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**How to distinguish rain from hail using radar: A cunning, variational method**

Robin Hogan Last Minute Productions Inc.

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Outline Increasingly in active remote sensing (radar and lidar), many instruments are being deployed together, and individual instruments may measure many variables We want to retrieve an “optimum” estimate of the state of the atmosphere that is consistent with all the measurements But most algorithms use at most only two instruments/variables and don’t take proper account of instrumental errors The “variational” approach (a.k.a. optimal estimation theory) is standard in data assimilation and passive sounding, but has only recently been applied to radar retrieval problems It is mathematically rigorous and takes full account of errors Straightforward to add extra constraints and extra instruments In this talk, it will be applied to polarization radar measurements of rain rate and hail intensity Met Office recently commissioned new polarization radar A variational retrieval is a very useful step towards assimilation of polarization data

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**Active sensing Passive sensing**

No attenuation With attenuation Isolated weighting functions (or Jacobians) so don’t need to bother with variational methods? With attenuation (e.g. spaceborne lidar) weighting functions are broader: variational method required Passive sensing Radiance at a particular wavelength has contributions from large range of heights A variational method is used to retrieve the temperature profile

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**Chilbolton 3GHz radar: Z**

We need to retrieve rain rate for accurate flood forecasts Conventional radar estimates rain-rate R from radar reflectivity factor Z using Z=aRb Around a factor of 2 error in retrievals due to variations in raindrop size and number concentration Attenuation through heavy rain must be corrected for, but gate-by-gate methods are intrinsically unstable Hail contamination can lead to large overestimates in rain rate

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**Chilbolton 3GHz radar: Zdr**

Differential reflectivity Zdr is a measure of drop shape, and hence drop size: Zdr = 10 log10 (ZH /ZV) In principle allows rain rate to be retrieved to 25% Can assist in correction for attenuation But Too noisy to use at each range-gate Needs to be accurately calibrated Degraded by hail Drop 1 mm ZV 3 mm ZH 4.5 mm ZDR = 0 dB (ZH = ZV) Drop shape is directly related to drop size: larger drops are less spherical Hence the combination of Z and ZDR can provide rain rate to ~25%. ZDR = 1.5 dB (ZH > ZV) ZDR = 3 dB (ZH >> ZV)

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**Chilbolton 3GHz radar: fdp**

Differential phase shift fdp is a propagation effect caused by the difference in speed of the H and V waves through oblate drops Can use to estimate attenuation Calibration not required Low sensitivity to hail But Need high rain rate Low resolution information: need to take derivative but far too noisy to use at each gate: derivative can be negative! How can we make the best use of the Zdr and fdp information? phase shift

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**Using Zdr and fdp for rain**

Useful at low and high R Differential attenuation allows accurate attenuation correction but difficult to implement Need accurate calibration Too noisy at each gate Degraded by hail Zdr Calibration not required Low sensitivity to hail Stable but inaccurate attenuation correction Need high R to use Must take derivative: far too noisy at each gate fdp

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**Simple Zdr method Observations Retrieval**

Use Zdr at each gate to infer a in Z=aR1.5 Measurement noise feeds through to retrieval Noise much worse in operational radars Noisy or Negative Zdr Retrieval Noisy or no retrieval Rainrate Lookup table

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**+ Smoothness constraints**

Variational method Start with a first guess of coefficient a in Z=aR1.5 Z/R implies a drop size: use this in a forward model to predict the observations of Zdr and fdp Include all the relevant physics, such as attenuation etc. Compare observations with forward-model values, and refine a by minimizing a cost function: + Smoothness constraints Observational errors are explicitly included, and the solution is weighted accordingly For a sensible solution at low rainrate, add an a priori constraint on coefficient a

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How do we solve this? The best estimate of x minimizes a cost function: At minimum of J, dJ/dx=0, which leads to: The least-squares solution is simply a weighted average of m and b, weighting each by the inverse of its error variance Can also be written in terms of difference of m and b from initial guess xi: Generalize: suppose I have two estimates of variable x: m with error sm (from measurements) b with error sb (“background” or “a priori” knowledge of the PDF of x)

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**The Gauss-Newton method**

We often don’t directly observe the variable we want to retrieve, but instead some related quantity y (e.g. we observe Zdr and fdp but not a) so the cost function becomes H(x) is the forward model predicting the observations y from state x and may be complex and non-analytic: difficult to minimize J Solution: linearize forward model about a first guess xi The x corresponding to y=H(x), is equivalent to a direct measurement m: …with error: y Observation y x xi+2 xi+1 xi (or m)

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**Substitute into prev. equation:**

If it is straightforward to calculate y/x then iterate this formula to find the optimum x If we have many observations and many variables to retrieve then write this in matrix form: The matrices and vectors are defined by: Where the Hessian matrix is State vector, a priori vector and observation vector Error covariance matrices of observations and background The Jacobian

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**xi+1= xi+A-1{HTR-1[y-H(xi)]**

Finding the solution New ray of data First guess of x In this problem, the observation vector y and state vector x are: Forward model Predict measurements y and Jacobian H from state vector x using forward model H(x) Compare measurements to forward model Has the solution converged? 2 convergence test No Gauss-Newton iteration step Predict new state vector: xi+1= xi+A-1{HTR-1[y-H(xi)] +B-1(b-xi)} where the Hessian is A=HTR-1H+B-1 Yes Calculate error in retrieval The solution error covariance matrix is S=A-1 Proceed to next ray

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First guess of a First guess: a =200 everywhere Rainrate Use difference between the observations and forward model to predict new state vector (i.e. values of a), and iterate

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Final iteration Zdr and fdp are well fitted by forward model at final iteration of minimization of cost function Rainrate Retrieved coefficient a is forced to vary smoothly Prevents random noise in measurements feeding through into retrieval (which occurs in the simple Zdr method)

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A ray of data Zdr and fdp are well fitted by the forward model at the final iteration of the minimization of the cost function The scheme also reports the error in the retrieved values Retrieved coefficient a is forced to vary smoothly Represented by cubic spline basis functions Prevents random noise in the measurements feeding through into the retrieval

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**Representing a 50-point function by 10 control points**

Enforcing smoothness In range: cubic-spline basis functions Rather than state vector x containing “a” at every range gate, it is the amplitude of smaller number of basis functions Cubic splines solution is continuous in itself, its first and second derivatives Fewer elements in x more efficient! Representing a 50-point function by 10 control points In azimuth: Two-pass Kalman smoother First pass: use one ray as a constraint on the retrieval at the next (a bit like an a priori) Second pass: repeat in the reverse direction, constraining each ray both by the retrieval at the previous ray, and by the first-pass retrieval from the ray on the other side

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**Enforcing smoothness 1 Cubic-spline basis functions**

Let state vector x contain the amplitudes of a set of basis functions Cubic splines ensure that the solution is continuous in itself and its first and second derivatives Fewer elements in x more efficient! Forward model Convert state vector to high resolution: xhr=Wx Predict measurements y and high-resolution Jacobian Hhr from xhr using forward model H(xhr) Convert Jacobian to low resolution: H=HhrW Representing a 50-point function by 10 control points The weighting matrix

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**Enforcing smoothness 2 Background error covariance matrix**

To smooth beyond the range of individual basis functions, recognise that errors in the a priori estimate are correlated Add off-diagonal elements to B assuming an exponential decay of the correlations with range The retrieved a now doesn’t return immediately to the a priori value in low rain rates Kalman smoother in azimuth Each ray is retrieved separately, so how do we ensure smoothness in azimuth as well? Two-pass solution: First pass: use one ray as a constraint on the retrieval at the next (a bit like an a priori) Second pass: repeat in the reverse direction, constraining each ray both by the retrieval at the previous ray, and by the first-pass retrieval from the ray on the other side

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**Full scan from Chilbolton**

Observations Retrieval Note: validation required! Forward-model values at final iteration are essentially least-squares fits to the observations, but without instrument noise

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**Response to observational errors**

Nominal Zdr error of ±0.2 dB Additional random error of ±1 dB

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**What if we use only Zdr or fdp ?**

Retrieved a Retrieval error Zdr and fdp Very similar retrievals: in moderate rain rates, much more useful information obtained from Zdr than fdp Zdr only Where observations provide no information, retrieval tends to a priori value (and its error) fdp only fdp only useful where there is appreciable gradient with range

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Heavy rain and hail Difficult case: differential attenuation of 1 dB and differential phase shift of 80º Observations Retrieval

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**How is hail retrieved? Hail is nearly spherical**

High Z but much lower Zdr than would get for rain Forward model cannot match both Zdr and fdp First pass of the algorithm Increase error on Zdr so that rain information comes from fdp Hail is where Zdrfwd-Zdr > 1.5 dB and Z > 35 dBZ Second pass of algorithm Use original Zdr error At each hail gate, retrieve the fraction of the measured Z that is due to hail, as well as a. Now the retrieval can match both Zdr and fdp

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**Retrieved hail fraction**

Distribution of hail Retrieved a Retrieval error Retrieved hail fraction Retrieved rain rate much lower in hail regions: high Z no longer attributed to rain Can avoid false-alarm flood warnings Use Twomey method for smoothness of hail retrieval

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Enforcing smoothness 3 Twomey matrix, for when we have no useful a priori information Add a term to the cost function to penalize curvature in the solution: ld2x/dr2 (where r is range and l is a smoothing coefficient) Implemented by adding “Twomey” matrix T to the matrix equations

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Summary New scheme achieves a seamless transition between the following separate algorithms: Drizzle. Zdr and fdp are both zero: use a-priori a coefficient Light rain. Useful information in Zdr only: retrieve a smoothly varying a field (Illingworth and Thompson 2005) Heavy rain. Use fdp as well (e.g. Testud et al. 2000), but weight the Zdr and fdp information according to their errors Weak attenuation. Use fdp to estimate attenuation (Holt 1988) Strong attenuation. Use differential attenuation, measured by negative Zdr at far end of ray (Smyth and Illingworth 1998) Hail occurrence. Identify by inconsistency between Zdr and fdp measurements (Smyth et al. 1999) Rain coexisting with hail. Estimate rain-rate in hail regions from fdp alone (Sachidananda and Zrnic 1987) Could be applied to new Met Office polarization radars Testing required: higher frequency higher attenuation! Hogan (2007, J. Appl. Meteorol. Climatology)

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**Conclusions and ongoing work**

Variational methods have been described for retrieving cloud, rain and hail, from combined active and passive sensors Appropriate choice of state vector and smoothness constraints ensures the retrievals are accurate and efficient Could provide the basis for cloud/rain data assimilation Ongoing work: cloud Test radiance part of cloud retrieval using geostationary-satellite radiances from Meteosat/SEVIRI above ground-based radar & lidar Retrieve properties of liquid-water layers, drizzle and aerosol Incorporate microwave radiances for deep precipitating clouds Apply to A-train data and validate using in-situ underflights Use to evaluate forecast/climate models Quantify radiative errors in representation of different sorts of cloud Ongoing work: rain Validate the retrieved drop-size information, e.g. using a distrometer Apply to operational C-band (5.6 GHz) radars: more attenuation! Apply to other radar problems, e.g. the radar refractivity method Scotland England Lake district Isle of Wight France

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