7.1 Remote Sensing

Passive vs Active Remote Sensing

Remote sensing instruments are broadly classified by whether they provide their own source of illumination.

Planck Function and Brightness Temperature

The Planck function describes the spectral radiance emitted by a blackbody at temperature T. It is the fundamental relationship connecting temperature to emitted radiation and underpins all thermal remote sensing.

The brightness temperature $T_b$ is defined as the temperature a blackbody would need to have to emit the observed radiance at a given frequency. It is obtained by inverting the Planck function:

For real (non-blackbody) surfaces and atmospheres, $T_b < T_{physical}$ because emissivity $\epsilon < 1$. Over ocean in the microwave, $\epsilon \approx 0.5$, making brightness temperatures much colder than SST, while land surfaces have $\epsilon \approx 0.9-0.95$.

Radar Meteorology

Weather radar is the primary tool for detecting and quantifying precipitation in real time. It operates by transmitting microwave pulses (typically S-band at 2.7-3.0 GHz, C-band at 5.3-5.7 GHz, or X-band at 9.3-9.5 GHz) and measuring the backscattered power from hydrometeors (raindrops, snowflakes, hail).

Radar Reflectivity Factor Z

The reflectivity factor Z represents the sum of the sixth power of the diameters of all hydrometeors in a unit volume, assuming Rayleigh scattering (particle diameter $D \ll \lambda$):

Weather Radar Equation

The received power from a volume of precipitation at range r is given by the weather radar equation:

Z-R Relationships

Reflectivity Z is empirically related to rainfall rate R (mm/hr) by power-law relationships:

Doppler Velocity

Doppler radar measures the radial velocity of hydrometeors by detecting the frequency shift of the returned signal. The Doppler shift is $\Delta f = 2v_r / \lambda$, where $v_r$ is the radial velocity component toward/away from the radar. The maximum unambiguous velocity is $v_{max} = \pm \lambda \cdot \text{PRF} / 4$, where PRF is the pulse repetition frequency. Doppler data reveals wind patterns including mesocyclones (tornado vortex signatures), convergence lines, wind shear, and the overall velocity-azimuth display (VAD) wind profile.

Sounder Channels and CO₂ Slicing

Infrared and microwave sounders use multiple channels with different absorption strengths to build vertical profiles of temperature and humidity. The CO₂ slicing technique retrieves cloud-top pressure by exploiting the difference in absorption between pairs of CO₂ channels:

Key Satellite Instruments

Fortran: Radar Reflectivity to Rainfall Rate Conversion

This Fortran program reads radar reflectivity data (in dBZ), applies selectable Z-R relationships for different precipitation types, and outputs estimated rainfall rates.

program radar_zr_conversion
  implicit none

  ! Z-R relationship parameters: Z = a * R^b
  ! We store coefficients for several common relationships
  integer, parameter :: dp = selected_real_kind(15, 307)
  integer, parameter :: nx = 200, ny = 200
  real(dp) :: dbz_field(nx, ny), rain_rate(nx, ny)
  real(dp) :: z_linear, R, a, b
  integer  :: i, j, method
  character(len=64) :: method_name

  ! Select Z-R relationship
  ! 1 = Marshall-Palmer (stratiform), 2 = Convective,
  ! 3 = Tropical (Rosenfeld),         4 = Snow (liquid equiv)
  method = 1

  select case(method)
    case(1)
      a = 200.0_dp;  b = 1.6_dp
      method_name = 'Marshall-Palmer (stratiform)'
    case(2)
      a = 300.0_dp;  b = 1.4_dp
      method_name = 'Convective'
    case(3)
      a = 250.0_dp;  b = 1.2_dp
      method_name = 'Tropical (Rosenfeld)'
    case(4)
      a = 2000.0_dp; b = 2.0_dp
      method_name = 'Snow (liquid equivalent)'
    case default
      a = 200.0_dp;  b = 1.6_dp
      method_name = 'Default Marshall-Palmer'
  end select

  write(*,'(A,A)') 'Z-R relationship: ', trim(method_name)
  write(*,'(A,F8.1,A,F4.2)') 'Z = ', a, ' * R^', b

  ! Generate synthetic radar reflectivity field for demonstration
  ! In practice, read from radar data file (e.g., Level-II NEXRAD)
  call random_seed()
  do j = 1, ny
    do i = 1, nx
      call random_number(dbz_field(i, j))
      dbz_field(i, j) = dbz_field(i, j) * 60.0_dp - 10.0_dp  ! -10 to 50 dBZ
    end do
  end do

  ! Convert dBZ to rainfall rate R [mm/hr]
  ! Z[mm^6/m^3] = 10^(dBZ/10), then R = (Z/a)^(1/b)
  do j = 1, ny
    do i = 1, nx
      if (dbz_field(i, j) < 0.0_dp) then
        rain_rate(i, j) = 0.0_dp   ! Below noise floor
      else
        z_linear = 10.0_dp ** (dbz_field(i, j) / 10.0_dp)
        R = (z_linear / a) ** (1.0_dp / b)
        rain_rate(i, j) = R
      end if
    end do
  end do

  ! Print statistics
  write(*,'(A)')       '--- Rainfall Rate Statistics ---'
  write(*,'(A,F10.4)') 'Max rain rate [mm/hr]: ', maxval(rain_rate)
  write(*,'(A,F10.4)') 'Mean rain rate [mm/hr]: ', &
       sum(rain_rate) / real(nx * ny, dp)
  write(*,'(A,I8)')    'Pixels with R > 10 mm/hr: ', &
       count(rain_rate > 10.0_dp)
  write(*,'(A,I8)')    'Pixels with R > 50 mm/hr: ', &
       count(rain_rate > 50.0_dp)

  ! Output a sample cross-section
  write(*,'(A)') '--- Sample row (j=100) ---'
  write(*,'(A)') '  i     dBZ     R[mm/hr]'
  do i = 1, min(20, nx)
    write(*,'(I4, F10.2, F10.3)') i, dbz_field(i, 100), rain_rate(i, 100)
  end do

end program radar_zr_conversion