Modeling Sonic and Density Logs From Resistivity
Resistivity is sometimes transformed into an apparent velocity log with a number of different equations:

Faust Method
This method is very old, but is useful in shallow rock sequences, especially clastics. You may need to determine new parameters for each major geologic horizon.
      1: Vc = KR1 * RESS ^ (1/KR2) * DEPTH ^ (1/KR3)
       2: DTCsyn = 10^6 / Vc

Where:
  Vc = compressional velocity (ft/sec)
   DTCsyn = synthetic (modelled) sonic travel time from Faust equation (usec/ft)
  KR1 = Faust constant (2000 to 3400 for depths in feet)
  RESS = resistivity from shallow investigation log (ohm-m}
  DEPTH = depth of layer (ft or m)
  KR2 and KR3 = 6.0 or as determined by regression analysis

NOTE: Constants given are for depths in FEET.
If depths are in meters, convert depth to feet. by multiplying depth in meters by 3.281.
To obtain DTCsyn in usec/m from DTC in usec/ft, divide by 3.281.

The Faust transform can be used when the sonic log is missing, and can be calibrated with offset well data, check shots, or vertical seismic profiles. The method does not account for gas effect.

Smith Method
This method uses a simple correlation between resistivity and sonic travel time:

_____ 3: DTCsyn = KR4 * (RESS ^ KR5)

Where:
   DTCsyn = synthetic (modelled) sonic travel time from Smith equation (usec/ft)
  KR4 = Smith constant (90 to 100 for depths in feet)
   RESS = resistivity from shallow investigation log (ohm-m}
  KR5 = -0.15 or as determined by regression analysis

NOTE: Constants given are for depths in FEET.
If depths are in meters, convert depth to feet. by multiplying depth in meters by 3.281.
To obtain DTCsyn in usec/m from DTC in usec/ft, divide by 3.281.

The method does not account for gas effect. You may need to determine new parameters for each major geologic horizon.

Fischer - Good Method
This method assumes a fairly sophisticated log analysis can be run on the well in question or on a nearby well. This is needed to obtain a list of water resistivity (RWA) versus depth. Since most sonic log problems are in shales due to bad hole or rock alteration, this calculation is usually possible and should be done continuously or at least zone by zone.

Similarly, the apparent RW in shale (RWSH) is needed, based on an estimate of the shale total porosity (BVWSH). This can be computed continuously or zone by zone from one of the following:

If neutron and density logs are both available and correct:

      4: BVWSH = (PHIDSH + PHINSH) / 2
      5: PHIt = (PHID + PHIN ) / 2

If density log is missing or bad:
      6: BVWSH = 0.95 * PHINSH
      7: PHIt = PHIN

Where the sonic log is behaving properly or from an offset well that is OK:
      8. BVWSH = (DTCSH - DTCMA) / (DTCW - DTCMA)
      9. PHIt = (DTC - DTCMA) / (DTCW - DTCMA)

Then, for each shale zone:
      10: RWSH = (BVWSH ^ M) * RSH / A

And, for each clean zone:
_____11: RWA = (PHIt ^ M) * RESD / A

For all digitized intervals or computation layers:
_____12: Vshg = (GR - GR0) / (GR100 - GR0)
      13: Vshs = (SP - SP0) / (SP100 - SP0)
      14: Vsh = Min (Vshg, Vshs)
      15: RMIX = 1 / (Vsh / RWSH + (1 - Vsh) / RWA)
      16: DTCsyn = DTCMA + (DTCW - DTCMA) * (A * RMIX / RESD) ^ (1/M)
      17: DTCyn = Min (DTCsyn, DTC)
      18: DENSsyn = DENSMA + (DENSW - DENSMA) * (A * RMIX / RESD) ^ (1/M)
      19: DENSsyn = Min (DENSsyn, DENS)

When the zone is 100% shale, this equation should return a reasonable travel time. If it doesn't match the log where it is believed to be good, then adjust RWSH or Vsh. In clean zones, adjust DELTMA or RWA if needed. When zones are hydrocarbon bearing, RWA and RESD will both be too high, and the result will be close to correct, but may give a DELTmod that is too low (too high a velocity) or a DENSmod that is too high.

To overcome some of this effect, you could substitute the shallow resistivity RESS for RESD and RMF@FT for RWA. You may still need to calibrate the RMF@FT with its own RMFA equation:
      20: RMFA = (PHIt ^ M) * RESS / A
      21: RMIX = 1 / (Vsh / RWSH + (1 - Vsh) / RMFA)
      22: DTCsyn = DTCMA + (DTCW - DTCMA) * (A * RMIX / RESS) ^ (1/M)
      23: DTCsyn = Min (DTCsyn, DTC)
      24: DENSsyn = DENSMA + (DENSW - DENSMA) * (A * RMIX / RESS) ^ (1/M)
      25: DENSsyn = Min (DENSsyn, DENS)

Where:
   DTCsyn = synthetic (modelled) sonic travel time from Fischer and Good equation (usec/ft or usec/m)
   DENSsyn = synthetic (modelled) density from Fischer and Good equation (g/cc or kg/m3)
   DTC, DENS = original sonic or density log readings if available
   DTCSH, DTCMA, DYCW = compressional sonic travel time of shale, matrix, water values
   DENSSH, DENSMA, DENSW = density of shale, matrix, water values
   PHID, PHIN = density and neutron porosity
   A, M, N = Archie water saturation exponents
   Vsh, PHIt = shale volume and total porosity
   RWSH, RNFA, RNIX = resistivity of water in shale, nudfiltrate, and invaded zone
 

Neither method accounts for the effect of gas, which must be handled separately.