArray \
        = (1+erf((np.abs(β)-self.β_t)/self.σ_ξ))/2
    return (
        np.abs(β)*self.ξ_trend +
        (self.ξ_min + switch*(self.ξ_max-self.ξ_min))
    )

ExponentialActivation dataclass 

ExponentialActivation(
    type: str = "",
    label: str = "",
    ξ_min: float = 0.2,
    ξ_max: float = 1,
    ξ_trend: float = 0.05,
    β_t: float = pi / 4,
    σ_ξ: float = 0.1,
    n_ξ: float = 2,
    Δt_slabfailure: float = 0.05,
    do_slabfailure: bool = True,
)


              flowchart TD
              erosionfront.geomorphic.model.ExponentialActivation[ExponentialActivation]
              erosionfront.geomorphic.model.BaseStepModel[BaseStepModel]

                              erosionfront.geomorphic.model.BaseStepModel --> erosionfront.geomorphic.model.ExponentialActivation
                


              click erosionfront.geomorphic.model.ExponentialActivation href "" "erosionfront.geomorphic.model.ExponentialActivation"
              click erosionfront.geomorphic.model.BaseStepModel href "" "erosionfront.geomorphic.model.BaseStepModel"

Exponential-activation erosion speed model with switch at threshold β.

Parameters

See base class.

Attributes

See base class.

Returned by: Used by:

Methods:

  • __post_init__ –

    Select the model type and assign its short-form label.

  • dHlsdφ_fn_Δφx –

    Derivative of ls H_{∂φ/∂x} wrt ∂φ/∂x for EA model.

  • dHlsdφ_fn_Δφz –

    Derivative of ls H_{∂φ/∂z} wrt ∂φ/∂z for EA model.

  • ξ_fn_β –

    Surface-normal erosion speed ξ(β) for EA model with step at β_t.

__post_init__ 

__post_init__() -> None

Select the model type and assign its short-form label.

Attributes

label: str Short-form model type string.

Source code in erosionfront/geomorphic/model.py 
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def __post_init__(self) -> None:
    """
    Select the model type and assign its short-form label.

    Attributes
    ----------
    label: str
        Short-form model type string.
    """
    if self.type not in Models._value2member_map_.keys():
        raise ValueError("Invalid model type")
    model_labels: dict = {
        Models.ISO: "ISO",
        Models.STEP: "STP",
        Models.EA: "EA"
    }
    self.label = model_labels[self.type]

dHlsdφ_fn_Δφx 

dHlsdφ_fn_Δφx(
    Δφ_x: float | NDArray | MaskedArray, Δφ_z: float | NDArray | MaskedArray
) -> float | NDArray | MaskedArray

Derivative of ls H_{∂φ/∂x} wrt ∂φ/∂x for EA model.

Parameters

Δφ_x: float | NDArray | MaskedArray Finite difference approximation of ∂φ/∂x. Δφ_z: float | NDArray | MaskedArray Finite difference approximation of ∂φ/∂z.

Returns

float | NDArray | MaskedArray: Derivative of ls H_{∂φ/∂x} wrt ∂φ/∂x.

Source code in erosionfront/geomorphic/model.py 
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def dHlsdφ_fn_Δφx(
        self, 
        Δφ_x: float | NDArray | MaskedArray,
        Δφ_z: float | NDArray | MaskedArray,
    ) -> float | NDArray | MaskedArray:
    """
    Derivative of ls H_{∂φ/∂x} wrt ∂φ/∂x for EA model.

    Parameters
    ----------
    Δφ_x: float | NDArray | MaskedArray
        Finite difference approximation of ∂φ/∂x.
    Δφ_z: float | NDArray | MaskedArray
        Finite difference approximation of ∂φ/∂z.

    Returns
    -------
    float | NDArray | MaskedArray:
        Derivative of ls H_{∂φ/∂x} wrt ∂φ/∂x.
    """
    ξ_tp = (self.ξ_max-self.ξ_min)
    ξ_bs = self.ξ_min
    β_t = self.β_t
    tan = np.tan
    tanβ = Δφ_x/Δφ_z
    secβ = np.sqrt(1+tanβ**2)
    cosβ = 1/secβ
    exp = np.exp
    return (
        ((
            2*ξ_tp*tan(β_t)**2 * exp(-tan(β_t)**2/tanβ**2)
            + (ξ_bs + ξ_tp*exp(-tan(β_t)**2/tanβ**2))*cosβ**2*tanβ**4
        ) / (cosβ*tanβ**3)) * np.heaviside(tanβ, 0)
    )

dHlsdφ_fn_Δφz 

dHlsdφ_fn_Δφz(
    Δφ_x: float | NDArray | MaskedArray, Δφ_z: float | NDArray | MaskedArray
) -> float | NDArray | MaskedArray

Derivative of ls H_{∂φ/∂z} wrt ∂φ/∂z for EA model.

Parameters

Δφ_x: float | NDArray | MaskedArray Finite difference approximation of ∂φ/∂x. Δφ_z: float | NDArray | MaskedArray Finite difference approximation of ∂φ/∂z.

Returns

float | NDArray | MaskedArray: Derivative of ls H_{∂φ/∂z} wrt ∂φ/∂z.

Source code in erosionfront/geomorphic/model.py 
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def dHlsdφ_fn_Δφz(
        self, 
        Δφ_x: float | NDArray | MaskedArray,
        Δφ_z: float | NDArray | MaskedArray,
    ) -> float | NDArray | MaskedArray:
    """
    Derivative of ls H_{∂φ/∂z} wrt ∂φ/∂z for EA model.

    Parameters
    ----------
    Δφ_x: float | NDArray | MaskedArray
        Finite difference approximation of ∂φ/∂x.
    Δφ_z: float | NDArray | MaskedArray
        Finite difference approximation of ∂φ/∂z.

    Returns
    -------
    float | NDArray | MaskedArray:
        Derivative of ls H_{∂φ/∂z} wrt ∂φ/∂z.
    """
    ξ_tp = (self.ξ_max-self.ξ_min)
    ξ_bs = self.ξ_min
    β_t = self.β_t
    tan = np.tan
    tanβ = Δφ_x/Δφ_z
    secβ = np.sqrt(1+tanβ**2)
    cosβ = 1/secβ
    exp = np.exp
    return (
        ((-ξ_bs
         - ξ_tp*exp(-tan(β_t)**2/tanβ**2)
         + 2*ξ_tp*exp(-tan(β_t)**2/tanβ**2)*tan(β_t)**2
            / (1 - cosβ**2)
        ) * cosβ) * np.heaviside(tanβ, 0)
    )

ξ_fn_β 

ξ_fn_β(β: float | NDArray | MaskedArray) -> float | NDArray | MaskedArray

Surface-normal erosion speed ξ(β) for EA model with step at β_t.

Parameters

β: float | NDArray | MaskedArray Surface tilt angle(s).

Attributes

References β_t, σ_ξ, ξ_min, ξ_max.

Returns

float | NDArray | MaskedArray: Surface-normal erosion speed(s) ξ(β) for EA model.

Source code in erosionfront/geomorphic/model.py 
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def ξ_fn_β(
        self, 
        β: float | NDArray | MaskedArray,
    ) -> float | NDArray | MaskedArray:
    """
    Surface-normal erosion speed ξ(β) for EA model with step at β_t.

    Parameters
    ----------
    β: float | NDArray | MaskedArray
        Surface tilt angle(s).

    Attributes
    ----------
    References β_t, σ_ξ, ξ_min, ξ_max.

    Returns
    -------
    float | NDArray | MaskedArray:
        Surface-normal erosion speed(s) ξ(β) for EA model.
    """
    def xi_fn(
            β_: float | NDArray | MaskedArray
        ) -> float | NDArray | MaskedArray:
        return (
            self.ξ_min 
            + 
            (self.ξ_max-self.ξ_min)
            * np.exp(
                -np.abs((np.tan(self.β_t)/np.tan(np.abs(β_))))**self.n_ξ
            )
        )

    xi_: float | NDArray | MaskedArray = xi_fn(β)
    π: float = np.pi
    if type(β) is np.ndarray or type(β) is MaskedArray:
        xi_[np.abs(β)<=1e-10] = self.ξ_min      #type: ignore
        # If there are overhanging points in the β array,
        #   limit at ξ(π/2)
        if xi_[np.abs(β)>=π/2].shape[0]>0:      #type: ignore
            xi_[np.abs(β)>=π/2] = (             #type: ignore
                xi_fn(π/2) 
                    # * (1 + ((np.abs(β[np.abs(β)>=π/2])-π/2)/(π/2))*0 )
            )
    else:
        xi_ = (
            self.ξ_min if np.abs(β)<=1e-10
            # 1e-10 if np.abs(β)<=0.05
            else (
                xi_fn(π/2) 
                    # * (1 + ((np.abs(β)-π/2)/(π/2))*0 )
                if np.abs(β)>=π/2 else xi_
            )
        )
    return xi_

choose_model 

choose_model(parameters: dict) -> Isotropic | Step | ExponentialActivation

Choose erosion model.

Parameters

dict: Dictionary of keyword arguments to be passed to instantiate a model class. Choice of model type is specified in the parameters['type'] element.

Returns

Isotropic | Step | ExponentialActivation: One of these erosion model classes. Instantiation takes **parameters as keyword arguments.

Source code in erosionfront/geomorphic/model.py 
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def choose_model(parameters: dict,) -> (
          Isotropic 
        | Step
        | ExponentialActivation
    ):
    """
    Choose erosion model.

    Parameters
    ----------
    dict: 
        Dictionary of keyword arguments to be passed to instantiate
        a model class. Choice of model type is specified in
        the `parameters['type']` element.

    Returns
    -------
    Isotropic | Step | ExponentialActivation:
        One of these erosion model classes.
        Instantiation takes `**parameters` as keyword arguments.
    """
    models: dict = {
        Models.ISO : Isotropic,
        Models.STEP : Step,
        Models.EA : ExponentialActivation
    }
    return models[parameters["type"]](**parameters)