optimising mct (NOT TESTED)

src/lisflood/hydrological_modules/mct.py

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+import numpy as np
+
+
+def MCTRouting_single(
+ V00, q10, q01, q00, ql, q0mm, Cm0, Dm0, dt, xpix, s0, Balv, ANalv, Nalv
+):
+ """
+ This function implements Muskingum-Cunge-Todini routing method for a single pixel.
+ References:
+ Todini, E. (2007). A mass conservative and water storage consistent variable parameter Muskingum-Cunge approach. Hydrol. Earth Syst. Sci.
+ (Chapter 5)
+ Reggiani, P., Todini, E., & Meißner, D. (2016). On mass and momentum conservation in the variable-parameter Muskingum method. Journal of Hydrology, 543, 562–576. https://doi.org/10.1016/j.jhydrol.2016.10.030
+ (Appendix B)
+
+ :param V00: channel storage volume at t (beginning of computation step)
+ :param q10: O(t) - outflow (x+dx) at time t
+ :param q01: I(t+dt) - inflow (x) at time t+dt
+ :param q00: I(t) - inflow (x) at time t
+ :param ql: lateral flow over time dt [m3/s]
+ :param q0mm: I - average inflow during step dt
+ :param Cm0: Courant number at time t
+ :param Dm0: Reynolds number at time t
+ :param dt: time interval step
+ :param xpix: channel length
+ :param s0: channel slope
+ :param Balv: channel bankfull width
+ :param ANalv: angle of the riverbed side [rad]
+ :param Nalv: channel Manning roughness coefficient
+ :return:
+ q11: Outflow (x+dx) at O(t+dt)
+ V11: channel storage volume at t+dt
+ Cm1: Courant number at t+1 for state file
+ Dm1: Reynolds number at t+1 for state file
+ """
+
+ eps = 1e-06
+
+ # Calc O' first guess for the outflow at time t+dt
+ # O'(t+dt)=O(t)+(I(t+dt)-I(t))
+ q11 = q10 + (q01 - q00)
+
+ # check for negative and zero discharge values
+ # zero outflow is not allowed
+ if q11 < eps: # cmcheck <=0
+ q11 = eps
+
+ # calc reference discharge at time t
+ # qm0 = (I(t)+O(t))/2
+ # qm0 = (q00 + q10) / 2.
+
+ # Calc O(t+dt)=q11 at time t+dt using MCT equations
+ for i in range(2): # repeat 2 times for accuracy
+
+ # reference I discharge at x=0
+ qmx0 = (q00 + q01) / 2.0
+ if qmx0 < eps: # cmcheck ==0
+ qmx0 = eps
+ hmx0 = hoq(qmx0, s0, Balv, ANalv, Nalv)
+
+ # reference O discharge at x=1
+ qmx1 = (q10 + q11) / 2.0
+ if qmx1 < eps: # cmcheck ==0
+ qmx1 = eps
+ hmx1 = hoq(qmx1, s0, Balv, ANalv, Nalv)
+
+ # Calc riverbed slope correction factor
+ cor = 1 - (1 / s0 * (hmx1 - hmx0) / xpix)
+ sfx = s0 * cor
+ if sfx < (0.8 * s0):
+ sfx = 0.8 * s0 # In case of instability raise from 0.5 to 0.8
+
+ # Calc reference discharge time t+dt
+ # Q(t+dt)=(I(t+dt)+O'(t+dt))/2
+ qm1 = (q01 + q11) / 2.0
+ # cm
+ if qm1 < eps: # cmcheck ==0
+ qm1 = eps
+ # cm
+ hm1 = hoq(qm1, s0, Balv, ANalv, Nalv)
+ dummy, Ax1, Bx1, Px1, ck1 = qoh(hm1, s0, Balv, ANalv, Nalv)
+ if ck1 <= eps:
+ ck1 = eps
+
+ # Calc correcting factor Beta at time t+dt
+ Beta1 = ck1 / (qm1 / Ax1)
+ # calc corrected cell Reynolds number at time t+dt
+ Dm1 = qm1 / (sfx * ck1 * Bx1 * xpix) / Beta1
+ # corrected Courant number at time t+dt
+ Cm1 = ck1 * dt / xpix / Beta1
+
+ # Calc MCT parameters
+ den = 1 + Cm1 + Dm1
+ c1 = (-1 + Cm1 + Dm1) / den
+ c2 = (1 + Cm0 - Dm0) / den * (Cm1 / Cm0)
+ c3 = (1 - Cm0 + Dm0) / den * (Cm1 / Cm0)
+ c4 = (2 * Cm1) / den
+
+ # cmcheck
+ # Calc outflow q11 at time t+1
+ # Mass balance equation without lateral flow
+ # q11 = c1 * q01 + c2 * q00 + c3 * q10
+ # Mass balance equation that takes into consideration the lateral flow
+ q11 = c1 * q01 + c2 * q00 + c3 * q10 + c4 * ql
+
+ if q11 < eps: # cmcheck <=0
+ q11 = eps
+
+ #### end of for loop
+
+ # # cmcheck
+ # calc_t = xpix / ck1
+ # if calc_t < dt:
+ # print('xpix/ck1 < dt')
+
+ # k1 = dt / Cm1
+ # x1 = (1. - Dm1) / 2.
+
+ # Calc the corrected mass-conservative expression for the reach segment storage at time t+dt
+ # The lateral inflow ql is only explicitly accounted for in the mass balance equation, while it is not in the equation expressing
+ # the storage as a weighted average of inflow and outflow.The rationale of this approach lies in the fact that the outflow
+ # of the reach implicitly takes the effect of the lateral inflow into account.
+ if q11 == 0:
+ V11 = V00 + (q00 + q01 - q10 - q11) * dt / 2
+ else:
+ V11 = (1 - Dm1) * dt / (2 * Cm1) * q01 + (1 + Dm1) * dt / (2 * Cm1) * q11
+ # V11 = k1 * (x1 * q01 + (1. - x1) * q11) # MUST be the same as above!
+
+ ### calc integration on the control volume (pixel)
+ # calc average discharge outflow q1m for MCT channels during routing sub step dt
+ # Calculate average outflow using water balance for MCT channel grid cell over sub-routing step
+ q1mm = q0mm + ql + (V00 - V11) / dt
+
+ # cmcheck
+ # q1m cannot be smaller than eps or it will cause instability
+ if q1mm < eps:
+ q1mm = eps
+ V11 = V00 + (q0mm + ql - q1mm) * dt
+
+ # q11 Outflow at O(t+dt)
+ # q1m average outflow in time dt
+ # V11 water volume at t+dt
+ # Cm1 Courant number at t+dt for state files
+ # Dm1 Reynolds numbers at t+dt for state files
+ return q11, V11, q1mm, Cm1, Dm1
+
+
+def hoq(q, s0, Balv, ANalv, Nalv):
+ """Water depth h from discharge q.
+ Given a generic cross-section (rectangular, triangular or trapezoidal) and a steady-state discharge q=Q*, it computes
+ water depth (y), wet contour (Bx), wet area (Ax) and wave celerity (cel) using Newton-Raphson method.
+ Reference:
+ Reggiani, P., Todini, E., & Meißner, D. (2016). On mass and momentum conservation in the variable-parameter Muskingum method.
+ Journal of Hydrology, 543, 562–576. https://doi.org/10.1016/j.jhydrol.2016.10.030
+
+ :param q: steady-state river discharge [m3/s]
+ :param s0: river bed slope (tan B)
+ :param Balv : width of the riverbed [m]
+ :param ChanSdXdY : slope dx/dy of riverbed side
+ :param ANalv : angle of the riverbed side [rad]
+ :param Nalv : channel mannings coefficient n for the riverbed [s/m1/3]
+
+ :return:
+ y: water depth referred to the bottom of the riverbed [m]
+ """
+
+ alpha = 5.0 / 3.0 # exponent (5/3)
+ eps = 1.0e-06
+ max_tries = 1000
+
+ rs0 = np.sqrt(s0)
+ usalpha = 1.0 / alpha
+
+ # cotangent(angle of the riverbed side - dXdY)
+ if ANalv < np.pi / 2:
+ # triangular or trapezoid cross-section
+ c = cotan(ANalv)
+ else:
+ # rectangular corss-section
+ c = 0.0
+
+ # sin(angle of the riverbed side - dXdY)
+ if ANalv < np.pi / 2:
+ # triangular or trapezoid cross-section
+ s = np.sin(ANalv)
+ else:
+ # rectangular cross-section
+ s = 1.0
+
+ # water depth first approximation y0 based on steady state q
+ if Balv == 0:
+ # triangular cross-section
+ y = (Nalv * q / rs0) ** (3.0 / 8.0) * (2 / s) ** 0.25 / c ** (5.0 / 8.0)
+ else:
+ # rectangular cross-section and first approx for trapezoidal cross-section
+ y = (Nalv * q / (rs0 * Balv)) ** usalpha
+
+ if (Balv != 0) and (ANalv < np.pi / 2):
+ # trapezoid cross-section
+ y = (Nalv * q / rs0) ** usalpha * (Balv + 2.0 * y / s) ** 0.4 / (Balv + c * y)
+
+ for tries in range(1, max_tries):
+ # calc Q(y) for the different tries of y
+ q0, Ax, Bx, Px, cel = qoh(y, s0, Balv, ANalv, Nalv)
+ # Ax: wet area[m2]
+ # Bx: cross-section width at water surface[m]
+ # Px: cross-section wet contour [m]
+ # cel: wave celerity[m/s]
+
+ # this is the function we want to find the 0 for f(y)=Q(y)-Q*
+ fy = q0 - q
+ # calc first derivative of f(y) f'(y)=Bx(y)*cel(y)
+ dfy = Bx * cel
+ # calc update for water depth y
+ dy = fy / dfy
+ # update yt+1=yt-f'(yt)/f(yt)
+ y = y - dy
+ # stop loop if correction becomes too small
+ if np.abs(dy) < eps:
+ break
+
+ return y
+
+
+def qoh(y, s0, Balv, ANalv, Nalv):
+ """Discharge q from water depth h.
+ Given a generic river cross-section (rectangular, triangular and trapezoidal)
+ and a water depth (y [m]) referred to the bottom of the riverbed, it uses Manning’s formula to calculate:
+ q: steady-state discharge river discharge [m3/s]
+ a: wet area [m2]
+ b: cross-section width at water surface [m]
+ p: cross-section wet contour [m]
+ cel: wave celerity [m/s]
+ Reference: Reggiani, P., Todini, E., & Meißner, D. (2016). On mass and momentum conservation in the variable-parameter Muskingum method. Journal of Hydrology, 543, 562–576. https://doi.org/10.1016/j.jhydrol.2016.10.030
+
+ :param y: river water depth [m]
+ :param s0: river bed slope (tan B)
+ :param Balv : width of the riverbed [m]
+ :param ANalv : angle of the riverbed side [rad]
+ :param Nalv : channel mannings coefficient n for the riverbed [s/m1/3]
+
+ :return:
+ q,a,b,p,cel
+ """
+
+ alpha = 5.0 / 3.0 # exponent (5/3)
+
+ rs0 = np.sqrt(s0)
+ alpham = alpha - 1.0
+
+ # np.where(ANalv < np.pi/2, triang. or trapeiz., rectangular)
+ # cotangent(angle of the riverbed side - dXdY)
+ c = np.where(
+ ANalv < np.pi / 2,
+ # triangular or trapezoid cross-section
+ cotan(ANalv),
+ # rectangular cross-section
+ 0.0,
+ )
+ # sin(angle of the riverbed side - dXdY)
+ s = np.where(
+ ANalv < np.pi / 2,
+ # triangular or trapezoid cross-section
+ np.sin(ANalv),
+ # rectangular corss-section
+ 1.0,
+ )
+
+ a = (Balv + y * c) * y # wet area [m2]
+ b = Balv + 2.0 * y * c # cross-section width at water surface [m]
+ p = Balv + 2.0 * y / s # cross-section wet contour [m]
+ q = rs0 / Nalv * a**alpha / p**alpham # steady-state discharge [m3/s]
+ cel = (q / 3.0) * (5.0 / a - 4.0 / (p * b * s)) # wave celerity [m/s]
+
+ return q, a, b, p, cel
+
+
+def hoV(V, xpix, Balv, ANalv):
+ """Water depth h from volume V.
+ Given a generic river cross-section (rectangular, triangular and trapezoidal) and a river channel volume V,
+ it calculates the water depth referred to the bottom of the riverbed [m] (y).
+ Reference: Reggiani, P., Todini, E., & Meißner, D. (2016). On mass and momentum conservation in the variable-parameter Muskingum method. Journal of Hydrology, 543, 562–576. https://doi.org/10.1016/j.jhydrol.2016.10.030
+
+ :param V : volume of water in channel riverbed
+ :param xpix : dimension along the flow direction [m]
+ :param Balv : width of the riverbed [m]
+ :param ANalv : angle of the riverbed side [rad]
+
+ :return:
+ y : channel water depth [m]
+ """
+ eps = 1e-6
+
+ c = np.where(
+ ANalv < np.pi / 2, # angle of the riverbed side dXdY [rad]
+ cotan(ANalv), # triangular or trapezoidal cross-section
+ 0.0,
+ ) # rectangular cross-section
+
+ a = V / xpix # wet area [m2]
+
+ # np.where(c < 1.d-6, rectangular, triangular or trapezoidal)
+ y = np.where(
+ np.abs(c) < eps,
+ a / Balv, # rectangular cross-section
+ (-Balv + np.sqrt(Balv**2 + 4 * a * c)) / (2 * c),
+ ) # triangular or trapezoidal cross-section
+
+ return y
+
+
+def qoV(V, xpix, s0, Balv, ANalv, Nalv):
+ """Discharge q from river channel volume V.
+ Given a generic river cross-section (rectangular, triangular and trapezoidal)
+ and a water volume (V [m3]), it uses Manning’s formula to calculate the corresponding discharge (q [m3/s]).
+ Reference: Reggiani, P., Todini, E., & Meißner, D. (2016). On mass and momentum conservation in the variable-parameter Muskingum method. Journal of Hydrology, 543, 562–576. https://doi.org/10.1016/j.jhydrol.2016.10.030
+
+ :param V : volume of water in channel riverbed
+ :param xpix : dimension along the flow direction [m]
+ :param s0: river bed slope (tan B)
+ :param Balv : width of the riverbed [m]
+ :param ANalv : angle of the riverbed side [rad]
+ :param Nalv : channel mannings coefficient n for the riverbed [s/m1/3]
+
+ :return:
+ y : channel water depth [m]
+ """
+ y = hoV(V, xpix, Balv, ANalv)
+ q, a, b, p, cel = qoh(y, s0, Balv, ANalv, Nalv)
+ return q
+
+
+def cotan(x):
+ """There is no cotangent function in numpy"""
+ return np.cos(x) / np.sin(x)
+
+
+def rad_from_dxdy(dxdy):
+ """Calculate radians"""
+ rad = np.arctan(1 / dxdy)
+ # angle = np.rad2deg(rad)
+ return rad